Negative Questions MCQs for Sub-Topics of Topic 5: Construction

Question 1. Which of the following is NOT a basic geometric element used in compass and ruler constructions?

(A) Point

(B) Line Segment

(C) Sphere

(D) Circle

Question 2. When constructing a circle with a given radius, which tool is NOT essential for the basic construction itself (though might be used to define the initial radius length)?

(A) Compass

(B) Ruler (with markings)

(C) Pencil

(D) Paper

Question 3. Which of the following is NOT a correct way to measure a line segment of a specific length (say $5 \text{ cm}$) using a standard ruler?

(A) Place one endpoint at the $0 \text{ cm}$ mark and read the mark at $5 \text{ cm}$.

(B) Place one endpoint at the $1 \text{ cm}$ mark and read the mark at $6 \text{ cm}$.

(C) Place the ruler randomly and draw a segment that visually appears to be $5 \text{ cm}$.

(D) Use a compass opened to $5 \text{ cm}$ and mark two points on a line.

Question 4. Which of the following is NOT true about copying a line segment AB onto a line L starting at point P using a compass?

(A) The compass opening is set to the distance between A and B.

(B) The compass point is placed at P.

(C) An arc is drawn that intersects line L.

(D) The segment is copied at a $90^\circ$ angle to the original segment.

Question 5. Which statement is NOT true about a line segment AB?

(A) It has a definite finite length.

(B) It has two distinct endpoints.

(C) It is a part of a line.

(D) It extends infinitely in both directions.

Question 6. If you are given the diameter of a circle, which measurement do you NOT set the compass to for construction?

(A) The radius.

(B) Half the diameter.

(C) Twice the radius.

(D) The diameter itself.

Question 7. Which of the following properties does NOT define a unique circle?

(A) Centre and radius.

(B) Centre and one point on the circle.

(C) Any three non-collinear points.

(D) Circumference length alone.

Question 8. When constructing a line segment of a specific length using a ruler, which action is NOT recommended for accuracy?

(A) Using a sharp pencil tip.

(B) Viewing the scale perpendicularly.

(C) Starting the measurement from the '0' mark or another clear marking.

(D) Starting the measurement from the worn edge of the ruler.

Question 9. Which of the following is NOT a valid step when copying a line segment AB onto a line L starting at point P using a compass and ruler?

(A) Setting the compass radius to the length of AB.

(B) Placing the compass point at P and drawing an arc intersecting L.

(C) Drawing a line segment of arbitrary length from P on L.

(D) Measuring the angle of the segment AB with a protractor.

Question 10. Which statement is NOT true about a circle with radius R and center O?

(A) Every point on the circle is a distance R from O.

(B) A chord is a line segment connecting two points on the circle.

(C) The diameter is the shortest chord passing through the center.

(D) A radius is a segment from O to a point on the circle.

Question 11. Which tool is NOT used in the basic compass and ruler construction of a line segment of a specific length?

(A) Compass

(B) Ruler (with markings)

(C) Pencil

(D) Paper

Question 12. Copying a line segment using a compass and ruler does NOT guarantee preserving which property relative to the original segment?

(A) Length

(B) Orientation

(C) Position

(D) Distance between endpoints

Question 1. Which of the following angles cannot be constructed using only compass and ruler by combining or bisecting standard angles ($60^\circ, 90^\circ$)?

(A) $15^\circ$

(B) $75^\circ$

(C) $80^\circ$

(D) $105^\circ$

Question 2. When constructing a $60^\circ$ angle using a compass and ruler, which step is NOT necessary?

(A) Drawing a ray.

(B) Drawing an arc from the vertex intersecting the ray.

(C) Drawing a second arc from the intersection point with the same radius.

(D) Using a protractor to check the angle.

Question 3. Which statement is NOT true about bisecting an angle $\angle ABC$?

(A) You draw an arc from B intersecting BA and BC.

(B) You draw arcs of the same radius from the intersection points on BA and BC.

(C) The radius used for the final arcs must be less than half the distance between the intersection points on BA and BC.

(D) The ray connecting B to the intersection of the final arcs is the bisector.

Question 4. Which property is NOT used in the standard justification of the angle bisector construction?

(A) SSS congruence criterion.

(B) The property that points on the bisector are equidistant from the arms.

(C) Definition of an angle bisector.

(D) Angle sum property of a triangle.

Question 5. Which of the following angles is NOT obtained by simply bisecting constructible angles ($60^\circ, 90^\circ$, etc.) or combining them?

(A) $7.5^\circ$ ($60/8$ or $15/2$)

(B) $10^\circ$

(C) $15^\circ$ ($60/4$ or $30/2$)

(D) $22.5^\circ$ ($90/4$)

Question 6. Which step is NOT correct when constructing a $120^\circ$ angle using a compass and ruler?

(A) Draw a ray OA.

(B) Draw an arc from O intersecting OA at B.

(C) With B as center and a *different* radius, draw an arc intersecting the first arc at C.

(D) Join O to C.

Question 7. A point P is on the angle bisector of $\angle XYZ$. Which statement is NOT necessarily true?

(A) $\angle XYP = \angle ZYP$

(B) P is equidistant from ray YX and ray YZ.

(C) P is equidistant from point X and point Z.

(D) The distance from P to YX is equal to the distance from P to YZ.

Question 8. Which angle is NOT obtained by bisecting a straight angle ($180^\circ$) or bisecting the resulting angles?

(A) $90^\circ$

(B) $45^\circ$

(C) $22.5^\circ$

(D) $60^\circ$

Question 9. To construct a $90^\circ$ angle at a point P on a line, which method does NOT use compass and ruler in a standard way?

(A) Constructing the perpendicular bisector of a segment centered at P.

(B) Drawing arcs from P, then from intersections, to find a point equidistant from the intersections.

(C) Using a set square to draw a perpendicular line at P.

(D) Bisecting a $180^\circ$ angle at P.

Question 10. Which angle combination does NOT result in a constructible angle using standard compass and ruler methods?

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

(B) $90^\circ - 30^\circ$

(C) $180^\circ - 15^\circ$

(D) $90^\circ + 10^\circ$

Question 1. When constructing a perpendicular to a line at a point P on the line, which step is NOT correct?

(A) Draw arcs of the same radius from P intersecting the line at A and B.

(B) Draw arcs from A and B with radii less than AP, intersecting at Q.

(C) Draw arcs from A and B with radii greater than AP, intersecting at Q.

(D) Join P to Q.

Question 2. Which statement is NOT true about the perpendicular bisector of a line segment AB?

(A) It is perpendicular to AB.

(B) It passes through the midpoint of AB.

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

(D) It is parallel to AB.

Question 3. Which tool is NOT typically used in the standard compass and ruler construction of a perpendicular bisector?

(A) Compass

(B) Ruler (straight edge)

(C) Protractor

(D) Pencil

Question 4. When constructing a perpendicular from a point P outside a line, which step is NOT part of the standard method?

(A) Draw an arc from P intersecting the line at two points.

(B) Draw arcs from the intersection points with a radius equal to the distance between them.

(C) Draw arcs from the intersection points with equal radii on the opposite side of the line from P.

(D) Join P to the intersection of the arcs.

Question 5. Which property is NOT used in the standard justification for constructing a perpendicular bisector?

(A) The property that points on the bisector are equidistant from the endpoints.

(B) Triangle congruence criteria (like SSS or SAS).

(C) The definition of a midpoint.

(D) The angle in a semicircle is $90^\circ$.

Question 6. How many lines perpendicular to a given line can NOT be drawn through a point not on the line?

(A) Zero

(B) One

(C) Two

(D) Infinite

Question 7. Which of the following is NOT equivalent to constructing a $90^\circ$ angle at a point on a line?

(A) Constructing a perpendicular to the line at that point.

(B) Bisecting a straight angle ($180^\circ$) at that point.

(C) Applying the perpendicular bisector construction to a segment centered at that point.

(D) Combining a $60^\circ$ angle and a $30^\circ$ angle such that they form a $90^\circ$ angle.

Question 8. When constructing the perpendicular bisector of segment AB, why is using a radius *less than or equal to* half of AB NOT a valid step?

(A) The arcs would not intersect away from the segment, preventing the finding of the two necessary points for the bisector line.

(B) The resulting line would not be perpendicular.

(C) The resulting line would not pass through the midpoint.

(D) The construction would require a protractor.

Question 9. Which of the following is NOT a direct application of constructing perpendiculars or perpendicular bisectors?

(A) Constructing an altitude of a triangle.

(B) Finding the circumcenter of a triangle.

(C) Constructing a square (involving $90^\circ$ angles).

(D) Dividing a line segment into a given ratio internally.

Question 10. In the construction of a perpendicular from a point P 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 is NOT involved in forming these congruent triangles?

(A) The segment PA (from P to one intersection point).

(B) The segment PB (from P to the other intersection point).

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

(D) The segment from P to a point *not* on the line AB.

Question 1. Which of the following angle relationships does NOT guarantee that two lines are parallel when intersected by a transversal?

(A) Corresponding angles are equal.

(B) Alternate interior angles are equal.

(C) Consecutive interior angles are supplementary.

(D) Vertically opposite angles are equal.

Question 2. When constructing a line parallel to a given line through an external point using angle copying, which tool is NOT typically used in the standard compass and ruler method?

(A) Compass

(B) Ruler/Straight edge

(C) Protractor

(D) Pencil

Question 3. Which statement is NOT true about constructing a line parallel to line 'l' through point P not on 'l' using the corresponding angles method?

(A) A transversal is drawn through P intersecting 'l'.

(B) A corresponding angle at the intersection on 'l' is copied.

(C) The copied angle is constructed at P on the alternate interior side of the transversal.

(D) The line through P forming this angle is the parallel line.

Question 4. The construction of parallel lines relies on the converses of certain angle theorems/postulates. Which converse is NOT directly used in the standard methods?

(A) If corresponding angles are equal, lines are parallel.

(B) If alternate interior angles are equal, lines are parallel.

(C) If consecutive interior angles are supplementary, lines are parallel.

(D) If vertically opposite angles are equal, lines are parallel.

Question 5. Which statement is NOT true about parallel lines in Euclidean geometry?

(A) They lie in the same plane.

(B) They never intersect.

(C) The distance between them is constant.

(D) They always have different slopes.

Question 6. When copying an angle using compass and ruler, which measurement is NOT directly involved?

(A) The radius of the initial arc from the vertex.

(B) The straight-line distance between the points where the initial arc intersects the arms.

(C) The angle measure in degrees.

(D) The radius of the arc drawn from the new vertex.

Question 7. Which is NOT a valid initial step in constructing a line parallel to a given line 'l' through an external point P?

(A) Draw any line through P that intersects 'l'.

(B) Draw a perpendicular from P to 'l', intersecting at Q, then draw a perpendicular to PQ at P.

(C) Draw a circle centered at P that intersects 'l'.

(D) Draw a line through P that is parallel to 'l' by estimation.

Question 8. How many lines parallel to a given line can NOT be drawn through a point *on* the line?

(A) Zero (since the line itself is parallel to itself)

(B) One

(C) Two

(D) Infinite

Question 9. Which statement about the transversal used in parallel line construction is NOT true?

(A) It must intersect the given line.

(B) It must pass through the external point.

(C) It creates corresponding and alternate interior angles.

(D) It must be perpendicular to the given line.

Question 10. Which is NOT a reason why copying an angle using compass and ruler is preferred over using a protractor in formal geometric constructions?

(A) Compass and ruler methods are considered more fundamental (Euclidean tools).

(B) Compass and ruler methods are less prone to measurement errors compared to reading a protractor scale.

(C) Compass and ruler methods can construct angles that a standard protractor cannot measure precisely (e.g., $22.5^\circ$).

(D) Using a protractor is a significantly faster method.

Question 1. When dividing a line segment AB internally in the ratio $m:n$ using the standard construction method with a ray AC, which step is NOT correct?

(A) Draw a ray AC from A making an angle with AB.

(B) Mark $m+n$ equal parts on ray AC.

(C) Join the m-th point on AC to B.

(D) Draw a line through the m-th point on AC parallel to the line joining the (m+n)-th point to B.

Question 2. Which theorem is NOT directly used in the standard justification for dividing a line segment in a given ratio (internally)?

(A) Basic Proportionality Theorem (BPT).

(B) Properties of parallel lines cutting transversals proportionally.

(C) Similarity of triangles.

(D) Pythagoras Theorem.

Question 3. When dividing a line segment AB in the ratio $3:2$, which statement is NOT true about the standard construction?

(A) You mark 5 equal parts on the auxiliary ray.

(B) You connect the 5th point on the ray to B.

(C) You draw a line through the 2nd point on the ray parallel to the line connecting the 5th point to B.

(D) You draw a line through the 3rd point on the ray parallel to the line connecting the 5th point to B.

Question 4. Which of the following is NOT true about the internal division of a line segment AB at point P in the ratio $m:n$?

(A) P lies on the segment AB.

(B) $\frac{AP}{PB} = \frac{m}{n}$.

(C) AP + PB = AB.

(D) P lies outside the segment AB.

Question 5. Which tool is NOT essential for marking equal distances on the auxiliary ray in the line segment division construction?

(A) Compass

(B) Ruler (with markings)

(C) Pencil

(D) Paper

Question 6. Which statement is NOT true about the auxiliary ray AC used in the line segment division construction of AB in ratio $m:n$?

(A) It originates from point A.

(B) It makes an angle with AB.

(C) It must be perpendicular to AB.

(D) Points are marked on it using equal compass openings.

Question 7. If you are dividing a line segment into 'n' equal parts, which statement is NOT true about using the ratio division method?

(A) This is a specific case of dividing in the ratio $1:1:\dots:1$ (n times).

(B) You mark 'n' equal parts on the auxiliary ray.

(C) You connect the n-th point on the ray to one end of the segment.

(D) You draw parallel lines through all the marked points ($A_1, A_2, \dots, A_{n-1}$) to divide the segment.

Question 8. Which of the following is NOT a required ability for performing the standard line segment division construction?

(A) Drawing a line segment of a given length.

(B) Copying a line segment.

(C) Constructing parallel lines.

(D) Constructing perpendicular bisectors.

Question 9. In the justification for dividing AB in ratio $m:n$, the BPT is applied to $\triangle ABA_{m+n}$ and line $PA_m || BA_{m+n}$. Which ratio is NOT stated to be equal by BPT?

(A) $\frac{AP}{PB}$

(B) $\frac{AA_m}{A_m A_{m+n}}$

(C) $\frac{AB}{AP}$

(D) $\frac{BA_{m+n}}{PA_m}$ (ratio of parallel sides)

Question 10. Which statement is NOT true about the point P that divides segment AB internally in the ratio $m:n$?

(A) It is closer to A if m < n.

(B) It is closer to B if m > n.

(C) If m = n, P is the midpoint.

(D) If m = 0, P is the midpoint.

Question 1. Which of the following combinations of measurements does NOT uniquely determine a triangle?

(A) Three sides (SSS).

(B) Two sides and the included angle (SAS).

(C) Two angles and the included side (ASA).

(D) Three angles (AAA).

Question 2. When constructing a triangle using the SSS criterion with side lengths a, b, c, which condition is NOT necessary for a triangle to be possible?

(A) $a + b > c$

(B) $a + c > b$

(C) $b + c > a$

(D) $a + b = c$

Question 3. Which statement is NOT true about constructing a triangle using the SAS criterion (two sides and the included angle)?

(A) You draw one of the given sides as the base.

(B) You construct the given angle at one endpoint of the base.

(C) You mark the length of the second side along the arm of the constructed angle.

(D) The given angle is opposite to one of the given sides.

Question 4. When constructing a triangle using the ASA criterion (two angles and the included side), which statement is NOT correct?

(A) You draw the included side as the base.

(B) You construct the two given angles at the endpoints of the base.

(C) The sum of the two given angles must be less than $180^\circ$.

(D) The third vertex is found by bisecting the included side.

Question 5. Which tool is NOT used in the standard compass and ruler construction of basic triangles (SSS, SAS, ASA)?

(A) Compass

(B) Ruler (with markings)

(C) Protractor

(D) Pencil

Question 6. Which statement is NOT true about constructing a triangle given two angles and a non-included side (AAS)?

(A) You can find the third angle using the angle sum property.

(B) The problem can be converted into an ASA case.

(C) The given side is included between the two given angles.

(D) The construction is uniquely determined by the given information.

Question 7. Which of the following is NOT a valid set of side lengths for a triangle?

(A) $\{5 \text{ cm, } 5 \text{ cm, } 5 \text{ cm}\}$

(B) $\{3 \text{ cm, } 4 \text{ cm, } 5 \text{ cm}\}$

(C) $\{2 \text{ cm, } 2 \text{ cm, } 4 \text{ cm}\}$

(D) $\{6 \text{ cm, } 8 \text{ cm, } 10 \text{ cm}\}$

Question 8. When constructing a triangle using the SSS criterion, if the arcs from the two endpoints of the base with the given radii do not intersect, which statement is NOT true?

(A) The Triangle Inequality Theorem is violated for the given side lengths.

(B) The sum of the lengths of the two sides is less than or equal to the length of the base.

(C) A triangle with the given side lengths cannot be formed.

(D) The construction method itself is incorrect.

Question 9. Which is NOT a congruence criterion for triangles that guarantees unique construction?

(A) SSS

(B) SSA

(C) ASA

(D) RHS

Question 10. When constructing a triangle using the ASA criterion, the third vertex is found by the intersection of the arms of the two angles. Which statement is NOT true about this intersection?

(A) It must occur for a valid triangle to be formed.

(B) It lies on the same side of the base as the angles were constructed.

(C) It is the midpoint of the included side.

(D) Its distance from the base depends on the angles and the base length.

Question 1. Which statement is NOT true about the construction of an equilateral triangle with side length 's'?

(A) You can use the SSS criterion with sides s, s, s.

(B) You can use the ASA criterion with side s and $60^\circ$ angles at the endpoints.

(C) Drawing arcs of radius 's' from the endpoints of the base 's' is a valid method.

(D) You construct $90^\circ$ angles at the base endpoints and mark length 's' on the arms.

Question 2. Which property is NOT always true for an isosceles triangle?

(A) It has at least two equal sides.

(B) The angles opposite the equal sides are equal.

(C) All three angles are equal to $60^\circ$.

(D) The median to the base is also the altitude and angle bisector.

Question 3. When constructing a right-angled triangle using the RHS criterion (hypotenuse H, leg L), which condition is NOT necessary?

(A) One angle is $90^\circ$.

(B) The hypotenuse is opposite the $90^\circ$ angle.

(C) $H > L$.

(D) The other leg's length is given.

Question 4. Which statement is NOT true about the justification of constructing an equilateral triangle by drawing arcs of equal radius from the base endpoints?

(A) It relies on the definition of an equilateral triangle (all sides equal).

(B) The construction method creates three sides of equal length.

(C) The justification uses the angle sum property of a triangle to show all angles are $60^\circ$ without using SSS.

(D) It can be justified using SSS congruence if auxiliary lines are drawn.

Question 5. Which tool is NOT essential for constructing a specific basic triangle (like equilateral, isosceles given sides, right-angled given legs) if the side lengths are provided as segments?

(A) Compass

(B) Ruler (straight edge)

(C) Protractor

(D) Pencil

Question 6. When constructing an isosceles triangle given the base and base angles, which step is NOT correct?

(A) Draw the base.

(B) Construct the given angles at the endpoints of the base.

(C) Ensure the two constructed angles are equal.

(D) Ensure the sum of the two base angles is $180^\circ$.

Question 7. Which statement is NOT true about the RHS criterion for constructing a right-angled triangle?

(A) It requires a right angle.

(B) It requires the length of the hypotenuse.

(C) It requires the length of one leg.

(D) It requires the measures of the two acute angles.

Question 8. Which type of triangle is NOT a specific basic triangle mentioned for construction?

(A) Equilateral triangle.

(B) Isosceles triangle.

(C) Right-angled triangle.

(D) Obtuse-angled triangle (constructed under general cases).

Question 9. Which statement is NOT true about an isosceles triangle with base angles equal to $60^\circ$?

(A) It is also an equilateral triangle.

(B) All its angles are $60^\circ$.

(C) Its sides opposite the $60^\circ$ angles are equal.

(D) Its vertex angle is obtuse.

Question 10. When constructing an isosceles triangle given the base and vertex angle, which step is NOT necessary for the standard method?

(A) Drawing the base.

(B) Calculating the base angles.

(C) Bisecting the base.

(D) Constructing the calculated base angles at the endpoints of the base.

Question 1. When constructing a triangle given one side, one angle, and the sum of the other two sides (BC, $\angle B$, AB+AC), which step is NOT part of the standard method?

(A) Draw BC and construct $\angle B$.

(B) Mark a point D on the ray of $\angle B$ such that BD = AB+AC.

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

(D) Construct the perpendicular bisector of CD.

Question 2. When constructing a triangle given one side, one angle, and the difference of the other two sides (BC, $\angle B$, |AB-AC|), which step is NOT part of the standard method (assuming AB > AC)?

(A) Draw BC and construct $\angle B$.

(B) Mark a point D on the ray of $\angle B$ such that BD = AB-AC.

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

(D) Construct the perpendicular bisector of CD.

Question 3. When constructing a triangle given two angles and the perimeter ($\angle B, \angle C$, Perimeter), which step is NOT part of the standard method?

(A) Draw a segment equal to the perimeter.

(B) Construct angles equal to $\angle B$ and $\angle C$ at the ends of the segment.

(C) Construct angles equal to $(1/2)\angle B$ and $(1/2)\angle C$ at the ends of the segment.

(D) Construct perpendicular bisectors to find the vertices on the perimeter segment.

Question 4. When constructing a triangle given two sides and a median (AB, AC, median AD), which is NOT an intermediate step in the standard method involving forming a parallelogram?

(A) Extend the median AD to E such that AD = DE.

(B) Join C to E.

(C) Construct the perpendicular bisector of AB.

(D) Construct $\triangle ACE$ using SSS criterion with sides related to AB, AC, and $2 \cdot \text{AD}$.

Question 5. When constructing a triangle given two angles and an altitude ($\angle B, \angle C$, altitude AD), which step is NOT part of the standard method where D is on the base line?

(A) Draw a line and mark point D.

(B) Construct a perpendicular at D and mark A at the altitude length.

(C) Construct angles $\angle B$ and $\angle C$ at point D on the line.

(D) Construct angles at A related to $\angle B$ and $\angle C$ (e.g., $90^\circ - \angle B$) to find B and C on the line.

Question 6. Which basic construction is NOT primarily used in the advanced construction of a triangle given one side, one angle, and the sum or difference of the other two sides?

(A) Drawing a line segment.

(B) Constructing an angle.

(C) Constructing a perpendicular bisector.

(D) Constructing parallel lines.

Question 7. In the perimeter construction (given $\angle B, \angle C$, Perimeter), the justification relies on creating isosceles triangles (APB and AQC) by constructing half angles. Which statement is NOT true about these isosceles triangles?

(A) BP = BA

(B) CQ = CA

(C) $\angle APB = (1/2)\angle B$

(D) $\angle PAB = \angle PBA = (1/2)\angle B$

Question 8. When constructing a triangle given two angles ($\angle B, \angle C$) and an altitude (AD), which condition must NOT be met for the triangle to be possible?

(A) $\angle B > 0^\circ$

(B) $\angle C > 0^\circ$

(C) $\angle B + \angle C < 180^\circ$

(D) $\angle B + \angle C = 90^\circ$ (this is possible, it results in a right triangle)

Question 9. Which type of triangle construction problem is NOT discussed in the advanced cases listed?

(A) Given two sides and a median.

(B) Given two angles and an altitude.

(C) Given three altitudes.

(D) Given one side, one angle, and sum/difference of other two sides.

Question 10. In the construction of a triangle with difference of sides (|AB-AC|), if AC > AB, point D is marked on the extension of the ray from B backwards. Which statement is NOT true in this case?

(A) BD = AC - AB.

(B) The perpendicular bisector of CD is constructed.

(C) The vertex A lies on the segment BD.

(D) AC = AD due to A being on the perpendicular bisector.

Question 1. To construct a triangle similar to $\triangle ABC$ with a scale factor $m/n$, sharing vertex B and having BC' on BC, which tool is NOT typically used in the standard method?

(A) Compass

(B) Ruler (straight edge)

(C) Protractor

(D) Pencil

Question 2. When constructing a similar triangle with a scale factor $m/n$, which step is NOT correct?

(A) Draw a ray BX making an acute angle with BC.

(B) Mark $m+n$ equal parts on ray BX.

(C) Connect $B_n$ to C (if scaling down, m < n).

(D) Draw a line through $B_m$ parallel to $B_nC$ (if scaling down, m < n).

Question 3. Which theorem is NOT used in the standard justification for constructing similar triangles using the ray method?

(A) Basic Proportionality Theorem (BPT).

(B) AA Similarity Criterion.

(C) SAS Similarity Criterion (can be used if angle at B is common).

(D) Angle Bisector Theorem.

Question 4. If a similar triangle is constructed with a scale factor $k = m/n$, which statement is NOT true about the number of equal parts marked on the auxiliary ray BX?

(A) The number is at least m.

(B) The number is at least n.

(C) The number is m+n.

(D) The number is $\text{max}(m, n)$.

Question 5. When constructing a similar triangle with scale factor $k = 5/3$, which statement is NOT true?

(A) The constructed triangle is larger than the original.

(B) You mark 5 equal parts on the auxiliary ray.

(C) You connect $B_3$ to C and draw a parallel line through $B_5$.

(D) The new vertex C' lies on the segment BC.

Question 6. If a triangle $\triangle A'BC'$ is similar to $\triangle ABC$ with a scale factor $k$, which statement is NOT true?

(A) $\angle A' = \angle A$, $\angle B = \angle B$, $\angle C' = \angle C$.

(B) $\frac{A'B}{AB} = \frac{BC'}{BC} = \frac{A'C'}{AC} = k$.

(C) Ratio of perimeters = $k$.

(D) Ratio of areas = $k$.

Question 7. When constructing a similar triangle with scale factor $k < 1$ (scaling down), which statement is NOT true?

(A) The constructed triangle is smaller than the original.

(B) The new vertices lie on the interior of the sides of the original triangle (starting from the common vertex).

(C) The denominator 'n' of the scale factor $m/n$ (in simplest form) determines the total number of equal parts on the auxiliary ray.

(D) You connect $B_m$ to C and draw a parallel line through $B_n$.

Question 8. Which property is NOT used in the justification for similar triangle construction using parallel lines?

(A) Parallel lines create equal corresponding angles.

(B) Parallel lines create equal alternate interior angles.

(C) Parallel lines cut transversals proportionally.

(D) The sum of angles in a triangle is $180^\circ$.

Question 9. If a similar triangle is constructed with a scale factor $k$, which statement is NOT true about the ratio of altitudes?

(A) The ratio of corresponding altitudes is k.

(B) The ratio of corresponding medians is k.

(C) The ratio of corresponding angle bisectors is k.

(D) The ratio of corresponding angles is k.

Question 10. When constructing similar triangles using a common vertex and a ray with marked divisions, the ray BX should make an acute angle with BC. Which statement is NOT a reason for this?

(A) It makes the construction visually clearer.

(B) It ensures that the points marked on BX are distinct from the segment BC.

(C) An obtuse angle would make the BPT inapplicable.

(D) It is a standard practice to avoid complex diagrams.

Question 1. Which of the following combinations of measurements does NOT uniquely determine a general quadrilateral?

(A) Four sides and one diagonal.

(B) Three sides and two included angles.

(C) Two adjacent sides and three angles.

(D) Four angles.

Question 2. When constructing a general quadrilateral given four sides and a diagonal, say AB, BC, CD, DA, and diagonal AC, which step is NOT part of the standard method?

(A) Construct triangle ABC using SSS criterion.

(B) Construct triangle ADC using SSS criterion.

(C) Construct triangle ABC using SAS criterion.

(D) Ensure vertices B and D are on opposite sides of the diagonal AC.

Question 3. Which property is NOT always true for a parallelogram?

(A) Opposite sides are equal.

(B) Opposite angles are equal.

(C) Diagonals are equal.

(D) Diagonals bisect each other.

Question 4. When constructing a rectangle given its length and width, which construction ability is NOT required?

(A) Drawing a line segment of a given length.

(B) Constructing a $90^\circ$ angle.

(C) Constructing an angle bisector.

(D) Marking a point at a specific distance on a line.

Question 5. Which statement is NOT true about constructing a rhombus given the lengths of its two diagonals?

(A) The diagonals bisect each other at right angles.

(B) You draw one diagonal and construct its perpendicular bisector.

(C) You mark the full length of the second diagonal on the perpendicular bisector from the midpoint.

(D) The endpoints of the diagonals are the vertices of the rhombus.

Question 6. Which statement is NOT true about constructing a square given its side length 's'?

(A) You can use the property that all sides are 's' and all angles are $90^\circ$.

(B) You can construct a rhombus with side 's' and one angle $90^\circ$.

(C) You can construct a rectangle with length and width 's'.

(D) You only need to know the perimeter to construct a unique square.

Question 7. Which combination of measurements is NOT sufficient to uniquely construct a parallelogram?

(A) Two adjacent sides and the included angle.

(B) Two adjacent sides and a diagonal.

(C) Both diagonals and the angle between them.

(D) All four side lengths.

Question 8. Which statement is NOT true about dividing a quadrilateral into triangles for construction?

(A) A diagonal divides a quadrilateral into two triangles.

(B) If you can construct the triangles, you can construct the quadrilateral.

(C) The sum of the angles in a quadrilateral is equal to the sum of the angles in the two triangles it is divided into.

(D) The diagonal is always the shortest line segment connecting opposite vertices.

Question 9. Which property is NOT true for a square?

(A) All sides are equal.

(B) All angles are $90^\circ$.

(C) Diagonals are perpendicular.

(D) Diagonals are unequal.

Question 10. Which of the following quadrilaterals does NOT always have diagonals that bisect each other?

(A) Parallelogram

(B) Rectangle

(C) Rhombus

(D) Trapezium

Question 1. To construct a tangent to a circle at a point P on the circle (center O), which step is NOT correct?

(A) Draw the radius OP.

(B) Construct a line parallel to OP through P.

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

(D) Extend OP beyond P and construct a $90^\circ$ angle at P.

Question 2. When constructing tangents to a circle from an external point P (center O), which step is NOT part of the standard method?

(A) Join O to P.

(B) Find the midpoint of OP.

(C) Draw a circle with diameter OP.

(D) Draw a line parallel to OP through the center O.

Question 3. Which property is NOT used in the justification of constructing tangents from an external point P to a circle?

(A) The angle in a semicircle is $90^\circ$.

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

(C) The lengths of tangents from an external point are equal.

(D) Alternate interior angles are equal.

Question 4. How many tangents can NOT be drawn to a circle from a point *inside* the circle?

(A) Zero (you cannot draw any tangents)

(B) One

(C) Two

(D) Infinite

Question 5. Which statement is NOT true about the tangents drawn from an external point to a circle?

(A) The lengths of the tangent segments are equal.

(B) The line segment from the center to the external point bisects the angle between the tangents.

(C) The line segment from the center to the external point bisects the chord joining the points of contact at right angles.

(D) The angle between the two tangents is always $90^\circ$.

Question 6. When constructing tangents to a circle from an external point P, which circle is NOT involved in the standard construction method?

(A) The original circle.

(B) A circle centered at O with radius OP.

(C) A circle centered at M (midpoint of OP) with radius OM.

(D) A circle with diameter OP.

Question 7. If you are asked to construct tangents from an external point P to a circle without being given the center, which action is NOT a correct first step?

(A) Finding the center of the circle (e.g., using perpendicular bisectors of chords).

(B) Drawing a secant through P that intersects the circle at two points.

(C) Drawing a line through P and trying to construct a perpendicular from the estimated center.

(D) Drawing a line through P and trying to make it touch the circle at just one point by estimation.

Question 8. Which statement is NOT true about a tangent to a circle at a point P on the circle?

(A) It is perpendicular to the radius OP.

(B) It is the only line that passes through P and intersects the circle only at P.

(C) It passes through the center O.

(D) It forms a $90^\circ$ angle with the radius at P.

Question 9. Which of the following is NOT a method to construct a tangent to a circle?

(A) Constructing a line perpendicular to the radius at the point of contact.

(B) From an external point, drawing a line that forms an acute angle with the line joining the external point to the center.

(C) Using the property that the angle in a semicircle is $90^\circ$ for tangents from an external point.

(D) Using the property that the angle between a tangent and a chord is equal to the angle in the alternate segment (if center is not given).

Question 10. If two tangents from an external point meet at an angle $\theta$, which statement is NOT true about the quadrilateral formed by the center, the external point, and the points of contact?

(A) Two angles are $90^\circ$ (at points of contact).

(B) The angle at the external point is $\theta$.

(C) The angle at the center is $\theta$.

(D) The sum of opposite angles is $180^\circ$.

Question 1. Which of the following is NOT a primary role of justification in geometric constructions?

(A) To prove the mathematical correctness of the construction.

(B) To explain the steps of the construction procedure.

(C) To demonstrate understanding of underlying geometric principles.

(D) To show that the construction is accurately drawn in practice.

Question 2. Which of the following is NOT a type of statement used in formal geometric justification?

(A) Definitions.

(B) Axioms or postulates.

(C) Previously proven theorems.

(D) Opinions or beliefs about the figure.

Question 3. Which congruence criterion is NOT typically the primary one used to justify the angle bisector construction?

(A) SSS

(B) SAS

(C) ASA

(D) RHS

Question 4. Which property is NOT directly used in the standard justification of the perpendicular bisector construction?

(A) Any point on the bisector is equidistant from the endpoints.

(B) Triangle congruence (e.g., SSS or SAS).

(C) The fact that it passes through the midpoint and is perpendicular.

(D) Properties of similar triangles.

Question 5. Which of the following is NOT a converse theorem used in justifying parallel line constructions?

(A) If corresponding angles are equal, lines are parallel.

(B) If alternate interior angles are equal, lines are parallel.

(C) If vertically opposite angles are equal, lines are parallel.

(D) If consecutive interior angles are supplementary, lines are parallel.

Question 6. Which theorem is NOT directly used in the standard justification for dividing a line segment in a given ratio?

(A) Basic Proportionality Theorem (BPT).

(B) Properties of parallel lines cutting transversals proportionally.

(C) Triangle similarity criteria.

(D) Pythagorean Theorem.

Question 7. Which statement is NOT true about using measurement to verify a construction?

(A) It can help check for drawing errors.

(B) It provides numerical evidence of the result.

(C) It constitutes a formal geometric proof.

(D) It indicates whether the physical construction was performed accurately.

Question 8. Which of the following is NOT a characteristic of a formal geometric proof?

(A) It is based on definitions, axioms, postulates, and proven theorems.

(B) It uses logical deduction.

(C) Each step is supported by a valid reason.

(D) It relies on precise measurements of the figure.

Question 9. Which geometric principle is NOT a fundamental building block used in proving theorems and justifying constructions?

(A) Undefined terms (like point, line, plane).

(B) Definitions.

(C) Conjectures (statements believed to be true but not yet proven).

(D) Axioms or Postulates.

Question 10. Which statement is NOT true about the role of justification in learning geometry?

(A) It helps understand why geometric methods work.

(B) It develops logical thinking skills.

(C) It is less important than just being able to follow the steps.

(D) It connects the practical steps of construction to the theoretical basis of geometry.