Single Best Answer MCQs for Sub-Topics of Topic 5: Construction

Question 1. Which instrument is primarily used to construct a circle with a given radius?

(A) Ruler

(B) Protractor

(C) Compass

(D) Set square

Question 2. If the diameter of a circle is given as 14 cm, what should be the compass setting (radius) to construct this circle?

(A) 14 cm

(B) 28 cm

(C) 7 cm

(D) 3.5 cm

Question 3. To construct a line segment of length 6.5 cm using a ruler, where should you place the starting point of the ruler's scale?

(A) At the 0 mark

(B) At the 1 mm mark

(C) At any convenient mark, then subtract

(D) At the 1 cm mark

Question 4. To copy a line segment AB using a compass and ruler, what is the first step?

(A) Draw a line L longer than AB.

(B) Set the compass width to the length of AB.

(C) Mark a point P on line L.

(D) Draw an arc from point P.

Question 5. When constructing a circle with a compass, the stationary point of the compass indicates the:

(A) Radius

(B) Diameter

(C) Centre

(D) Arc

Question 6. If you want to construct a line segment exactly 8 cm long, which instrument guarantees the most accurate length measurement?

(A) A ruler marked in centimetres

(B) A divider

(C) A protractor

(D) A set square

Question 7. To copy a line segment of length 'l', you set the compass to length 'l'. If you then mark two points on a line using this compass setting, the distance between them will be:

(A) Less than l

(B) Equal to l

(C) Greater than l

(D) Twice l

Question 8. Which of the following statements is true about constructing a circle with a given diameter?

(A) You set the compass to the diameter length.

(B) You set the compass to half the diameter length.

(C) You set the compass to twice the diameter length.

(D) The diameter is irrelevant for compass setting.

Question 9. When constructing a line segment of a specific length using a ruler, it is advisable to start measuring from the '0' mark because:

(A) It is the international standard.

(B) The edge of the ruler might be worn or damaged.

(C) It makes calculations easier.

(D) All rulers are manufactured perfectly from the edge.

Question 10. What is the purpose of copying a line segment?

(A) To measure its length accurately.

(B) To transfer its length to another location.

(C) To find its midpoint.

(D) To draw a line parallel to it.

Question 11. If you are given a radius of 5 cm, how many circles can you construct with this radius?

(A) Exactly one

(B) Two

(C) A finite number

(D) An infinite number

Question 12. When copying a line segment AB onto a line L starting from point P, you place the compass point at P and draw an arc intersecting L. What does the intersection point represent?

(A) The centre of a circle

(B) The midpoint of AB

(C) The other endpoint of the copied segment

(D) A random point on L

Question 13. What is the minimum number of points required to define a line segment?

(A) One

(B) Two

(C) Three

(D) Infinite

Question 14. When constructing a circle, the distance from the centre to any point on the circle is called the:

(A) Diameter

(B) Chord

(C) Radius

(D) Arc length

Question 15. If you need to construct a line segment of length $x$, using a ruler marked in millimetres, what is the smallest unit you can accurately measure?

(A) 1 cm

(B) 1 mm

(C) 0.5 cm

(D) 0.1 mm

Question 16. Which of the following cannot be used to construct a line segment of a specific length?

(A) Ruler

(B) Compass (in conjunction with a ruler or another segment)

(C) Protractor

(D) Divider

Question 17. To construct a circle with diameter $D$, the compass opening should be:

(A) $D$

(B) $2D$

(C) $D/2$

(D) $\sqrt{D}$

Question 18. When copying a line segment, which property are you essentially preserving?

(A) Orientation

(B) Position

(C) Length

(D) Colour

Question 19. A point is a basic geometric element that has:

(A) Length only

(B) Area only

(C) Position only

(D) Volume only

Question 20. A line segment is a part of a line that has:

(A) One endpoint

(B) Two endpoints

(C) Infinite length

(D) No endpoints

Question 21. Which instrument is essential for drawing a straight line segment?

(A) Compass

(B) Protractor

(C) Ruler (or straight edge)

(D) Divider

Question 22. To construct a circle with a given radius, you need to know:

(A) The diameter

(B) The circumference

(C) The position of the centre and the radius

(D) The area

Question 23. Copying a line segment is useful for:

(A) Finding the slope of the segment.

(B) Creating parallel lines.

(C) Constructing other geometric figures that require specific lengths.

(D) Measuring the segment in different units.

Question 24. If you construct a circle with a radius of 4 cm, what is its diameter?

(A) 2 cm

(B) 4 cm

(C) 8 cm

(D) 16 cm

Question 25. Which tool is used to measure the distance between two points on a line segment?

(A) Protractor

(B) Compass and ruler

(C) Set square

(D) Only compass

Question 1. To construct an angle of $60^\circ$ using a compass and ruler, you first draw a ray. What is the next step?

(A) Draw an arc from the endpoint of the ray, intersecting the ray.

(B) Draw a line perpendicular to the ray.

(C) Set the compass to a specific angle.

(D) Mark a point on the ray.

Question 2. When constructing a $90^\circ$ angle at a point on a line, how many arcs do you typically draw using a compass?

(A) Two

(B) Three

(C) Four

(D) Five

Question 3. To construct a $120^\circ$ angle, you essentially construct two $60^\circ$ angles adjacent to each other on the same ray. From which points do you draw the second main arc?

(A) From the vertex of the angle.

(B) From the point where the initial arc intersects the ray.

(C) From the point that gives the $60^\circ$ mark.

(D) From any random point.

Question 4. To construct a $30^\circ$ angle, you first construct a $60^\circ$ angle and then:

(A) Bisect the $60^\circ$ angle.

(B) Construct a perpendicular bisector.

(C) Construct another $30^\circ$ angle adjacent to it.

(D) Divide the $60^\circ$ angle into three equal parts.

Question 5. Which of the following angles cannot be constructed using *only* a compass and ruler (without a protractor) by standard methods?

(A) $22.5^\circ$

(B) $40^\circ$

(C) $75^\circ$

(D) $15^\circ$

Question 6. To construct an angle bisector of $\angle ABC$, you draw an arc from B intersecting BA and BC at points D and E respectively. What is the next step?

(A) Draw a line segment DE.

(B) From D and E, draw arcs with the same radius intersecting each other.

(C) From D, draw an arc towards the interior of the angle.

(D) Connect D to E.

Question 7. The construction of an angle bisector relies on the property that any point on the angle bisector is equidistant from:

(A) The vertex of the angle.

(B) The two arms of the angle.

(C) The intersection points on the arms.

(D) Any point inside the angle.

Question 8. To construct a $45^\circ$ angle, you first construct a $90^\circ$ angle and then:

(A) Bisect one of the arms.

(B) Bisect the $90^\circ$ angle.

(C) Add a $45^\circ$ angle to it.

(D) Subtract $45^\circ$ from it.

Question 9. Which of the following angles can be obtained by bisecting a $90^\circ$ angle?

(A) $30^\circ$

(B) $45^\circ$

(C) $60^\circ$

(D) $180^\circ$

Question 10. To construct a $150^\circ$ angle, you can construct a $180^\circ$ straight angle and then construct a $30^\circ$ angle adjacent to it on the same line, but *inside* the angle. Alternatively, you can construct a $90^\circ$ angle and add another angle. What angle should you add?

(A) $30^\circ$

(B) $45^\circ$

(C) $60^\circ$

(D) $75^\circ$

Question 11. The justification for constructing an angle bisector involves proving the congruence of two triangles formed by connecting the intersection points and the vertex to a point on the bisector. Which congruence criterion is typically used?

(A) SSS

(B) SAS

(C) ASA

(D) RHS

Question 12. Which angle is half of $150^\circ$?

(A) $30^\circ$

(B) $45^\circ$

(C) $60^\circ$

(D) $75^\circ$

Question 13. To construct a $75^\circ$ angle, you can combine standard constructions. Which combination works?

(A) $60^\circ + 15^\circ$

(B) $90^\circ - 15^\circ$

(C) Bisecting $150^\circ$

(D) All of the above

Question 14. What is the maximum number of angles you can construct using a single compass and ruler starting from a ray and performing only angle bisections?

(A) A finite number

(B) An infinite number

(C) Limited to multiples of $15^\circ$

(D) Limited to angles whose measure is a power of 2 times a base angle.

Question 15. When bisecting an angle, the two resulting angles are:

(A) Supplementary

(B) Complementary

(C) Equal in measure

(D) Always acute

Question 16. To construct an angle of $105^\circ$, you can combine standard constructions. Which combination is valid?

(A) $60^\circ + 45^\circ$

(B) $90^\circ + 15^\circ$

(C) Both (A) and (B)

(D) Neither (A) nor (B)

Question 17. What is the first step when bisecting any angle?

(A) Draw a line segment.

(B) Draw an arc intersecting both arms of the angle from the vertex.

(C) Measure the angle with a protractor.

(D) Draw a ray inside the angle.

Question 18. The justification for angle bisector construction relies on the SSS congruence criterion. Which sides are congruent in the triangles formed?

(A) Two sides are given, the third is common.

(B) One side is given, two are constructed equal.

(C) All three sides are constructed to be equal.

(D) Only two sides are equal.

Question 19. Can a $1^\circ$ angle be constructed using only compass and ruler?

(A) Yes, by dividing $60^\circ$ into 60 parts.

(B) No, not by standard compass and ruler constructions.

(C) Yes, by repeatedly bisecting standard angles.

(D) Only if you have a very accurate compass.

Question 20. What is the result of bisecting a straight angle ($180^\circ$)?

(A) An acute angle

(B) An obtuse angle

(C) A right angle ($90^\circ$)

(D) A reflex angle

Question 21. When constructing a $60^\circ$ angle, you set the compass to an arbitrary radius, draw an arc, and then from the intersection point on the ray, draw another arc with the *same* radius. Why must the radius be the same?

(A) To ensure the arcs intersect at all.

(B) To create an equilateral triangle, guaranteeing the $60^\circ$ angle.

(C) To make the construction look neat.

(D) It doesn't have to be the same.

Question 22. To construct a $135^\circ$ angle, you can combine standard angles. Which combination is valid?

(A) $90^\circ + 45^\circ$

(B) $180^\circ - 45^\circ$

(C) Bisecting $270^\circ$ reflex angle

(D) All of the above

Question 23. A point on the angle bisector is 5 cm away from one arm of the angle. What is its distance from the other arm?

(A) Less than 5 cm

(B) Exactly 5 cm

(C) More than 5 cm

(D) Cannot be determined.

Question 24. Which of the following is NOT a standard angle that can be constructed directly or by simple bisections/combinations of $60^\circ$ and $90^\circ$ with compass and ruler?

(A) $7.5^\circ$

(B) $10^\circ$

(C) $15^\circ$

(D) $30^\circ$

Question 25. The ray that divides an angle into two equal angles is called the:

(A) Median

(B) Altitude

(C) Perpendicular bisector

(D) Angle bisector

Question 1. To construct a perpendicular to a line at a point P on the line, you first draw arcs of the same radius from P, intersecting the line at two points, say A and B. What is the next step?

(A) Draw an arc from A with a radius greater than AP.

(B) Draw an arc from P with a larger radius.

(C) Draw a line segment AB.

(D) Measure the angle with a protractor.

Question 2. When constructing a perpendicular to a line from a point outside the line, P, you first draw an arc from P intersecting the line at two points, say C and D. What is the next step?

(A) Draw a line segment CD.

(B) From C and D, draw arcs with the same radius below the line, intersecting at a point Q.

(C) Draw an arc from P intersecting the line at one point.

(D) Connect P to the midpoint of CD.

Question 3. The perpendicular bisector of a line segment:

(A) Passes through the midpoint of the segment.

(B) Is perpendicular to the segment.

(C) Contains all points equidistant from the endpoints of the segment.

(D) All of the above.

Question 4. To construct the perpendicular bisector of a line segment AB, you draw arcs of the same radius (greater than half the length of AB) from A and B. These arcs intersect at two points. What do you do next?

(A) Connect the two intersection points.

(B) Connect one intersection point to A.

(C) Connect one intersection point to the midpoint of AB.

(D) Draw a line through A and B.

Question 5. The construction of a perpendicular to a line at a point on the line essentially constructs an angle of:

(A) $60^\circ$

(B) $90^\circ$

(C) $120^\circ$

(D) $180^\circ$

Question 6. The justification for the perpendicular bisector construction relies on which congruence criterion to show that any point on the bisector is equidistant from the endpoints?

(A) SSS

(B) SAS

(C) ASA

(D) RHS

Question 7. When constructing a perpendicular from a point outside a line, the line segment connecting the point to the line segment intersection on the line forms the base of two congruent triangles. Which segment acts as a common side?

(A) The line segment on the original line between intersection points.

(B) The perpendicular segment itself.

(C) The line segment connecting the point to one of the intersection points.

(D) No common side is involved.

Question 8. If a point lies on the perpendicular bisector of a segment AB, then:

(A) It is closer to A than to B.

(B) It is closer to B than to A.

(C) Its distance from A is equal to its distance from B (PA = PB).

(D) Its distance from the line containing AB is zero.

Question 9. To construct a perpendicular to a line from a point on the line, why must the initial arcs from the point on the line have the same radius?

(A) To ensure they intersect.

(B) To create a symmetrical setup where the point is the centre of a circle segment.

(C) It's just a preference, any radius works.

(D) To make the resulting perpendicular longer.

Question 10. When constructing the perpendicular bisector of segment AB, why must the radius of the arcs drawn from A and B be greater than half the length of AB?

(A) To ensure the arcs intersect.

(B) To make the bisector longer.

(C) To make the angle $90^\circ$.

(D) To ensure the bisector passes through the midpoint.

Question 11. A line segment has exactly one perpendicular bisector. Is this statement true or false?

(A) True

(B) False

(C) Depends on the length of the segment

(D) Depends on the position of the segment

Question 12. The construction of a perpendicular bisector divides the line segment into:

(A) Two unequal parts.

(B) Two equal parts.

(C) Three equal parts.

(D) Parts depending on the radius used.

Question 13. To construct a perpendicular from a point P *on* a line, which is equivalent to constructing a $90^\circ$ angle, you create two points equidistant from P on the line. This sets up a situation similar to bisecting what kind of angle?

(A) An acute angle

(B) An obtuse angle

(C) A straight angle ($180^\circ$)

(D) A reflex angle

Question 14. The justification for constructing a perpendicular from a point outside a line often involves showing that the constructed line is the altitude of an isosceles triangle or proving congruence using SSS or SAS.

(A) True

(B) False

(C) Only SSS is used

(D) Only SAS is used

Question 15. If a point lies on the perpendicular bisector of a segment AB, then:

(A) $\text{XA} > \text{XB}$

(B) $\text{XA} < \text{XB}$

(C) $\text{XA} = \text{XB}$

(D) $\text{XA} + \text{XB} = \text{AB}$

Question 16. The point where the perpendicular bisectors of the sides of a triangle meet is called the:

(A) Incenter

(B) Centroid

(C) Orthocenter

(D) Circumcenter

Question 17. How many lines perpendicular to a given line can be drawn through a point not on the line?

(A) Zero

(B) One

(C) Two

(D) Infinite

Question 18. The construction of a perpendicular to a line from a point outside the line is used to find the shortest distance from the point to the line. This shortest distance is along the perpendicular segment.

(A) True

(B) False

(C) Only true if the point is directly above the line

(D) Only true for horizontal lines

Question 19. If you need to construct a right angle at a specific point on a line, the method involving drawing arcs of equal radius from the point on the line is essentially a method for:

(A) Copying an angle

(B) Bisecting a straight angle

(C) Constructing an equilateral triangle

(D) Dividing a segment

Question 20. Which geometric figure is formed when you construct the perpendicular bisector of a line segment using the intersection of two arcs from the endpoints?

(A) A square

(B) An isosceles triangle

(C) A rhombus (formed by the endpoints and the two intersection points)

(D) A circle

Question 21. The justification for constructing a perpendicular from a point on the line involves proving that the angle formed is $90^\circ$. This is often done by showing congruence of triangles formed by connecting the intersection points and the point on the line to the point created by the final arc intersection.

(A) True

(B) False

(C) Only for specific angles

(D) Only if using a protractor

Question 22. A perpendicular line makes an angle of _____ degrees with the original line.

(A) 45

(B) 60

(C) 90

(D) 180

Question 23. The construction of a perpendicular bisector is a key step in constructing the circumcircle of a triangle.

(A) True

(B) False

(C) Only for right-angled triangles

(D) Only for equilateral triangles

Question 24. When constructing a perpendicular from a point *outside* a line, the initial arc from the point P must:

(A) Intersect the line at exactly one point.

(B) Intersect the line at exactly two points.

(C) Be parallel to the line.

(D) Pass through the point P.

Question 25. Which of the following is NOT a property of a perpendicular bisector of segment AB?

(A) Every point on it is equidistant from A and B.

(B) It is the axis of symmetry for segment AB.

(C) It passes through A and B.

(D) It forms a $90^\circ$ angle with segment AB.

Question 1. To construct a line parallel to a given line 'l' through a point P not on 'l', one common method involves constructing equal:

(A) Adjacent angles

(B) Vertically opposite angles

(C) Corresponding angles or alternate interior angles

(D) Supplementary angles

Question 2. In the method of constructing a parallel line using corresponding angles, you first draw a transversal line 't' intersecting the given line 'l' at point Q and passing through P. Then you copy the angle formed by 'l' and 't' at point P. Where is this copied angle positioned relative to point P and the transversal 't'?

(A) Alternate interior side

(B) Same side interior

(C) Corresponding side

(D) Vertically opposite side

Question 3. If you construct a line parallel to line 'l' through point P using the alternate interior angles method, you copy the angle formed by the transversal and line 'l'. Where do you copy this angle at point P?

(A) On the corresponding side of the transversal.

(B) On the alternate interior side of the transversal.

(C) On the same side interior of the transversal.

(D) It doesn't matter where you copy it.

Question 4. The property that states if a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel is a fundamental postulate (or theorem) used in parallel line construction. What is this property sometimes called?

(A) Angle Sum Property

(B) Pythagoras Theorem

(C) Converse of Corresponding Angles Postulate

(D) Alternate Interior Angles Theorem

Question 5. To copy an angle using a compass and ruler, you essentially transfer the arc shape and distance between intersection points on the arc. This process is crucial for constructing parallel lines using angle properties.

(A) True

(B) False

(C) Only needed for $90^\circ$ angles

(D) Only needed for $60^\circ$ angles

Question 6. Which of the following is a valid initial step in constructing a parallel line through a point not on the line?

(A) Draw a perpendicular from the point to the line.

(B) Draw any line through the point that intersects the given line.

(C) Draw a circle centered at the point.

(D) Measure the distance from the point to the line.

Question 7. When using the corresponding angles method, after drawing the transversal and the initial arc at Q, you draw a similar arc centered at P with the same radius. Why must the radius be the same?

(A) To ensure the arcs intersect the lines.

(B) To accurately transfer the "spread" of the angle.

(C) It doesn't have to be the same.

(D) To make the parallel line longer.

Question 8. In the alternate interior angles method, after the initial arc from Q and a similar arc from P, you measure the distance between the intersection points on the arc at Q using the compass. Where do you transfer this distance?

(A) On the arc centered at Q.

(B) On the initial arc centered at P.

(C) Along the transversal from P.

(D) On the given line 'l'.

Question 9. The converse of the Alternate Interior Angles Theorem states that if a transversal intersects two lines such that a pair of alternate interior angles is equal, then the lines are:

(A) Perpendicular

(B) Intersecting

(C) Parallel

(D) Coincident

Question 10. How many lines parallel to a given line can be drawn through a point not on the line?

(A) Zero

(B) One

(C) Two

(D) Infinite

Question 11. The construction of parallel lines is based on the properties of angles formed by a transversal intersecting two lines. Which angles are relevant?

(A) Corresponding angles

(B) Alternate interior angles

(C) Both (A) and (B)

(D) Adjacent angles

Question 12. In the method using corresponding angles, the newly constructed angle at P must be equal to the corresponding angle at Q to ensure parallelism. This equality is achieved by:

(A) Measuring with a protractor.

(B) Setting the compass width equal to the distance between the arc intersection points at Q and transferring it to P's arc.

(C) Using a set square.

(D) Visually estimating the angle.

Question 13. Consider constructing a parallel line to line AB through point P using alternate interior angles. You draw a transversal through P intersecting AB at Q. You identify the alternate interior angle at Q. Where is the corresponding alternate interior angle at P located?

(A) On the same side of the transversal as the angle at Q, between the lines.

(B) On the opposite side of the transversal as the angle at Q, between the lines.

(C) Outside the lines, on the same side of the transversal.

(D) Outside the lines, on the opposite side of the transversal.

Question 14. The construction of parallel lines is an application of which fundamental concept?

(A) Congruence of triangles

(B) Properties of angles formed by a transversal

(C) Similarity of triangles

(D) Area calculations

Question 15. Can you construct a line parallel to a given line through a point *on* the line?

(A) Yes, there is a unique parallel line.

(B) Yes, there are infinite parallel lines.

(C) No, this construction is for points *not* on the line.

(D) Yes, the line itself is considered parallel to itself.

Question 16. Which pair of angles, if equal when a transversal cuts two lines, guarantee the lines are parallel?

(A) Adjacent angles

(B) Linear pair

(C) Corresponding angles

(D) Vertically opposite angles

Question 17. When constructing parallel lines using corresponding angles, the final step is to draw a line through the point P and the intersection point of the transferred arc. What does this line represent?

(A) The transversal

(B) The original line

(C) The parallel line

(D) A perpendicular line

Question 18. The method of constructing a parallel line using alternate interior angles requires accurately copying an angle. Which tools are used for this copying process?

(A) Ruler and Protractor

(B) Compass and Set Square

(C) Ruler and Compass

(D) Only Set Square

Question 19. If you use the property that the sum of interior angles on the same side of a transversal is $180^\circ$ to check if two lines are parallel, this property is the converse of which theorem/postulate?

(A) Corresponding Angles Postulate

(B) Alternate Interior Angles Theorem

(C) Same Side Interior Angles Theorem

(D) Angle Sum Property of a triangle

Question 20. Which tool is NOT directly involved in the standard compass and ruler construction of a parallel line through a point?

(A) Compass

(B) Ruler/Straight edge

(C) Protractor

(D) Pencil

Question 21. Parallel lines are lines that:

(A) Intersect at a $90^\circ$ angle.

(B) Meet at a single point.

(C) Never intersect and are in the same plane.

(D) Always have the same length.

Question 1. To divide a line segment AB in the ratio m:n internally, the basic construction method involves drawing a ray AC from A making an angle with AB, and marking points on this ray. How many points should be marked on ray AC?

(A) m

(B) n

(C) m+n

(D) m-n

Question 2. When dividing a line segment AB in the ratio 3:2, you draw a ray AC and mark points $A_1, A_2, A_3, A_4, A_5$ on AC such that $AA_1 = A_1A_2 = \dots = A_4A_5$. Which point on AC should you connect to B first?

(A) $A_1$

(B) $A_2$

(C) $A_3$

(D) $A_5$

Question 3. To divide a line segment AB in the ratio m:n, after connecting the $(m+n)^{th}$ point on ray AC to B, you draw a line through the $m^{th}$ point on AC parallel to the line connecting $(m+n)^{th}$ point to B. This parallel line intersects AB at the required point P. What property ensures that P divides AB in the ratio m:n?

(A) Angle Bisector Theorem

(B) Converse of Basic Proportionality Theorem (Thales Theorem)

(C) Pythagoras Theorem

(D) Congruence of triangles

Question 4. To divide a line segment of length 10 cm in the ratio 3:2, what is the total number of equal parts you divide the ray AC into?

(A) 3

(B) 2

(C) 5

(D) 10

Question 5. If a point P divides line segment AB in the ratio m:n, then $\text{AP/PB} = \text{m/n}$. This is the definition of internal division.

(A) True

(B) False

(C) Only if m=n

(D) Only if m+n = length of AB

Question 6. When dividing a segment AB in the ratio m:n, is it necessary for ray AC to make an acute angle with AB?

(A) Yes, always.

(B) No, any angle can be used, as long as C is not on the line AB.

(C) Only obtuse angles work.

(D) The angle must be $90^\circ$.

Question 7. The segments marked on ray AC ($AA_1, A_1A_2, \dots$) must be of equal length. Which tool is typically used to mark these equal lengths?

(A) Ruler

(B) Protractor

(C) Compass

(D) Set square

Question 8. Justification for dividing a line segment in a given ratio relies heavily on the concept of similar triangles formed by parallel lines and transversals. Which specific theorem is most relevant?

(A) Pythagoras Theorem

(B) Midpoint Theorem

(C) Basic Proportionality Theorem (BPT) / Thales Theorem

(D) Area theorem

Question 9. If you want to divide a line segment AB into 5 equal parts, using the construction method, you should divide ray AC into how many equal parts?

(A) 2

(B) 3

(C) 4

(D) 5

Question 10. Consider dividing segment AB in ratio m:n. You draw ray AC and mark $m+n$ points $A_1, \dots, A_{m+n}$. You connect $A_{m+n}$ to B. Then you draw a line through $A_m$ parallel to $A_{m+n}B$. This line intersects AB at P. According to BPT, $\frac{\text{AP}}{\text{PB}} = \frac{AA_m}{A_m A_{m+n}}$. Since $AA_m$ consists of m equal parts and $A_m A_{m+n}$ consists of n equal parts, $\frac{AA_m}{A_m A_{m+n}} = \frac{m}{n}$. This justifies the construction.

(A) True

(B) False

(C) Only if m=n

(D) Only if the angle CAB is acute

Question 11. Can a line segment be divided in the ratio 1:0 using this method?

(A) Yes

(B) No

(C) Only if the segment length is 1

(D) Only if m and n are integers

Question 12. To divide a line segment AB in the ratio m:n, you can also draw two rays AX and BY parallel to each other, making angles with AB on opposite sides, and mark points on them. On ray AX, you mark m equal points, and on ray BY, you mark n equal points. Which point on AX should be connected to which point on BY?

(A) The $m^{th}$ point on AX to the $n^{th}$ point on BY.

(B) The $1^{st}$ point on AX to the $1^{st}$ point on BY.

(C) The $m^{th}$ point on AX to B.

(D) A to the $n^{th}$ point on BY.

Question 13. The division of a line segment in a given ratio is an example of applying properties of:

(A) Isosceles triangles

(B) Right-angled triangles

(C) Parallel lines and transversals

(D) Circles and tangents

Question 14. If a line segment is divided in the ratio 1:1, the point of division is the:

(A) Endpoint

(B) Quarter point

(C) Midpoint

(D) Three-quarter point

Question 15. When dividing a segment AB in ratio m:n, you draw a ray AC. Why is it specified that C is not on the line AB?

(A) To avoid confusion.

(B) To create a triangle or a non-collinear setup needed for parallel line theorem.

(C) It doesn't affect the construction.

(D) To make the diagram larger.

Question 16. The segments $AA_1, A_1A_2, \dots$ marked on ray AC are made equal using a compass. Is it necessary to measure their length with a ruler?

(A) Yes, always.

(B) No, as long as the compass setting is kept constant for all markings.

(C) Only for the first segment.

(D) Only if the ratio is 1:1.

Question 17. If segment AB is 15 cm long and divided in the ratio 2:3 at point P, what are the lengths of AP and PB?

(A) AP = 2 cm, PB = 3 cm

(B) AP = 6 cm, PB = 9 cm

(C) AP = 9 cm, PB = 6 cm

(D) AP = 7.5 cm, PB = 7.5 cm

Question 18. The construction for dividing a line segment in a given ratio relies on the fact that parallel lines cut transversals proportionally. Which property is being applied?

(A) Angles are equal.

(B) Lengths of parallel lines are equal.

(C) Ratios of corresponding segments on transversals are equal.

(D) Areas are proportional.

Question 19. When dividing a line segment AB in ratio m:n, which congruence criterion is used in the justification?

(A) SSS

(B) SAS

(C) ASA

(D) None of the above (Similarity is used, not congruence)

Question 20. Can you divide a line segment in a ratio like $1:\sqrt{2}$ using compass and ruler?

(A) Yes

(B) No

(C) Only if you can construct $\sqrt{2}$ length

(D) Only if you can construct the ratio itself on the ray

Question 21. Dividing a line segment internally in the ratio m:n means finding a point P *on* the segment AB such that P lies between A and B.

(A) True

(B) False

(C) Only if m=n

(D) Only if m and n are positive

Question 22. If the ratio is 5:0, the point P would coincide with:

(A) Point A

(B) Point B

(C) The midpoint

(D) It's not possible to divide in this ratio internally.

Question 23. The construction for dividing a line segment requires the ability to:

(A) Construct perpendiculars

(B) Construct parallel lines

(C) Bisect angles

(D) Construct tangents

Question 24. If you are dividing a segment in the ratio m:n, the total number of segments on the auxiliary ray is relevant for the construction steps. What is this number?

(A) The smaller of m and n.

(B) The larger of m and n.

(C) The sum m+n.

(D) The difference |m-n|.

Question 25. The justification of the line segment division construction directly uses which geometric concept?

(A) Area proportionality

(B) Length equality

(C) Angle equality

(D) Proportional sides in similar triangles

Question 26. To divide a 20 cm line segment in the ratio 3:1, how many equal parts should the auxiliary ray be marked into?

(A) 3

(B) 1

(C) 4

(D) 2

Question 1. To construct a triangle given the lengths of its three sides (SSS Criterion), you first draw one side as the base. What are the next steps?

(A) Measure the angles with a protractor.

(B) From each endpoint of the base, draw arcs with radii equal to the other two sides, intersecting each other.

(C) Draw perpendiculars from the endpoints of the base.

(D) Bisect the base.

Question 2. When constructing a triangle with sides 5 cm, 6 cm, and 12 cm, you will find that the arcs drawn from the endpoints of the base (say 12 cm) with radii 5 cm and 6 cm will not intersect. This indicates:

(A) An error in measurement.

(B) The triangle is a right-angled triangle.

(C) A triangle cannot be formed with these side lengths (Triangle Inequality Theorem is violated).

(D) The triangle is isosceles.

Question 3. To construct a triangle given two sides and the included angle (SAS Criterion), you first draw one of the given sides as the base. What is the next step?

(A) Measure the angle opposite to the given angle.

(B) Construct the given included angle at one endpoint of the base.

(C) Draw an arc from the other endpoint of the base.

(D) Draw a perpendicular from the midpoint of the base.

Question 4. In constructing a triangle using SAS, if the given sides are 'a' and 'b' and the included angle is $\theta$, which statement is correct?

(A) Draw side 'a', then angle $\theta$ at one end, then side 'b' along the angle arm.

(B) Draw side 'a', then angle $\theta$ at the other end, then side 'b' along the angle arm.

(C) Draw angle $\theta$, then side 'a' along one arm, then side 'b' along the other arm.

(D) Draw side 'a', then side 'b', then angle $\theta$ anywhere.

Question 5. To construct a triangle given two angles and the included side (ASA Criterion), you first draw the included side as the base. What is the next step?

(A) Construct the two given angles at the two endpoints of the base.

(B) Construct one given angle at one endpoint and the other angle at any point.

(C) Construct the angles first, then the side.

(D) Find the third angle using angle sum property.

Question 6. When constructing a triangle using ASA, if the given angles are $\alpha$ and $\beta$ and the included side is 's', the constructed triangle will have vertices where the arms of the two angles meet, provided:

(A) $\alpha + \beta = 180^\circ$

(B) $\alpha + \beta > 180^\circ$

(C) $\alpha + \beta < 180^\circ$

(D) $\alpha = \beta$

Question 7. To construct a triangle given two angles and one side (AAS Criterion), if the given side is NOT included between the angles, you first need to:

(A) Construct the given side.

(B) Use the angle sum property of a triangle to find the third angle, which *is* included with the given side.

(C) Construct one of the angles.

(D) Draw a perpendicular.

Question 8. In AAS construction, if angles are $A$ and $B$, and side $a$ (opposite to angle A) is given, you can find angle $C = 180^\circ - (A+B)$. Then the problem reduces to which construction type?

(A) SSS

(B) SAS

(C) ASA

(D) RHS

Question 9. To construct a triangle with sides 7 cm, 8 cm, and 9 cm, using the SSS method, how many intersection points are there for the two major arcs drawn from the endpoints of the base?

(A) Zero

(B) One

(C) Two

(D) Infinite

Question 10. Given sides 5 cm and 7 cm and an included angle of $60^\circ$, which criterion is used for construction?

(A) SSS

(B) SAS

(C) ASA

(D) AAS

Question 11. Given angles $40^\circ$ and $80^\circ$ and the included side 6 cm, which criterion is used for construction?

(A) SSS

(B) SAS

(C) ASA

(D) AAS

Question 12. Given angles $50^\circ$ and $70^\circ$ and a side of 8 cm opposite the $70^\circ$ angle, which criterion is directly applied?

(A) SSS

(B) SAS

(C) ASA

(D) AAS

Question 13. When constructing a triangle using SSS, what condition must the side lengths satisfy for a triangle to be possible?

(A) Sum of any two sides is equal to the third side.

(B) Sum of any two sides is less than the third side.

(C) Sum of any two sides is greater than the third side.

(D) Product of any two sides is greater than the third side.

Question 14. To construct a triangle with sides 3 cm, 4 cm, and 5 cm, what type of triangle is it likely to be?

(A) Equilateral

(B) Isosceles

(C) Right-angled (since $3^2 + 4^2 = 5^2$)

(D) Obtuse-angled

Question 15. When constructing a triangle using SAS, the included angle must be between the two given sides.

(A) True

(B) False

(C) It can be any angle

(D) It must be a right angle

Question 16. To construct a triangle using ASA, the included side must be between the two given angles.

(A) True

(B) False

(C) It can be any side

(D) It must be the longest side

Question 17. If you are given two angles of a triangle as $70^\circ$ and $120^\circ$, can you construct such a triangle?

(A) Yes

(B) No, because the sum of two angles is greater than $180^\circ$.

(C) Yes, but only if a side length is also given.

(D) Yes, if the third angle is $0^\circ$.

Question 18. The minimum number of measurements required to uniquely construct a triangle is typically:

(A) Two

(B) Three

(C) Four

(D) Five

Question 19. Which of the following combinations of measurements will NOT uniquely determine a triangle?

(A) Three sides (if triangle inequality holds)

(B) Two sides and any angle

(C) Two angles and one side

(D) Two sides and the included angle

Question 20. In SSS construction, if the two arcs intersect at two points, why do we usually pick only one intersection point above the base line?

(A) The other point would form a congruent triangle reflected across the base.

(B) The other point is invalid.

(C) One point gives an acute triangle, the other an obtuse triangle.

(D) The compass setting was wrong.

Question 21. When constructing a triangle using ASA, the third vertex is found by:

(A) Drawing an arc from the midpoint of the base.

(B) The intersection of the two constructed angle arms.

(C) Drawing a perpendicular bisector.

(D) Connecting the two endpoints of the base.

Question 22. If you are given two sides and a non-included angle (SSA case), it is sometimes possible to construct two different triangles. This ambiguity is known as the ambiguous case of the Sine Rule, but it's relevant to construction possibilities.

(A) True

(B) False

(C) Only if the angle is $90^\circ$

(D) Only if the sides are equal

Question 1. To construct an equilateral triangle with a given side length 'a', you draw the base of length 'a'. What are the next steps?

(A) Draw arcs of radius 'a' from both endpoints of the base, intersecting above the base.

(B) Construct angles of $60^\circ$ at both endpoints of the base.

(C) Both (A) and (B) describe valid methods.

(D) Draw a perpendicular bisector of the base.

Question 2. The justification for the compass and ruler construction of an equilateral triangle relies on the fact that by drawing arcs of radius equal to the base side from both endpoints, the intersection point forms a triangle where all three sides are equal to the given side length. This satisfies the definition of an equilateral triangle.

(A) True

(B) False

(C) Only if the base is horizontal

(D) Only if the side length is an integer

Question 3. In an isosceles triangle construction where the base and equal sides are given, you use the SSS criterion. If the base is 'b' and the equal sides are 'a', you draw the base of length 'b'. What radius do you set the compass to for the arcs from the endpoints?

(A) b

(B) a

(C) (a+b)/2

(D) |a-b|

Question 4. To construct an isosceles triangle given the base and base angles, which criterion are you using?

(A) SSS

(B) SAS

(C) ASA

(D) RHS

Question 5. To construct a right-angled triangle when the hypotenuse and the length of one side are given (RHS Criterion), you typically draw the given side as the base. Then you construct a right angle at one endpoint of the base. Where is the third vertex located?

(A) On the perpendicular line at a distance equal to the hypotenuse from the endpoint where the right angle was made.

(B) On an arc drawn from the other endpoint of the base with a radius equal to the hypotenuse, intersecting the perpendicular line.

(C) At the intersection of the perpendicular line and the angle bisector.

(D) At the midpoint of the hypotenuse.

Question 6. The justification for the RHS construction involves showing that the triangle formed by the given side, the perpendicular, and the hypotenuse satisfies the given conditions and is a right-angled triangle with the specified hypotenuse. Which theorem confirms this?

(A) Triangle Inequality Theorem

(B) Angle Sum Property

(C) Pythagoras Theorem

(D) Basic Proportionality Theorem

Question 7. Every equilateral triangle is also an isosceles triangle.

(A) True

(B) False

(C) Only sometimes

(D) Depends on the side length

Question 8. To construct an isosceles triangle with base 6 cm and base angles $50^\circ$ each, what is the measure of the third angle (vertex angle)?

(A) $50^\circ$

(B) $60^\circ$

(C) $80^\circ$

(D) $100^\circ$

Question 9. To construct a right-angled triangle with hypotenuse 10 cm and one side 6 cm, you can draw the 6 cm side. At one end, construct a $90^\circ$ angle. From the other end of the 6 cm side, draw an arc with radius 10 cm. Where does this arc intersect the perpendicular?

(A) At one point

(B) At two points

(C) It will not intersect

(D) It will be tangent to the perpendicular

Question 10. The angles in an equilateral triangle are always:

(A) $90^\circ, 45^\circ, 45^\circ$

(B) $60^\circ, 60^\circ, 60^\circ$

(C) $90^\circ, 30^\circ, 60^\circ$

(D) Varying based on side length

Question 11. When constructing an isosceles triangle given the base and base angles, the sum of the two given base angles must be:

(A) Equal to $180^\circ$

(B) Greater than $180^\circ$

(C) Less than $180^\circ$

(D) Equal to $90^\circ$

Question 12. The RHS congruence criterion is specifically for constructing which type of triangle?

(A) Equilateral

(B) Isosceles

(C) Obtuse-angled

(D) Right-angled

Question 13. If you construct an equilateral triangle, you have automatically constructed three angles of:

(A) $30^\circ$

(B) $45^\circ$

(C) $60^\circ$

(D) $90^\circ$

Question 14. To construct an isosceles triangle with a base of 7 cm and equal sides of 5 cm, you would use which construction criterion?

(A) SSS

(B) SAS

(C) ASA

(D) RHS

Question 15. When using the RHS criterion to construct a right-angled triangle, what must be given besides the right angle?

(A) Both legs

(B) Hypotenuse and one leg

(C) Both acute angles

(D) Any two sides

Question 16. The justification for constructing an equilateral triangle with $60^\circ$ angles at the base relies on the fact that if two angles of a triangle are $60^\circ$, the third angle must also be $60^\circ$ (by angle sum property), making it equiangular and thus equilateral (sides opposite equal angles are equal).

(A) True

(B) False

(C) Only for triangles with integer side lengths

(D) Only for triangles with integer angle measures

Question 17. In an isosceles triangle constructed using the base and base angles, the two arms of the angles constructed on the base will intersect at the:

(A) Midpoint of the base

(B) Vertex opposite the base

(C) Circumcenter

(D) Incenter

Question 18. Can you construct a right-angled triangle using the RHS criterion if the hypotenuse length is equal to or less than the given side length?

(A) Yes

(B) No, because the hypotenuse must be strictly greater than either leg.

(C) Only if the other leg is zero.

(D) Only if the given side is zero.

Question 19. The simplest type of triangle to construct given its side length is a/an:

(A) Isosceles triangle

(B) Scalene triangle

(C) Equilateral triangle

(D) Right-angled triangle

Question 20. To construct an isosceles triangle given the base and vertex angle, you first draw the base. Then you calculate the base angles using the angle sum property. If the vertex angle is $\theta$ and the base angles are $\alpha$, then $2\alpha + \theta = 180^\circ$. You then construct the base angles $\alpha$ at the endpoints of the base using the ASA criterion method.

(A) True

(B) False

(C) Only if $\theta = 90^\circ$

(D) Only if the base angles are $60^\circ$

Question 21. When constructing a right-angled triangle using the RHS criterion, one of the angles is already fixed at $90^\circ$. The other two angles will always be:

(A) Equal

(B) Complementary (sum to $90^\circ$)

(C) Supplementary (sum to $180^\circ$)

(D) Both acute and equal

Question 1. To construct a triangle ABC where side BC, $\angle B$, and the sum of the other two sides AB + AC are given, you first draw BC and construct $\angle B$. Along the ray BX (where $\angle B$ is formed), you cut off a segment BD equal to AB + AC. Then you join D to C. What is the next key construction step?

(A) Draw a line parallel to BC through A.

(B) Construct the perpendicular bisector of CD.

(C) Bisect $\angle BDC$.

(D) Draw a perpendicular from B to CD.

Question 2. In the construction of triangle ABC given BC, $\angle B$, and AB + AC, after constructing the perpendicular bisector of CD, it intersects BD at point A. The justification involves showing that A is equidistant from C and D (AC = AD) because A lies on the perpendicular bisector of CD. Then since BD = BA + AD and BD = AB + AC, it follows that BA + AD = AB + AC, and since AD = AC, this holds true.

(A) True

(B) False

(C) Only if $\angle B = 90^\circ$

(D) Only if BC = AB + AC

Question 3. To construct a triangle ABC where side BC, $\angle B$, and the difference of the other two sides AB - AC (or AC - AB) are given, you first draw BC and construct $\angle B$. Along the ray BX (where $\angle B$ is formed), you cut off a segment BD equal to the given difference. If AB > AC, you cut BD on the same side as BC. If AC > AB, you extend CB and cut BD on the opposite side. After joining D to C, what is the next key construction step?

(A) Construct the angle bisector of $\angle C$.

(B) Draw a line parallel to BD through C.

(C) Construct the perpendicular bisector of CD.

(D) Draw a circle with radius BD centered at B.

Question 4. In the construction of triangle ABC given BC, $\angle B$, and AB - AC, the perpendicular bisector of CD intersects BX at A. The justification involves showing AC = AD because A is on the perpendicular bisector. If AB > AC, then AB = AD + DB = AC + DB. So AB - AC = DB, which was given. This confirms the construction.

(A) True

(B) False

(C) Only if $\angle C = 90^\circ$

(D) Only if AB = AC

Question 5. To construct a triangle ABC given two angles $\angle B$ and $\angle C$ and the perimeter (AB + BC + CA), you first draw a line segment PQ equal to the perimeter. Then you construct angles equal to $\angle B$ and $\angle C$ at P and Q respectively. These angles are usually half of the original angles. Why half?

(A) To ensure the triangle fits within the perimeter line segment.

(B) Because the construction forms isosceles triangles where the base angles are related to the required angles.

(C) It simplifies the parallel line construction.

(D) It is an arbitrary choice.

Question 6. In the perimeter construction (given $\angle B$, $\angle C$, Perimeter), after constructing angles $(1/2)\angle B$ at P and $(1/2)\angle C$ at Q on the segment PQ = Perimeter, the intersection of the angle arms gives vertex A. Then you construct perpendicular bisectors of PA and QA. These bisectors intersect PQ at points B and C. The justification relies on which property?

(A) Points on angle bisector are equidistant from arms.

(B) Points on perpendicular bisector are equidistant from endpoints of segment.

(C) Angle sum property of a triangle.

(D) Similarity of triangles.

Question 7. When constructing a triangle given two sides and a median, say sides AB, AC and median AD, the strategy often involves constructing a related figure first. A common approach is to extend AD to E such that AD = DE, and then join C to E. What shape is formed by ABEC?

(A) Square

(B) Rhombus

(C) Rectangle

(D) Parallelogram

Question 8. In the construction with two sides (AB, AC) and median (AD), forming parallelogram ABEC, we know AB = CE and AC = BE. Also, the diagonals bisect each other at D. Triangle ACE can be constructed because its sides are related to the given information. What are the lengths of the sides of triangle ACE?

(A) AB, AC, 2*AD

(B) AB, AC, AD

(C) AB, BC, AD

(D) AC, BE, AD

Question 9. To construct a triangle given two angles and an altitude, say $\angle B$, $\angle C$, and altitude AD to BC. You can first draw a line and a point D on it. Then construct a perpendicular at D. The vertex A lies on this perpendicular at a distance equal to the altitude. To find B and C, you use the given angles. $\angle ABD$ and $\angle ACD$ are part of the triangle. How are $\angle B$ and $\angle C$ used?

(A) Construct $\angle DAB = 90^\circ - \angle B$ and $\angle DAC = 90^\circ - \angle C$ at A on the perpendicular.

(B) Construct $\angle DBA = \angle B$ and $\angle DCA = \angle C$ at D on the original line.

(C) Construct angles equal to $\angle B$ and $\angle C$ at A on the perpendicular.

(D) Construct lines parallel to the perpendicular at angles $\angle B$ and $\angle C$.

Question 10. In the construction of a triangle given two angles and an altitude, say $\angle B$, $\angle C$, altitude AD. After drawing the line, point D, the perpendicular at D, and marking A such that AD = altitude, you construct $\angle DAB = 90^\circ - \angle B$ and $\angle DAC = 90^\circ - \angle C$. The rays forming these angles intersect the original line at B and C. This construction works because in right triangles ABD and ACD, $\angle B = 90^\circ - \angle DAB$ and $\angle C = 90^\circ - \angle DAC$.

(A) True

(B) False

(C) Only if the altitude is one of the sides

(D) Only if $\angle B = \angle C$

Question 11. When constructing a triangle given one side, one angle, and the sum of the other two sides, the point A (the third vertex) is found by the intersection of the perpendicular bisector of CD (where D is on the extended base ray) and the ray from B forming the given angle. The key property is that any point on the perpendicular bisector is equidistant from the endpoints C and D.

(A) True

(B) False

(C) This applies to the difference case, not sum.

(D) This applies to the perimeter case.

Question 12. Consider constructing $\triangle ABC$ with BC = 6 cm, $\angle B = 60^\circ$, and AB + AC = 10 cm. You draw BC = 6 cm, construct $\angle B = 60^\circ$. Along the ray from B, mark D such that BD = 10 cm. Join DC. Which construction helps find A?

(A) Angle bisector of $\angle BDC$

(B) Perpendicular bisector of CD

(C) Angle bisector of $\angle BCD$

(D) Altitude from B to CD

Question 13. For constructing $\triangle ABC$ given BC, $\angle B$, and AB - AC (assume AB > AC), you draw BC and $\angle B$. Cut BD = AB - AC from the ray BX. Join DC. The perpendicular bisector of DC intersects BX at A. Triangle ADC is formed. What kind of triangle is ADC in this construction?

(A) Equilateral

(B) Isosceles (with AD = AC)

(C) Right-angled

(D) Scalene

Question 14. To construct a triangle ABC given $\angle B$, $\angle C$, and perimeter AB + BC + CA, after drawing the perimeter segment PQ and angles $(1/2)\angle B$ and $(1/2)\angle C$ at P and Q to find A, you need to find B and C on PQ. B is the intersection of PQ and the perpendicular bisector of PA. C is the intersection of PQ and the perpendicular bisector of QA. This is because any point on the perpendicular bisector of PA is equidistant from P and A (BP = BA). Similarly, CQ = CA. Since BP + BC + CQ = PQ and PQ = AB + BC + CA, substituting BP = BA and CQ = CA gives BA + BC + CA = AB + BC + CA.

(A) True

(B) False

(C) Only if $\angle B = \angle C$

(D) Only if the perimeter is an integer

Question 15. In the construction of a triangle given two angles $\angle B$ and $\angle C$ and altitude AD (where D is on BC), the sum of angles $\angle B$ and $\angle C$ must be:

(A) Equal to $90^\circ$

(B) Greater than $90^\circ$

(C) Less than $180^\circ$

(D) Equal to $180^\circ$

Question 16. The construction of a triangle when two sides and a median are given is often reduced to which basic triangle construction?

(A) SSS

(B) SAS

(C) ASA

(D) RHS

Question 17. To construct $\triangle ABC$ given BC, $\angle B$, and AC - AB (assume AC > AB), you draw BC, construct $\angle B$. Cut BD = AC - AB from the ray BX *extended backwards* from B. Join DC. Where will the perpendicular bisector of DC intersect the ray BX to find A?

(A) Between B and D

(B) Beyond D on the extended ray

(C) Between B and C

(D) It will not intersect the ray.

Question 18. In the perimeter construction given $\angle B$, $\angle C$, and perimeter, the angles constructed at P and Q on the perimeter line PQ are $\angle APQ = (1/2)\angle B$ and $\angle AQP = (1/2)\angle C$. Why is this done?

(A) To make $\triangle APB$ and $\triangle AQC$ isosceles.

(B) To make $\triangle ABC$ isosceles.

(C) To make PQ equal to the perimeter.

(D) To ensure $\angle PAQ = 180^\circ - (\angle B + \angle C)$.

Question 19. When constructing a triangle given two angles and an altitude, the altitude is always perpendicular to one of the sides. If the altitude from A to BC is given, it means the point D lies on the line containing BC.

(A) True

(B) False

(C) Only if the triangle is acute

(D) Only if the triangle is obtuse

Question 20. Can you construct a triangle given three medians?

(A) Yes

(B) No

(C) Only specific types of triangles

(D) Requires additional information

Question 21. The construction of a triangle when one side, one angle, and the difference of the other two sides are provided involves creating an isosceles triangle outside (or inside) the main triangle based on the difference length. The perpendicular bisector of the third side of this isosceles triangle finds the vertex A.

(A) True

(B) False

(C) This method is for the sum case.

(D) This method is for the perimeter case.

Question 22. In the perimeter construction (given $\angle B$, $\angle C$, Perimeter), if $\angle B + \angle C \geq 180^\circ$, can you form the triangle?

(A) Yes

(B) No, because the sum of angles in a triangle must be $180^\circ$.

(C) Only if the perimeter is very large.

(D) Only if the angles are $90^\circ$ each.

Question 23. The construction of a triangle given two sides and a median relies on the properties of parallelograms formed by extending the median.

(A) True

(B) False

(C) It relies on properties of trapezoids.

(D) It relies on properties of rhombi.

Question 24. When constructing a triangle given two angles and an altitude, if the sum of the given angles is $90^\circ$, what type of triangle is it?

(A) Equilateral

(B) Isosceles

(C) Right-angled

(D) Obtuse-angled

Question 25. The construction method for triangles given one side, one angle, and the sum/difference of other two sides transforms the problem into finding a point on the perpendicular bisector of a segment.

(A) True

(B) False

(C) It transforms it into finding a point on an angle bisector.

(D) It transforms it into finding a point on an altitude.

Question 1. To construct a triangle similar to a given triangle ABC with a scale factor $k$ (a rational number), you first draw one side of the original triangle, say BC. On a ray BX making an acute angle with BC, you mark points. The number of points to mark depends on the scale factor.

(A) True

(B) False

(C) Only if $k > 1$

(D) Only if $k < 1$

Question 2. If the scale factor is $m/n$, where m and n are positive integers, and you are scaling from $\triangle ABC$ to $\triangle A'BC'$, with B as the common vertex and BC on the same line as BC'. You draw ray BX and mark $m+n$ equal points $B_1, B_2, \dots, B_{m+n}$ on BX. To find C', you connect which point on BX to C?

(A) $B_m$

(B) $B_n$

(C) $B_{m+n}$

(D) $B_1$

Question 3. If the scale factor is $3/5$ (less than one), you construct ray BX and mark 5 points. You connect $B_5$ to C. To find C' on BC, you draw a line through $B_3$ parallel to $B_5$C. This line intersects BC at C'. What ensures that BC'/BC = 3/5?

(A) Angle Bisector Theorem

(B) Converse of Midpoint Theorem

(C) Basic Proportionality Theorem (BPT)

(D) Pythagoras Theorem

Question 4. If the scale factor is $5/3$ (greater than one), you construct ray BX and mark 5 points. You connect $B_3$ to C. To find C' on the extension of BC, you draw a line through $B_5$ parallel to $B_3$C. This line intersects the extension of BC at C'. What ensures that BC'/BC = 5/3?

(A) Congruence of triangles

(B) Similarity of triangles (specifically, $\triangle BB_3C \sim \triangle BB_5C'$)

(C) Area ratios

(D) Perpendicularity

Question 5. In the construction of a similar triangle with scale factor $m/n$, the maximum number of equal parts the ray BX is divided into is:

(A) m

(B) n

(C) m+n

(D) Max(m, n)

Question 6. When the scale factor is greater than one, the constructed similar triangle will be _____ than the original triangle.

(A) Smaller

(B) Larger

(C) Congruent

(D) Rotated

Question 7. When the scale factor is less than one, the constructed similar triangle will be _____ than the original triangle.

(A) Smaller

(B) Larger

(C) Congruent

(D) Translated

Question 8. Justification of similar triangle construction (using BPT) shows that if a line is parallel to one side of a triangle intersecting the other two sides, it divides the two sides proportionally. By constructing parallel lines, we create a smaller or larger triangle whose sides are in the desired ratio to the original triangle's sides.

(A) True

(B) False

(C) Only for equilateral triangles

(D) Only for right-angled triangles

Question 9. In the construction of a similar triangle, after finding C' on BC (or its extension), you need to find A'. You draw a line through C' parallel to AC. This line intersects BA (or its extension) at A'. Why is this step necessary?

(A) To ensure $\angle BA'C' = \angle BAC$.

(B) To ensure $\triangle A'BC'$ is similar to $\triangle ABC$ by AA or AAA similarity criterion.

(C) To make A'C' parallel to AC.

(D) All of the above.

Question 10. If the scale factor is $1/1$, the constructed triangle will be:

(A) Smaller

(B) Larger

(C) Congruent to the original triangle

(D) A single point

Question 11. To construct parallel lines in the similar triangle construction, you can use the method of copying angles or using set squares. The compass and ruler method for copying angles is typically preferred for formal construction.

(A) True

(B) False

(C) Only if the angle is $90^\circ$

(D) Only if the angle is $60^\circ$

Question 12. If the scale factor is $m/n$, and m > n, you connect the $n^{th}$ point on the ray BX to C and draw a parallel line through the $m^{th}$ point. The new triangle is constructed by extending the original sides.

(A) True

(B) False

(C) Only if the triangle is acute

(D) Only if the triangle is obtuse

Question 13. If the scale factor is $m/n$, and m < n, you connect the $n^{th}$ point on the ray BX to C and draw a parallel line through the $m^{th}$ point. The new triangle is constructed *inside* the original triangle.

(A) True

(B) False

(C) Only if m and n are prime

(D) Only if m = n-1

Question 14. The ray BX, used for marking equal divisions, must make an acute angle with BC. Why?

(A) To make the construction easier to see.

(B) To ensure the points on BX are distinctly separated from the points on BC.

(C) An obtuse angle wouldn't work.

(D) It's a convention, any angle works as long as it's not $0^\circ$ or $180^\circ$.

Question 15. The justification for similarity ($ \triangle A'BC' \sim \triangle ABC $) after the construction with scale factor $m/n$ using BPT relies on the fact that $\frac{BC'}{BC} = \frac{BA'}{BA} = \frac{m}{n}$ and $\angle B$ is common to both triangles (SAS similarity) or that $\angle BA'C' = \angle BAC$ and $\angle BC'A' = \angle BCA$ (AA similarity due to parallel lines A'C' || AC).

(A) True

(B) False

(C) Only if $m=n$

(D) Only if the triangle is isosceles

Question 16. If the scale factor is 2, the area of the constructed triangle will be how many times the area of the original triangle?

(A) 2

(B) 4

(C) 1/2

(D) 1/4

Question 17. If the scale factor is $1/3$, the ratio of the perimeters of the new triangle to the original triangle is:

(A) 1:3

(B) 3:1

(C) 1:9

(D) 9:1

Question 18. Can you construct a triangle similar to a given triangle with a scale factor that is an irrational number (e.g., $\sqrt{2}$)?

(A) Yes, by constructing the length $\sqrt{2}$ and then applying the ratio division concept.

(B) No, the method requires rational scale factors.

(C) Only if the original triangle sides are irrational.

(D) Only if using a calculator.

Question 19. When constructing a similar triangle, the corresponding angles of the new triangle are _____ to the corresponding angles of the original triangle.

(A) Proportional

(B) Supplementary

(C) Equal

(D) Complementary

Question 20. The construction of similar triangles is a direct application of which theorem/postulate?

(A) Pythagoras Theorem

(B) Angle Sum Property

(C) Basic Proportionality Theorem (Thales Theorem)

(D) SSS Congruence

Question 21. If you construct a similar triangle with scale factor $k$, where $k > 1$, the vertices A' and C' will lie on the _____ of BA and BC respectively.

(A) Interior

(B) Extension

(C) Midpoints

(D) Perpendicular bisectors

Question 22. If you construct a similar triangle with scale factor $k$, where $k < 1$, the vertices A' and C' will lie on the _____ of BA and BC respectively.

(A) Interior

(B) Extension

(C) Outside

(D) Perpendicular lines

Question 23. The method of constructing similar triangles using a common vertex and a ray with marked divisions is based on ensuring the corresponding sides are in the required ratio.

(A) True

(B) False

(C) It's based on ensuring corresponding angles are equal.

(D) It's based on ensuring the areas are proportional.

Question 1. To construct a general quadrilateral, you need a certain number of independent measurements. What is the minimum number of measurements typically required for a unique construction?

(A) Three

(B) Four

(C) Five

(D) Six

Question 2. Which set of measurements is sufficient to uniquely construct a quadrilateral ABCD?

(A) 4 sides

(B) 3 angles

(C) 2 sides and 2 angles

(D) 4 sides and 1 diagonal

Question 3. To construct a parallelogram, you can use properties like opposite sides being equal and parallel, or opposite angles being equal, or diagonals bisecting each other. If you are given two adjacent sides and the included angle, which criterion for triangle construction is often used as a step?

(A) SSS

(B) SAS

(C) ASA

(D) RHS

Question 4. A rectangle is a parallelogram with all angles equal to $90^\circ$. To construct a rectangle given its length and width, say 'l' and 'w', you can draw a side of length 'l', construct $90^\circ$ angles at both ends, and mark points on the perpendiculars at distance 'w'. Which construction is essential?

(A) Angle bisector

(B) Perpendicular construction

(C) Parallel line construction (though implied by perpendiculars)

(D) Both B and C are fundamentally involved.

Question 5. A rhombus is a parallelogram with all four sides equal. To construct a rhombus given the side length and one angle, you can draw a side, construct the given angle, draw the adjacent side (equal to the first), and then find the fourth vertex using arcs of radius equal to the side length from the endpoints of the two sides.

(A) True

(B) False

(C) You need both diagonals to construct a rhombus.

(D) A rhombus construction always uses $90^\circ$ angles.

Question 6. A square is a rhombus with one angle equal to $90^\circ$, or a rectangle with adjacent sides equal. To construct a square with a given side length 's', you draw a side 's', construct $90^\circ$ angles at both ends, and mark points at distance 's' on the perpendiculars. This construction is a specific case of which general quadrilateral construction?

(A) SSS

(B) SAS

(C) 4 sides and 1 angle

(D) 2 adjacent sides and included angle

Question 7. To construct a general quadrilateral given 4 sides and a diagonal, you first construct one of the triangles formed by the diagonal and three sides using the SSS criterion. Then you use the remaining two sides to locate the fourth vertex by drawing intersecting arcs.

(A) True

(B) False

(C) You need 5 sides for this.

(D) You need 2 diagonals.

Question 8. The conditions for unique construction of a quadrilateral include cases like 4 sides and a diagonal, 3 sides and 2 included angles, etc. These conditions essentially break down the quadrilateral into one or more triangles which can be uniquely constructed.

(A) True

(B) False

(C) Unique construction is always possible with any 4 measurements.

(D) Unique construction is never possible.

Question 9. To construct a parallelogram given two adjacent sides and a diagonal, you can construct the triangle formed by the two adjacent sides and the diagonal using SSS. Then use the parallelogram properties (opposite sides equal) to find the fourth vertex using intersecting arcs.

(A) True

(B) False

(C) You need the angle between the diagonal and a side.

(D) This method only works for rectangles.

Question 10. To construct a rhombus given the lengths of its two diagonals, you draw one diagonal and find its midpoint. Then you construct the perpendicular bisector of the first diagonal. The other diagonal lies on this bisector. You mark half the length of the second diagonal on the bisector on either side of the midpoint. The endpoints are the other two vertices of the rhombus.

(A) True

(B) False

(C) Rhombus diagonals are perpendicular but don't bisect each other.

(D) Rhombus diagonals are equal.

Question 11. Can you uniquely construct a quadrilateral if you are given the lengths of all four sides?

(A) Yes, always.

(B) No, not uniquely (it can be a parallelogram, trapezoid, kite, etc., with varying angles).

(C) Only if it's a square or rhombus.

(D) Only if one angle is also given.

Question 12. To construct a square given the length of its diagonal, you can draw the diagonal, construct its perpendicular bisector, and mark half the diagonal length on the bisector on either side of the midpoint. These four endpoints form the square.

(A) True

(B) False

(C) This construction is for a rhombus, not a square.

(D) You need the side length, not the diagonal.

Question 13. Which specific quadrilateral is uniquely determined by its four sides and one angle?

(A) Parallelogram (given 2 adjacent sides and included angle)

(B) Rhombus (given side and one angle)

(C) Rectangle (given length and width, implies $90^\circ$ angle)

(D) All of the above are uniquely determined by this combination.

Question 14. To construct a parallelogram given two adjacent sides and the angle between them, you use the SAS criterion to construct the first triangle (formed by the two sides and the diagonal opposite the angle). Then use parallel line or side length properties to find the fourth vertex.

(A) True

(B) False

(C) You only need the two sides, the angle is irrelevant.

(D) You need a diagonal instead of the angle.

Question 15. A unique quadrilateral can be constructed if you are given:

(A) 3 sides and 1 angle

(B) 2 diagonals

(C) 3 angles and 1 side

(D) 4 sides and 1 diagonal

Question 16. The construction of a rectangle given its perimeter (P) and one side (s) is possible because from these, you can deduce the length of the adjacent side ($w = P/2 - s$). Then you construct it using the length and width and $90^\circ$ angles.

(A) True

(B) False

(C) You need the area, not the perimeter.

(D) You need a diagonal.

Question 17. To construct a rhombus given one side length and one diagonal, you draw the diagonal as the base of two isosceles triangles with the given side length as the equal sides. Constructing these two triangles using SSS criterion (diagonal length, side length, side length) will locate the other two vertices of the rhombus.

(A) True

(B) False

(C) This only works if the given diagonal is the shorter one.

(D) This only works if the given diagonal is the longer one.

Question 18. Which of the following is NOT a condition for uniquely constructing a quadrilateral?

(A) Four sides and one diagonal

(B) Three sides and two included angles

(C) Four sides and one angle

(D) Two adjacent sides and three angles

Question 19. To construct a square given its perimeter (P), you first find the side length (s = P/4). Then construct the square using the side length and $90^\circ$ angles.

(A) True

(B) False

(C) You need the area to find the side length.

(D) Perimeter is insufficient information.

Question 20. When constructing a parallelogram given the diagonals and the angle between them, you use the property that diagonals bisect each other. You construct the point of intersection, draw the diagonals at the given angle, and mark half the length of each diagonal from the intersection point along the diagonal lines. Connecting these endpoints forms the parallelogram.

(A) True

(B) False

(C) Only for rectangles.

(D) Only for rhombi.

Question 21. The construction of any polygon with more than 3 sides can often be broken down into the construction of:

(A) Circles

(B) Triangles

(C) Parallel lines

(D) Perpendiculars

Question 1. To construct a tangent to a circle at a point P on the circle, given the center O, you first draw the radius OP. What is the next step?

(A) Draw a line parallel to OP through P.

(B) Construct a line perpendicular to OP at P.

(C) Draw a circle with center P.

(D) Connect O to any other point on the circle.

Question 2. The construction of a tangent at a point on the circle relies on which geometric property?

(A) The tangent is parallel to the radius at the point of contact.

(B) The tangent is perpendicular to the radius at the point of contact.

(C) The tangent passes through the center of the circle.

(D) The tangent forms an acute angle with the radius.

Question 3. To construct tangents to a circle from a point P outside the circle, you first join O (center) to P. Then you construct the perpendicular bisector of OP to find its midpoint M. What is the next crucial step?

(A) Draw a circle with center P and radius OP.

(B) Draw a circle with center O and radius OP.

(C) Draw a circle with center M and radius OM (or MP), which passes through O and P.

(D) Draw a line parallel to OP through M.

Question 4. In the construction of tangents from an external point P, the circle with center M (midpoint of OP) and radius OM intersects the given circle at two points, say Q and R. Why are PQ and PR the required tangents?

(A) Because $\angle OQP$ and $\angle ORP$ are angles in a semicircle, they are $90^\circ$. A line from the external point P to a point on the circle is tangent if it is perpendicular to the radius at that point (OQ and OR are radii). Therefore, PQ $\perp$ OQ and PR $\perp$ OR.

(B) Because M is the midpoint of OP.

(C) Because $\triangle OQP$ and $\triangle ORP$ are congruent.

(D) By definition of a tangent.

Question 5. How many tangents can be drawn to a circle from an external point?

(A) One

(B) Two

(C) Infinite

(D) Zero

Question 6. The length of the two tangents drawn from an external point to a circle are:

(A) Always unequal

(B) Always equal

(C) Equal only if the point is on the circle

(D) Equal only if the circle passes through the point

Question 7. To construct a pair of tangents to a circle from an external point such that the angle between the tangents is $60^\circ$. If O is the center, P is the external point, and A, B are points of contact, then $\angle APB = 60^\circ$. In quadrilateral OAPB, $\angle OAP = \angle OBP = 90^\circ$. The sum of angles in a quadrilateral is $360^\circ$. So, $\angle AOB + \angle OAP + \angle APB + \angle OBP = 360^\circ$. This gives $\angle AOB + 90^\circ + 60^\circ + 90^\circ = 360^\circ$. What is the value of $\angle AOB$?

(A) $60^\circ$

(B) $120^\circ$

(C) $150^\circ$

(D) $180^\circ$

Question 8. To construct tangents with a specific angle between them (say $60^\circ$), you first calculate the angle between the radii to the points of contact. This angle is $180^\circ - (\text{angle between tangents})$. So, for $60^\circ$ between tangents, the angle between radii is $120^\circ$. You construct this angle at the center O. The arms of this angle intersect the circle at the points of contact. What is the next step?

(A) Draw perpendiculars at these points of contact.

(B) Draw lines from O to these points.

(C) Connect the points of contact.

(D) Draw lines parallel to the radii.

Question 9. The justification for constructing tangents from an external point relies on the property that the angle in a semicircle is $90^\circ$, leading to the conclusion that the radius is perpendicular to the constructed line at the point of contact.

(A) True

(B) False

(C) Tangent justification uses parallel lines.

(D) Tangent justification uses angle bisectors.

Question 10. How many tangents can be drawn to a circle from a point *inside* the circle?

(A) One

(B) Two

(C) Infinite

(D) Zero

Question 11. To construct a tangent at a point on the circle without using the center, you can draw any chord PQ passing through P. Then take a point R on the major arc PQ. Construct $\angle RQP$. Construct an angle equal to $\angle RQP$ at P on the opposite side of the chord PQ. The arm of this new angle forms the tangent. This construction relies on the property that the angle between a tangent and a chord through the point of contact is equal to the angle in the alternate segment.

(A) True

(B) False

(C) This method is only for finding the center.

(D) This method is incorrect.

Question 12. When constructing tangents from an external point P, the line segment OP is the diameter of the second circle drawn. The centre of the second circle is the midpoint of OP.

(A) True

(B) False

(C) OP is the radius of the second circle.

(D) The second circle is centered at O.

Question 13. If you need to construct a pair of tangents such that they are parallel to each other, how many such pairs are possible for a given circle?

(A) One

(B) Two

(C) Infinite

(D) Zero

Question 14. The justification for constructing tangents from an external point P often involves proving the congruence of triangles $\triangle OQP$ and $\triangle ORP$ using which criterion?

(A) SSS

(B) SAS

(C) ASA

(D) RHS (since OQ=OR (radii), OP=OP (common), $\angle OQP = \angle ORP = 90^\circ$)

Question 15. To construct a tangent at a point on the circle without using the center, you need to construct a line perpendicular to the radius OP at P. This is the same method used to construct a $90^\circ$ angle at a point on a line.

(A) True

(B) False

(C) Only if the point is on the x-axis.

(D) Only if the radius is an integer.

Question 16. In the construction of tangents from an external point P, the points of intersection of the second circle with the original circle are the points of tangency. This is because the angle subtended by the diameter OP at these intersection points is $90^\circ$, making the lines OQ and OR perpendicular to PQ and PR respectively.

(A) True

(B) False

(C) The points of intersection are the center of the original circle.

(D) The points of intersection are the external point P.

Question 17. If the distance of an external point P from the center O is less than the radius of the circle, can you construct tangents from P to the circle?

(A) Yes, two tangents.

(B) Yes, one tangent.

(C) No, the point is inside the circle.

(D) Yes, infinite tangents.

Question 18. To construct tangents to a circle with a specific angle $\theta$ between them, you need to construct an angle of $180^\circ - \theta$ at the center. If $\theta = 90^\circ$, the angle at the center is:

(A) $0^\circ$

(B) $90^\circ$

(C) $180^\circ$

(D) $270^\circ$

Question 19. The line segment from the center of a circle to the point of tangency is always _____ to the tangent at that point.

(A) Parallel

(B) Perpendicular

(C) At $60^\circ$

(D) Bisected

Question 20. If you are asked to construct tangents from an external point P to a circle without being given the center, what is the first step you should take?

(A) Draw a line through P.

(B) Find the center of the circle.

(C) Draw a secant through P.

(D) Measure the distance from P to the circle.

Question 21. The construction of tangents from an external point involves creating a right angle relative to the radius at the point of contact. This right angle is formed by which lines?

(A) The radius and the line segment from the external point to the point of contact.

(B) The two radii to the points of contact.

(C) The two tangents.

(D) The line connecting the center to the external point and a radius.

Question 22. If two tangents are inclined at an angle of $90^\circ$ to each other, the quadrilateral formed by the center, the external point, and the two points of contact is a:

(A) Rhombus

(B) Rectangle

(C) Square

(D) Trapezium

Question 1. What is the primary role of justification in geometric constructions?

(A) To make the construction look complicated.

(B) To prove that the constructed figure meets the required properties.

(C) To show alternative construction methods.

(D) To measure the lengths and angles of the construction.

Question 2. Justification in geometric constructions typically involves applying:

(A) Advanced calculus theorems.

(B) Probability calculations.

(C) Basic geometric principles (axioms, postulates, theorems) and logical reasoning.

(D) Statistical analysis.

Question 3. When justifying the angle bisector construction, which geometric principle is most directly applied?

(A) SAS congruence criterion.

(B) Properties of parallel lines.

(C) Angle sum property of a triangle.

(D) The fact that any point on the bisector is equidistant from the arms (and this property is *proven* using congruence).

Question 4. The justification for constructing a perpendicular bisector involves proving that the constructed line is both perpendicular to the segment and passes through its midpoint. This often involves demonstrating that points on the bisector are equidistant from the segment endpoints, which can be shown using triangle congruence.

(A) True

(B) False

(C) Perpendicular bisector justification only involves proving perpendicularity.

(D) Perpendicular bisector justification only involves proving it passes through the midpoint.

Question 5. The justification for constructing parallel lines using corresponding or alternate interior angles relies on the converses of the theorems related to these angles when a transversal intersects two lines.

(A) True

(B) False

(C) It relies on the theorems themselves, not their converses.

(D) Parallel line justification uses Pythagoras theorem.

Question 6. Justification of dividing a line segment in a given ratio directly uses which theorem?

(A) Pythagoras Theorem

(B) Angle Sum Property

(C) Basic Proportionality Theorem (Thales Theorem)

(D) Midpoint Theorem

Question 7. When justifying triangle constructions (SSS, SAS, ASA, AAS), the primary tool is:

(A) Similarity of triangles

(B) Congruence criteria of triangles

(C) Properties of circles

(D) Properties of quadrilaterals

Question 8. The justification for constructing an equilateral triangle by drawing arcs of equal radius from the base endpoints relies on the SSS congruence criterion when comparing parts of the construction, ultimately showing all sides of the formed triangle are equal to the initial side.

(A) True

(B) False

(C) Justification for equilateral triangle uses ASA.

(D) Justification for equilateral triangle uses RHS.

Question 9. Justification for constructing tangents to a circle from an external point often involves using the property that the angle in a semicircle is $90^\circ$, leading to the conclusion that the radius is perpendicular to the constructed line at the point of contact.

(A) True

(B) False

(C) Tangent justification uses parallel lines.

(D) Tangent justification uses angle bisectors.

Question 10. Verifying the accuracy of a construction using measurement tools like a ruler or protractor:

(A) Constitutes a formal geometric justification.

(B) Does not constitute a formal geometric justification, but can indicate if the construction was performed correctly.

(C) Proves the underlying geometric principle.

(D) Is unnecessary if the steps were followed correctly.

Question 11. A theorem is a statement that:

(A) Is accepted without proof.

(B) Can be proven using axioms, postulates, and previously proven theorems.

(C) Is a description of a geometric figure.

(D) Is only used in construction, not justification.

Question 12. An axiom or postulate is a statement that is:

(A) Derived from other statements.

(B) Accepted as true without proof.

(C) Used only in measurement.

(D) Disproven in higher geometry.

Question 13. The justification of geometric constructions helps to understand:

(A) How to draw faster.

(B) Why the construction method works based on established geometric facts.

(C) The history of geometry.

(D) How to use a protractor more accurately.

Question 14. Which of the following is NOT typically a tool for formal geometric justification?

(A) Congruence criteria

(B) Properties of parallel lines

(C) Measuring with a ruler.

(D) Basic definitions (e.g., definition of perpendicular lines, angle bisector)

Question 15. Justification involves a sequence of logical steps, where each step is supported by a definition, axiom, postulate, or previously proven theorem. This process is a core part of deductive reasoning in geometry.

(A) True

(B) False

(C) Justification is based on inductive reasoning.

(D) Justification is based on probability.

Question 16. If a construction produces a figure that looks correct when measured, does it mean the construction is formally justified?

(A) Yes, measurement is the ultimate proof.

(B) No, measurement provides evidence of accuracy but not a formal proof based on geometry principles.

(C) Only if the measurement is extremely precise.

(D) Only if multiple measurements are taken.

Question 17. Which of the following congruence criteria is frequently used in justifications related to angle bisectors and perpendicular bisectors?

(A) ASA

(B) AAS

(C) SSS

(D) All of these can be used depending on the specific setup.

Question 18. When justifying the construction of a $60^\circ$ angle, you demonstrate that the triangle formed by the vertex and the intersection points of the arcs is equilateral. This uses the fact that the sides are constructed equal, and then applies SSS congruence or properties of angles in a triangle.

(A) True

(B) False

(C) Justification for $60^\circ$ uses parallel lines.

(D) Justification for $60^\circ$ angle is not required.

Question 19. The primary goal of learning geometric constructions is to be able to draw figures accurately using tools. Justification is a secondary, less important skill.

(A) True

(B) False (Understanding *why* the construction works is as important as performing it).

(C) Justification is only for advanced mathematics.

(D) Accuracy in drawing is the only important skill.

Question 20. The justification for dividing a line segment in a given ratio relies on the proportionality of segments created by parallel lines intersecting transversals. This is a fundamental concept from similar triangles or Thales Theorem.

(A) True

(B) False

(C) It relies on congruence of triangles.

(D) It relies on properties of circles.

Question 21. When justifying the construction of a $90^\circ$ angle at a point on a line, one might show that the angle is formed by bisecting a straight angle using a perpendicular bisector construction applied to a segment on the straight line centered at the point. This creates two $90^\circ$ angles.

(A) True

(B) False

(C) $90^\circ$ justification uses SAS.

(D) $90^\circ$ justification uses parallel lines.

Question 22. Formal justification of a construction ensures that the method will always produce the desired result, regardless of the specific measurements used, provided the initial conditions are met.

(A) True

(B) False

(C) Justification only works for specific examples.

(D) Justification is only needed if the construction is difficult.

Question 23. Which of the following is a basic geometric postulate often used implicitly in constructions and their justifications?

(A) Through any two points, there is exactly one line.

(B) All right angles are equal.

(C) A circle may be described with any centre and any radius.

(D) All of the above are relevant postulates.

Question 24. Justification is crucial for developing logical thinking and proving mathematical statements. It elevates construction from merely following steps to understanding the underlying geometric principles.

(A) True

(B) False

(C) Logical thinking is not needed for geometry.

(D) Geometric principles are only for theory, not practice.