Matching Items MCQs for Sub-Topics of Topic 5: Construction

(i) Compass

(ii) Ruler

(iii) Line Segment

(iv) Radius

(v) Diameter

(vi) Circle

(a) A tool used to draw curves and transfer distances.

(b) A line segment passing through the center of a circle with endpoints on the circle.

(c) A set of points in a plane equidistant from a fixed point.

(d) A line segment with two distinct endpoints.

(e) A tool used to draw straight lines and measure length.

(f) A line segment from the center of a circle to any point on its circumference.

(i) Constructing a circle of given radius

(ii) Constructing a line segment of given length

(iii) Copying a line segment

(iv) Drawing a straight line

(v) Marking equal distances on a ray

(a) Ruler (with markings)

(b) Compass and Ruler (straight edge)

(c) Compass

(d) Ruler (straight edge) and Pencil

(e) Compass and Ruler (with markings)

(i) All radii of a circle are equal.

(ii) The diameter of a circle is twice its radius.

(iii) A circle has only one diameter.

(iv) A point inside a circle is closer to the center than any point on the circle.

(v) A chord passing through the center is the longest chord.

(vi) Two circles with the same radius are congruent.

(a) True

(b) False

(i) Has position but no dimension

(ii) A straight path extending infinitely in one direction

(iii) A straight path extending infinitely in both directions

(iv) A part of a line with a definite length

(v) Formed by two rays with a common endpoint

(vi) A flat surface extending infinitely

(a) Line

(b) Ray

(c) Point

(d) Angle

(e) Line Segment

(f) Plane

(i) Draw line L longer than AB.

(ii) Set compass to length AB.

(iii) Place compass point at P.

(iv) Draw arc intersecting L at Q.

(v) Join P and Q.

(a) To create the copied segment.

(b) To transfer the length of AB.

(c) To provide space for the copied segment.

(d) To establish the starting point of the copy.

(e) To mark the endpoint of the copied segment on L.

(i) Has definite length

(ii) Has one endpoint

(iii) Has no endpoints

(iv) All points are equidistant from a center

(v) Defined by two points

(a) Line

(b) Ray

(c) Line Segment

(d) Circle

(e) Line Segment

(i) $60^\circ$

(ii) $90^\circ$

(iii) Angle Bisector

(iv) $120^\circ$

(v) $30^\circ$

(vi) $45^\circ$

(a) Bisecting a $60^\circ$ angle.

(b) Drawing an arc from the vertex intersecting both arms.

(c) Constructing a $90^\circ$ angle and bisecting it.

(d) Constructing a $60^\circ$ angle and adding another adjacent $60^\circ$.

(e) Drawing an arc from the vertex and then another arc from the intersection on the ray with the same radius.

(f) Constructing perpendicular at a point on a line.

(i) $15^\circ$

(ii) $75^\circ$

(iii) $105^\circ$

(iv) $150^\circ$

(v) $22.5^\circ$

(vi) $135^\circ$

(a) $60^\circ + 45^\circ$ or $90^\circ + 15^\circ$.

(b) Bisecting a $45^\circ$ angle.

(c) $90^\circ + 45^\circ$ or $180^\circ - 45^\circ$.

(d) Bisecting a $30^\circ$ angle.

(e) $60^\circ + 15^\circ$ or $90^\circ - 15^\circ$.

(f) $180^\circ - 30^\circ$ or $90^\circ + 60^\circ$.

(i) Definition of angle bisector

(ii) Property of points on angle bisector

(iii) Justification of angle bisector construction

(iv) Result of bisecting a $180^\circ$ angle

(v) Result of bisecting a $60^\circ$ angle

(a) Creates two $90^\circ$ angles.

(b) Any point on the bisector is equidistant from the two arms of the angle.

(c) A ray that divides an angle into two equal angles.

(d) Creates two $30^\circ$ angles.

(e) Relies on triangle congruence (often SSS or ASA).

(i) $60^\circ$ bisected

(ii) $90^\circ$ bisected

(iii) $30^\circ$ bisected

(iv) $45^\circ$ bisected

(v) $60^\circ$ and $90^\circ$ combined and angle between them bisected

(a) $45^\circ$

(b) $15^\circ$

(c) $30^\circ$

(d) $75^\circ$ ($60 + 15$) or $105^\circ$ ($90 + 15$) depending on how combined

(e) $22.5^\circ$

(i) $0^\circ$

(ii) $45^\circ$

(iii) $90^\circ$

(iv) $135^\circ$

(v) $180^\circ$

(vi) $270^\circ$

(a) Right Angle

(b) Reflex Angle

(c) Acute Angle

(d) Straight Angle

(e) Zero Angle

(f) Obtuse Angle

(i) Equidistance from arms

(ii) Angles of an equilateral triangle

(iii) Angle in a semicircle

(iv) Angles on a straight line

(v) Angles opposite equal sides in isosceles triangle

(a) Constructing Angle Bisector

(b) Justifying Tangent from External Point

(c) Justifying Perpendicular at a Point

(d) Justifying Construction of Isosceles Triangle (given base angles)

(e) Justifying Construction of $60^\circ$ angle

(i) Perpendicular to a line from a point on the line

(ii) Perpendicular to a line from a point outside the line

(iii) Perpendicular Bisector of a line segment

(iv) Perpendicular from a vertex of a triangle to the opposite side

(v) Line forming a $90^\circ$ angle with another line

(a) To find the shortest distance from a point to a line.

(b) To divide a segment into two equal parts and form a right angle.

(c) To construct an altitude of the triangle.

(d) To construct a right angle at a specific point.

(e) Definition of a perpendicular line.

(i) Draw arcs from A and B with radius > AB/2

(ii) Join the two intersection points of the arcs

(iii) The line segment connecting the two intersection points and AB

(iv) The intersection point of the perpendicular bisector and AB

(v) Radius used in initial arcs

(a) Forms a rhombus with A and B.

(b) Ensures the arcs intersect.

(c) Locates the midpoint of AB.

(d) Creates the perpendicular bisector line.

(e) Forms four right triangles.

(i) Angle in a semicircle

(ii) SSS or SAS congruence

(iii) Bisecting a straight angle

(iv) Shortest distance from point to line

(v) Equidistance from endpoints of a segment

(a) Perpendicular from a point outside the line.

(b) Perpendicular at a point on the line (as a $90^\circ$ angle construction).

(c) Perpendicular Bisector.

(d) Justification of tangent from external point construction.

(e) Proving congruence of triangles in perpendicular constructions.

(i) Passes through midpoint only

(ii) Forms $90^\circ$ angle only

(iii) Passes through midpoint and forms $90^\circ$ angle

(iv) Passes through vertex and is perpendicular to opposite side

(v) Passes through vertex and midpoint of opposite side

(a) Altitude of a triangle

(b) Perpendicular line (not necessarily bisector)

(c) Median of a triangle

(d) Perpendicular Bisector of a segment

(e) Median of a segment

(i) Compass

(ii) Ruler/Straight edge

(iii) Pencil

(iv) Protractor

(v) Set Square

(a) Drawing straight lines connecting points.

(b) Marking points and drawing lines/arcs.

(c) Measuring or verifying a $90^\circ$ angle (not standard construction).

(d) Drawing arcs for intersection points.

(e) Can be used to draw perpendicular lines directly (not standard compass/ruler construction).

(i) Arc from P intersecting line at A, B

(ii) Arcs from A and B intersecting at Q

(iii) Line PQ

(iv) Segments PA and PB

(v) Segments QA and QB

(a) Radii of equal arcs.

(b) Points equidistant from P on the line.

(c) Points equidistant from P and equidistant from A and B.

(d) The constructed perpendicular line.

(e) Equal length radii of the initial arc.

(i) Corresponding Angles

(ii) Alternate Interior Angles

(iii) Consecutive Interior Angles

(iv) Vertically Opposite Angles

(v) Alternate Exterior Angles

(vi) Adjacent Angles on a line

(a) On opposite sides of the transversal, between the two lines.

(b) On the same side of the transversal, between the two lines.

(c) Angles sharing a vertex and a common arm, summing to $180^\circ$.

(d) Angles in the same relative position at each intersection.

(e) Angles opposite each other at an intersection, always equal.

(f) On opposite sides of the transversal, outside the two lines.

(i) Corresponding angles are equal

(ii) Alternate interior angles are equal

(iii) Consecutive interior angles are supplementary

(iv) Lines are in the same plane and never intersect

(v) Two lines perpendicular to the same line

(a) Definition of parallel lines.

(b) Converse of Alternate Interior Angle Theorem.

(c) A property used for parallel line construction.

(d) Converse of Corresponding Angles Postulate.

(e) Converse of Consecutive Interior Angle Theorem.

(i) Draw transversal PQ intersecting 'l' at Q

(ii) Draw arc centered at Q intersecting 'l' and PQ

(iii) Draw arc centered at P with same radius

(iv) Measure distance between intersection points on arc at Q

(v) Transfer measured distance onto arc at P

(vi) Draw line through P and the transferred point

(a) To establish the angle to be copied.

(b) To create the parallel line.

(c) To initiate the angle copying process at P.

(d) To establish a line that defines the angle and passes through P.

(e) To capture the 'width' of the angle.

(f) To define the second arm of the copied angle at P.

(i) Compass

(ii) Ruler/Straight edge

(iii) Protractor

(iv) Pencil

(v) Set Squares

(a) Drawing straight lines.

(b) Drawing arcs and transferring distances for angle copying.

(c) Marking points and drawing lines/arcs.

(d) Can be used for practical parallel line drawing, but not standard compass/ruler construction.

(e) Measuring angles, generally not used in pure constructions.

(i) Unique parallel line through an external point

(ii) Parallel lines cut transversals proportionally

(iii) Converse of Corresponding Angles Postulate

(iv) Parallel lines are equidistant

(v) Converse of Alternate Interior Angles Theorem

(a) Justifies the method using alternate interior angles.

(b) Implies only one solution exists for the construction.

(c) Justifies the method using corresponding angles.

(d) Relevant for constructions involving dividing segments or similar figures (BPT).

(e) Can be used as a basis for constructing parallel lines (e.g., using perpendiculars).

(i) Copying corresponding angles

(ii) Copying alternate interior angles

(iii) Copying angles in general

(iv) Using perpendiculars

(v) Using transversal

(a) Involves drawing a line that intersects the given line and passes through the external point.

(b) A method for parallel line construction where the copied angle is on the same side of the transversal, outside one line and inside the other.

(c) A fundamental technique required for angle-based parallel line constructions.

(d) A method for parallel line construction where the copied angle is on opposite sides of the transversal and between the lines.

(e) An alternative method for parallel lines based on the property that lines perpendicular to the same line are parallel.

(i) Draw ray AC at acute angle to AB

(ii) Mark $m+n$ equal points on AC

(iii) Join $A_{m+n}$ to B

(iv) Draw line through $A_m$ parallel to $A_{m+n}B$

(v) Intersection point P on AB

(a) To establish a line segment whose division will, by similarity, divide AB in the same ratio.

(b) To create a reference for proportional division.

(c) To find the point that divides AB in the ratio $m:n$.

(d) To apply the Basic Proportionality Theorem.

(e) To create a ray with a specific number of equal divisions.

(i) 1:1 (Midpoint)

(ii) 1:2

(iii) 2:1

(iv) 3:2

(v) 1:4

(vi) 4:1

(a) $A_3$ (for 3:?) or $A_2$ (for ?:2); $A_5$ (total parts), draw parallel through $A_3$.

(b) $A_2$ (total parts), draw parallel through $A_1$.

(c) $A_3$ (total parts), draw parallel through $A_2$.

(d) $A_2$ (for ?:1); $A_3$ (total parts), draw parallel through $A_2$.

(e) $A_5$ (total parts), draw parallel through $A_1$.

(f) $A_5$ (total parts), draw parallel through $A_4$.

(i) Basic Proportionality Theorem

(ii) Parallel lines

(iii) Equal segments on ray AC

(iv) Similar triangles

(v) Ratio $m:n$

(a) Ensures the ray AC is divided in the ratio $m:n$ from A.

(b) Provides the fundamental theorem for proportional division by parallel lines.

(c) The desired outcome of the construction.

(d) Created by the construction, allowing application of BPT or definition of similarity.

(e) The geometric figure drawn to apply BPT.

(i) Internal division

(ii) Ratio $m:n$

(iii) Point of division P

(iv) Total number of parts on auxiliary ray

(v) Use of compass for equal segments

(a) The point on the segment that divides it.

(b) The ratio of the lengths of the two resulting segments.

(c) Ensures the segments on the auxiliary ray are truly equal in length.

(d) The point P lies between the endpoints A and B.

(e) Equal to $m+n$ for ratio $m:n$.

(i) Dividing AB into 4 equal parts

(ii) Dividing AB in ratio 1:3

(iii) Dividing AB in ratio 3:1

(iv) Finding the midpoint of AB

(v) Dividing AB in ratio 2:3

(a) Ratio 1:1, total 2 parts on ray, parallel through $A_1$.

(b) Total 4 parts on ray, parallel through $A_1$.

(c) Total 4 parts on ray, parallel through $A_3$.

(d) Total 4 parts on ray, parallel lines through $A_1, A_2, A_3$.

(e) Total 5 parts on ray, parallel through $A_2$.

(i) Ruler (straight edge)

(ii) Compass

(iii) Pencil

(iv) Ruler (markings)

(v) Protractor

(a) Drawing the initial segment AB of given length.

(b) Drawing the ray AC and the lines connecting points.

(c) Drawing arcs to mark equal segments on AC and copying angles for parallel lines.

(d) Marking points and drawing lines/arcs.

(e) Not used in the standard compass and ruler method.

(i) SSS

(ii) SAS

(iii) ASA

(iv) AAS

(v) RHS

(a) Two angles and the included side.

(b) Three sides.

(c) Two sides and the included angle.

(d) Hypotenuse and one leg of a right-angled triangle.

(e) Two angles and a non-included side.

(i) Draw BC, construct $\angle B$, mark AB on arm.

(ii) Draw BC, construct $\angle B$, construct $\angle C$.

(iii) Draw BC, draw arc from B (radius AB), draw arc from C (radius AC).

(iv) Draw a line, construct $90^\circ$ at B, mark AB, draw arc from C (hypotenuse).

(v) Draw AB, construct $\angle A$, find $\angle C$ using angle sum, construct $\angle C$.

(a) SSS

(b) SAS

(c) ASA

(d) AAS (converted to ASA)

(e) RHS

(i) 3, 4, 5

(ii) 2, 3, 5

(iii) 4, 5, 10

(iv) 6, 6, 6

(v) 1, 2, 2

(vi) 7, 8, 16

(a) Can form a triangle.

(b) Cannot form a triangle.

(c) Can form a degenerate triangle (endpoints collinear).

(i) Any Triangle

(ii) Right-angled Triangle

(iii) Equilateral Triangle

(iv) Isosceles Triangle

(v) Scalene Triangle

(a) Length of one side.

(b) Lengths of three sides.

(c) Lengths of two sides and the included angle.

(d) Lengths of hypotenuse and one leg.

(e) Lengths of base and equal sides, or base and base angles.

(i) Ruler (with markings)

(ii) Compass

(iii) Protractor

(iv) Pencil

(v) Straight edge (no markings)

(a) Marking points and drawing lines/arcs.

(b) Drawing arcs to find vertex intersections and transferring lengths.

(c) Drawing straight lines between two points.

(d) Measuring/drawing angles (used if compass/ruler construction of angle is not feasible or required).

(e) Measuring/drawing line segments of specific lengths.

(i) SSS

(ii) SAS

(iii) ASA

(iv) AAS

(v) RHS

(a) 2 (one side, one angle are fixed at $90^\circ$ implicitly or explicitly).

(b) 3

(c) 3

(d) 3

(e) 3

(i) Equilateral Triangle (given side 'a')

(ii) Isosceles Triangle (base 'b', equal sides 'a')

(iii) Isosceles Triangle (base 'b', base angles $\alpha$)

(iv) Right Triangle (hypotenuse 'c', leg 'a')

(v) Right Triangle (legs 'a', 'b')

(a) Use SSS criterion with sides b, a, a.

(b) Use ASA criterion with base b and angles $\alpha$ at endpoints.

(c) Use SSS criterion with sides a, a, a OR ASA criterion with side a and $60^\circ$ angles.

(d) Draw leg 'a', construct $90^\circ$, draw arc from other end of 'a' with radius 'c'.

(e) Draw leg 'a', construct $90^\circ$, mark leg 'b' on perpendicular arm, join endpoints.

(i) All sides are constructed equal to 'a'.

(ii) Angles opposite equal sides are equal.

(iii) Angle in a semicircle is $90^\circ$.

(iv) $a^2 + b^2 = c^2$

(v) $180^\circ - 2\alpha = \theta$ (vertex angle)

(a) Justification of Isosceles Triangle (given base and vertex angle).

(b) Justification of Equilateral Triangle (SSS method).

(c) Justification of Right Triangle (RHS method).

(d) Property of Isosceles Triangles.

(e) Justification related to $90^\circ$ angle construction.

(i) All angles are $60^\circ$.

(ii) Has exactly one right angle.

(iii) Has exactly two equal sides.

(iv) All three sides are different lengths.

(v) Sum of squares of two sides equals square of the third.

(vi) Base angles are equal.

(a) Scalene Triangle

(b) Isosceles Triangle

(c) Equilateral Triangle

(d) Right-angled Triangle

(e) Isosceles Triangle

(f) Right-angled Triangle

(i) Circumcenter

(ii) Incenter

(iii) Centroid

(iv) Orthocenter

(v) Midpoint of Hypotenuse in Right Triangle

(a) Intersection of medians.

(b) Intersection of angle bisectors.

(c) Intersection of altitudes.

(d) Equidistant from vertices, center of circumcircle.

(e) Equidistant from the three vertices.

(i) Base BC and length of equal sides AB, AC

(ii) Base BC and base angles $\angle B, \angle C$

(iii) Equal sides AB, AC and included angle $\angle A$

(iv) Base BC and altitude AD to BC

(v) Equal sides AB, AC and base BC

(a) SSS criterion

(b) ASA criterion

(c) Requires additional steps to find vertices, e.g., construct BC, perp bisector of BC, arc from B with radius AB intersecting bisector at A.

(d) SAS criterion

(e) SSS criterion

(i) Any three side lengths can form an equilateral triangle.

(ii) Any three angles can form an equilateral triangle.

(iii) An isosceles triangle with a $60^\circ$ vertex angle is equilateral.

(iv) An isosceles triangle with a $60^\circ$ base angle is equilateral.

(v) A right triangle with equal legs is an isosceles right triangle.

(a) True

(b) False

(i) Given Side, Angle, Sum of other two sides (BC, $\angle B$, AB+AC)

(ii) Given Side, Angle, Difference of other two sides (BC, $\angle B$, AB-AC)

(iii) Given Two Sides and Median (AB, AC, AD)

(iv) Given Two Angles and Perimeter ($\angle B, \angle C$, AB+BC+CA)

(v) Given Two Angles and Altitude ($\angle B, \angle C$, AD)

(a) A parallelogram formed by extending the median.

(b) An isosceles triangle formed by the difference of sides.

(c) Two right triangles formed by the altitude.

(d) An isosceles triangle formed by the sum of sides.

(e) Two isosceles triangles formed by perpendicular bisectors on the perimeter line.

(i) Construction with Sum of Sides (BC, $\angle B$, AB+AC)

(ii) Construction with Difference of Sides (BC, $\angle B$, |AB-AC|)

(iii) Finding vertex A in Sum case (using perp bisector of CD)

(iv) Finding vertex A in Difference case (using perp bisector of CD)

(v) Relationship between constructed length and side sum/difference

(a) Point A lies on the perpendicular bisector of CD, thus AC = AD.

(b) Relies on creating an isosceles triangle where one side is the sum AB+AC (or difference |AB-AC|).

(c) Relies on creating an isosceles triangle where the base angles are half the original angles.

(d) BD = AB + AC by construction, and AC=AD justifies this.

(e) BD = |AB - AC| by construction, and AC=AD justifies this.

(i) Side, Angle, Sum of sides

(ii) Side, Angle, Difference of sides

(iii) Two Angles and Perimeter

(iv) Two Angles and Altitude

(v) Two Sides and Median

(a) Constructing perpendicular bisectors and angles.

(b) Constructing perpendiculars and angles.

(c) Constructing angles and perpendicular bisectors.

(d) Constructing angles and perpendicular bisectors.

(e) Constructing a triangle using SSS (for the intermediate parallelogram triangle).

(i) Calculating base angles $\frac{180^\circ - \theta}{2}$ where $\theta$ is vertex angle

(ii) Calculating $\angle BAD = 90^\circ - \angle B$ where AD is altitude to BC

(iii) Calculating angles $\frac{\angle B}{2}, \frac{\angle C}{2}$ for construction

(iv) Using known angles $\angle B, \angle C$ directly in construction

(v) No new angle calculation needed, angles given

(a) Construction given Two Angles and Perimeter.

(b) Construction given Two Sides and Included Angle (SAS).

(c) Construction given Isosceles Triangle Base and Vertex Angle.

(d) Construction given Two Angles and Altitude.

(e) Construction given Two Angles and Included Side (ASA).

(i) B is on the perpendicular bisector of AP

(ii) C is on the perpendicular bisector of AQ

(iii) Angles $\angle APQ = \angle PAB = (1/2)\angle B$

(iv) Angles $\angle AQP = \angle QAC = (1/2)\angle C$

(v) PQ = PB + BC + CQ

(a) $\triangle AQC$ is isosceles with CQ = CA.

(b) By construction on the line segment PQ.

(c) $\triangle APB$ is isosceles with BP = BA.

(d) By property of perpendicular bisector, BP = BA.

(e) By property of perpendicular bisector, CQ = CA.

(i) Given Side, Angle $\angle B$, Sum AB+AC

(ii) Given Side, Angle $\angle B$, Difference |AB-AC|

(iii) Given Two Angles $\angle B, \angle C$ and Perimeter

(iv) Given Two Angles $\angle B, \angle C$ and Altitude

(v) Triangle inequality for the intermediate triangle in Median case

(a) Sum of two sides of $\triangle ACE$ (sides AB, AC, 2AD) > third side. e.g., AB + AC > 2AD.

(b) Sum of given angles $\angle B + \angle C < 180^\circ$.

(c) Sum of given angles $\angle B + \angle C < 180^\circ$.

(d) Sum of given angles $\angle B + \angle C < 180^\circ$ and sum of two sides > third side.

(e) Sum of given angles $\angle B + \angle C < 180^\circ$ and the side > the difference of the other two sides.

(i) Scale Factor $k = 1$

(ii) Scale Factor $k > 1$

(iii) Scale Factor $k < 1$

(iv) Scale Factor $k = m/n, m > n$

(v) Scale Factor $k = m/n, m < n$

(a) The new triangle is larger than the original.

(b) The new triangle is congruent to the original.

(c) The vertices of the new triangle lie on the interior of the sides AB and BC.

(d) The vertices of the new triangle lie on the extensions of the sides AB and BC.

(e) The new triangle is smaller than the original.

(i) Draw ray BX

(ii) Mark $\text{max}(m, n)$ points on BX

(iii) Join $B_n$ to C (for $mn$)

(iv) Draw line through $B_m$ (for $mn$) parallel to the segment from (iii)

(v) Draw line through C' parallel to AC

(a) To establish a base for applying BPT/creating similar triangles.

(b) To find the position of vertex A'.

(c) To create a ray with the necessary total number of divisions.

(d) To locate the position of vertex C' on BC (or its extension).

(e) To create a line that is parallel to a side of a reference triangle.

(i) Basic Proportionality Theorem

(ii) AA Similarity

(iii) SAS Similarity

(iv) Ratio of Areas = $k^2$

(v) Ratio of Perimeters = $k$

(a) Used to show that if sides forming an angle are proportional and the angle is common, triangles are similar.

(b) Directly proves that the ratio of corresponding sides on the transversal and the base is equal to the ratio of segments on the auxiliary ray.

(c) Used to show that if two angles of one triangle are equal to two angles of another, they are similar.

(d) A property of similar figures derived from the side ratio, not the primary justification for the construction method itself.

(e) A property of similar figures derived from the side ratio.

(i) Scale Factor 2/3

(ii) Scale Factor 3/2

(iii) Scale Factor 1/2

(iv) Scale Factor 3/1

(v) Scale Factor 1

(a) Max 3 points. Connect $B_3$ to C. Parallel through $B_2$.

(b) Max 3 points. Connect $B_2$ to C. Parallel through $B_3$.

(c) Max 2 points. Connect $B_2$ to C. Parallel through $B_1$.

(d) Max 3 points. Connect $B_1$ to C. Parallel through $B_3$.

(e) Max 1 point. Connect $B_1$ to C. Parallel through $B_1$.

(i) Scale Factor 2/3

(ii) Scale Factor 3/2

(iii) Scale Factor 1/2

(iv) Scale Factor 3/1

(v) Scale Factor 1

(a) Max 3 points. Connect $B_3$ to C. Parallel through $B_2$.

(b) Max 3 points. Connect $B_2$ to C. Parallel through $B_3$.

(c) Max 2 points. Connect $B_2$ to C. Parallel through $B_1$.

(d) Max 3 points. Connect $B_1$ to C. Parallel through $B_3$.

(e) Max 1 point. Connect $B_1$ to C. Parallel through $B_1$.

(i) Compass

(ii) Ruler (straight edge)

(iii) Pencil

(iv) Ruler (markings)

(v) Protractor

(a) Drawing straight lines (sides of triangles, ray, transversals, parallel lines).

(b) Drawing arcs to mark equal divisions on the ray and copying angles for parallel lines.

(c) Marking points and drawing lines/arcs.

(d) Measuring the initial triangle side lengths (if not given directly as a segment).

(e) Not used in the standard compass and ruler method.

(i) $k < 1$

(ii) $k = 1$

(iii) $k > 1$

(iv) $k = 0.5$

(v) $k = 2$

(a) Vertices A' and C' are on the extensions of BA and BC respectively.

(b) The triangle is congruent to the original.

(c) Vertices A' and C' are on the interior of segments BA and BC respectively.

(d) Same as (c).

(e) Same as (a).

(i) Ratio of corresponding side lengths

(ii) Ratio of perimeters

(iii) Ratio of corresponding altitudes

(iv) Ratio of corresponding medians

(v) Ratio of areas

(vi) Ratio of corresponding angles

(a) $k^2$

(b) 1

(c) $k$

(d) $k$

(e) $k$

(f) $k$

(i) General Quadrilateral

(ii) Parallelogram

(iii) Rectangle

(iv) Rhombus

(v) Square

(a) Length of one side.

(b) Lengths of two adjacent sides and included angle.

(c) Lengths of four sides and one diagonal.

(d) Lengths of length and width.

(e) Lengths of side and one diagonal OR lengths of two diagonals.

(i) Constructing a Rectangle (given l and w)

(ii) Constructing a Rhombus (given side and one angle)

(iii) Constructing a Rhombus (given two diagonals)

(iv) Constructing a Square (given side 's')

(v) Constructing a Parallelogram (given adjacent sides and diagonal)

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

(b) Diagonals bisect each other at right angles.

(c) All sides are equal, and one angle is given.

(d) All sides are equal and all angles are $90^\circ$.

(e) Opposite sides are equal, and a diagonal divides it into two triangles.

(i) Diagonals bisect each other

(ii) Diagonals are perpendicular bisectors of each other

(iii) Diagonals are equal

(iv) Only one pair of opposite sides is parallel

(v) All angles are equal to $90^\circ$

(vi) All sides are equal

(a) Rhombus

(b) Rectangle

(c) Parallelogram, Rectangle, Rhombus, Square

(d) Rhombus, Square

(e) Trapezium

(f) Rectangle, Square

(i) AB=4, BC=5, CD=6, DA=7

(ii) AB=4, BC=5, CD=6, DA=7, AC=8

(iii) AB=4, BC=5, $\angle B=60^\circ$, $\angle C=70^\circ$, CD=6

(iv) AB=4, BC=5, $\angle B=90^\circ$, $\angle C=90^\circ$, $\angle D=90^\circ$

(v) AB=4, BC=4, CD=4, DA=4

(a) Unique Quadrilateral

(b) Not a unique quadrilateral (can form different shapes)

(c) Unique Rectangle (if implied by angles, or given l,w). Unique Square (if one side given).

(d) Unique Quadrilateral

(e) Not a unique quadrilateral (can be rhombus, square, kite, etc.)

(i) Parallelogram

(ii) Rectangle

(iii) Rhombus

(iv) Square

(v) Kite

(a) Two congruent isosceles triangles.

(b) Two congruent right-angled triangles.

(c) Two congruent triangles.

(d) Two pairs of congruent right-angled triangles (by both diagonals).

(e) Two congruent right-angled isosceles triangles (by both diagonals).

(i) Draw a diagonal, construct its perpendicular bisector, mark half of the other diagonal on the bisector.

(ii) Draw a side, construct $90^\circ$ angles at both ends, mark adjacent side length on perpendiculars.

(iii) Draw a side, construct an angle, mark adjacent side, draw arcs from endpoints with opposite side lengths.

(iv) Draw a side, construct an angle, mark adjacent side (same length), draw arcs of side length from endpoints.

(v) Construct two triangles using SSS with the diagonal as a common side.

(a) Rectangle

(b) Rhombus (given diagonals)

(c) General Quadrilateral (given 4 sides and 1 diagonal)

(d) Parallelogram (given adjacent sides and included angle)

(e) Rhombus (given side and one angle)

(i) Tangent at a point on the circle

(ii) Tangents from a point outside the circle

(iii) Tangents with a specific angle between them

(iv) Lengths of tangents from external point

(v) No tangent from point inside circle

(a) The line segment from the center to the point of contact is perpendicular to the tangent.

(b) Can be constructed by finding the external point where radii forming $180^\circ - \theta$ angle intersect perpendiculars at circumference.

(c) The angle in a semicircle is a right angle.

(d) All points on the circle are equidistant from the center, points inside are closer.

(e) Congruence of right-angled triangles formed by radii, tangents, and line to center.

(i) Join O to P

(ii) Construct perpendicular bisector of OP to find midpoint M

(iii) Draw circle with center M and radius OM

(iv) The intersection points Q, R of circle (M, OM) and original circle

(v) Join P to Q and P to R

(a) Locates the center of the auxiliary circle.

(b) Defines the segment which will be the diameter of the auxiliary circle.

(c) Creates the auxiliary circle that passes through O and P.

(d) Forms the tangent lines.

(e) Are the points of tangency.

(i) Two tangents can always be drawn from an external point.

(ii) The lengths of the two tangents are always equal.

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

(iv) The chord joining the points of contact is perpendicular to the line from the center to the external point.

(v) The line from the center to the external point bisects the chord joining the points of contact.

(a) True

(b) False

(i) Tangent at point on circle

(ii) Tangents from external point

(iii) Tangents with angle $\theta$ between them

(iv) Finding center of circle (if not given)

(v) Justifying tangent construction

(a) Constructing perpendicular bisectors of chords.

(b) Constructing a perpendicular at a point on a line.

(c) Constructing angle $180^\circ - \theta$.

(d) Constructing perpendicular bisector of a line segment (OP).

(e) Using geometric theorems and properties.

(i) Tangent

(ii) Secant

(iii) Chord

(iv) Point of contact

(v) External Point

(a) A line that intersects a circle at exactly one point.

(b) A point outside the boundary of the circle.

(c) The single point where a tangent touches a circle.

(d) A line segment connecting two points on a circle.

(e) A line that intersects a circle at two distinct points.

(i) Two tangents can be constructed.

(ii) Exactly one tangent can be constructed.

(iii) No tangent can be constructed.

(iv) Infinite tangents can be constructed.

(v) The point P coincides with the center O.

(a) OP < r

(b) OP = 0

(c) OP = r

(d) OP > r

(e) P is on the circle.

(f) P is outside the circle.

(i) Definition

(ii) Axiom/Postulate

(iii) Theorem

(iv) Proof

(v) Justification

(a) A statement accepted as true without proof.

(b) A statement proven from accepted facts.

(c) Explains the meaning of a geometric term.

(d) A logical sequence of steps demonstrating a statement is true.

(e) The process of proving that a construction meets its requirements.

(i) Angle Bisector

(ii) Perpendicular Bisector

(iii) Parallel Lines (using angles)

(iv) Dividing Line Segment (ratio $m:n$)

(v) Tangents from External Point

(vi) $60^\circ$ angle

(a) Converse of angle theorems (Corresponding/Alternate Interior).

(b) Angle in a semicircle is $90^\circ$.

(c) Basic Proportionality Theorem (BPT).

(d) Equidistance from two points / SSS congruence.

(e) Equidistance from arms of an angle / SSS or ASA congruence.

(f) Properties of an equilateral triangle.

(i) Justification proves the construction is mathematically sound.

(ii) Measurement is a formal justification.

(iii) Justification relies on logical deduction.

(iv) Axioms are proven statements used in justification.

(v) Theorems are unproven statements accepted as true.

(vi) Justification helps understand "why" a construction works.

(a) True

(b) False

(i) SSS Congruence

(ii) Converse of Corresponding Angles Postulate

(iii) BPT

(iv) Angle in a Semicircle

(v) Definition of perpendicular lines

(a) Dividing a line segment in a ratio.

(b) Constructing a perpendicular at a point on a line.

(c) Constructing the perpendicular bisector.

(d) Constructing parallel lines using corresponding angles.

(e) Constructing tangents from an external point.

(i) Learning to draw accurately

(ii) Understanding the underlying principles

(iii) Proving statements

(iv) Accepting fundamental truths

(v) Checking if a construction looks right

(a) Justification

(b) Axioms

(c) Construction

(d) Proof

(e) Measurement (as verification)

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

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

(iii) All right angles are equal.

(iv) Through two points, exactly one line exists.

(v) If two sides and included angle are equal, triangles are congruent.

(a) Postulate

(b) Theorem

(c) Axiom

(d) Congruence Criterion (Postulate or Theorem depending on system)

(e) Converse of a theorem