Class 7 Mathematics

Chapter 9 Solutions — Geometric Twins

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Solutions

Overview

Step-by-step NCERT solutions for Geometric Twins (Chapter 9, NCERT Class 7 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Geometric Twins textbook chapter.

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What these solutions cover

All 18 questions in Geometric Twins are solved in the PDF. Here's what's inside, exercise by exercise:

Geometric Twins — Congruent Figures

Check if the two figures are congruent. [Figure shows two V-shaped (angle) figures: the left one is a symmetric V with two equal-looking arms; the right one has arms of different lengths and a wider opening angle — they do not look the same size and shape.]
Circle the pairs that appear congruent. [Four pairs of figures are shown: Pair 1 — two leaf/droplet shapes of the same size; Pair 2 — two cloud shapes where one is larger than the other; Pair 3 — two identical spiky star/sun shapes; Pair 4 — two identical petal/leaf shapes.]
3.
What measurements would you take to create a figure congruent to a given:
- (a) Circle
- (b) Rectangle. Using this, state how would you check if two circles are congruent and if two rectangles are congruent.
How would we check if two figures like the one below are congruent? [The figure shows two line segments crossing each other at a point — like an X or asterisk shape.] Use this to identify whether each of the following pairs are congruent. [Pair 1 (top): two crossed-line figures that appear identical in arm lengths and crossing angle. Pair 2 (bottom): two crossed-line figures where the arm…

Congruence of Triangles — SSS Condition

Suppose ΔHEN is congruent to ΔBIG. List all the other correct ways of expressing this congruence.
Determine whether the triangles are congruent. If yes, express the congruence. [Left triangle RED: RE = 3.5 cm, ED = 5 cm, RD = 6 cm. Right triangle JMA: JM = 6 cm, MA = 5 cm, JA = 3.5 cm.]
In the figure, AB = AD and CB = CD. Can you identify any pair of congruent triangles? If yes, explain why they are congruent. Does AC divide ∠BAD and ∠BCD into two equal parts? Give reasons. [Kite-shaped figure with A at top, B at middle-left, C at bottom, D at middle-right. The dashed line AC is the diagonal.]
In the figure, are ΔDFE and ΔGED congruent to each other? It is given that DF = DG and FE = GE. [Figure shows an isosceles triangle with D at the top, F at bottom-left and G at bottom-right. E is the midpoint of FG, with DE drawn inside.]

Congruence of Triangles — ASA and SAS Conditions

Identify whether the triangles below are congruent. What conditions did you use to establish their congruence? Express the congruence. [ΔABC: AB = 7 cm (left side), ∠B = 47°, BC = 5 cm (base). ΔXYZ: XZ = 7 cm (right side), ∠Z = 47°, YZ = 5 cm (base).]
Given that CD and AB are parallel, and AB = CD. What are the other equal parts in this figure? [Figure shows two triangles meeting at point O (intersection of diagonals): D (top-left), C (top-right), O (center), A (bottom-left), B (bottom-right). DC has double tick marks, AB has double tick marks.] Are the two resulting triangles congruent? If so, express the congruence.
Given that ∠ABC = ∠DBC and ∠ACB = ∠DCB, show that ∠BAC = ∠BDC. Are the two triangles congruent? [Figure: kite/rhombus shape with B at left, C at right, A at top, D at bottom. BC is the common base. Angle marks show ∠ABC = ∠DBC (at B) and ∠ACB = ∠DCB (at C).]
Identify the equal parts in the following figure, given that ∠ABD = ∠DCA and ∠ACB = ∠DBC. [Trapezoid-like figure: A (top-left), D (top-right), B (bottom-left), C (bottom-right). Diagonals AC and BD cross each other. Angle arc marks: ∠ABD = ∠DCA and ∠ACB = ∠DBC (arcs at B and C).]

Angles of Isosceles and Equilateral Triangles

ΔAIR ≅ ΔFLY. Identify the corresponding vertices, sides and angles.
2.
Each of the following cases contains certain measurements taken from two triangles. Identify the pairs in which the triangles are congruent to each other, with reason. Express the congruence whenever they are congruent.
- (a) AB=DE, BC=EF, CA=DF
- (b) AB=EF, ∠A=∠E, AC=ED
- (c) AB=DF, ∠B=∠D=90°, AC=FE
- (d) ∠A=∠D, ∠B=∠E, AC=DF
- (e) AB=DF, ∠B=∠F, AC=DE
It is given that OB = OC, and OA = OD. Show that AB is parallel to CD. [Hint: AD is a transversal for these two lines. Are there any equal alternate angles?] [Figure: A (top-left) and C (top-right) connected through O (center); B (bottom-left) and D (bottom-right). OA=OD (single tick), OB=OC (double tick). Lines cross at O.]
ABCD is a square. Show that ΔABC ≅ ΔADC. Is ΔABC also congruent to ΔCDA? [Figure: square with A (top-left), B (top-right), C (bottom-right), D (bottom-left). Diagonal AC is drawn.]
Find ∠B and ∠C, if A is the centre of the circle. [Figure: Circle with centre A. Points B and C are on the circle. ∠BAC = 120°. Triangle ABC is formed with AB and AC as radii.]
Find the missing angles. As per the convention, all line segments marked with a single '|' are equal, those with a double '||' are equal, and those with a triple '|||' are equal. [Figure: rectangle with corners C (top-left), D (top-right), B (bottom-right), A (bottom-left); right-angle marks at C and B. On the top edge lie R then V (C–R–V–D); on the bottom edge lie K then L (A–K–L–B); U on the…

Sample

Worked examples

3 questions from Geometric Twins, solved step by step.

Congruence of Triangles — SSS Condition

Suppose ΔHEN is congruent to ΔBIG. List all the other correct ways of expressing this congruence.

When we write ΔHEN ≅ ΔBIG, it means the vertices correspond in order: H↔B, E↔I, N↔G.

Any correct re-expression must keep the same correspondence. We can cyclically permute or reverse the order of vertices in both triangles together.

The six valid ways (all expressing the same correspondence) are:

ΔHEN ≅ ΔBIG

ΔENH ≅ ΔIGB

ΔNHE ≅ ΔGBI

ΔHNE ≅ ΔBGI

ΔNEH ≅ ΔGIB

ΔEHN ≅ ΔIBG

Answer: The other correct ways are: ΔENH ≅ ΔIGB, ΔNHE ≅ ΔGBI, ΔHNE ≅ ΔBGI, ΔNEH ≅ ΔGIB, ΔEHN ≅ ΔIBG.

Congruence of Triangles — ASA and SAS Conditions

Identify whether the triangles below are congruent. What conditions did you use to establish their congruence? Express the congruence. [ΔABC: AB = 7 cm (left side), ∠B = 47°, BC = 5 cm (base). ΔXYZ: XZ = 7 cm (right side), ∠Z = 47°, YZ = 5 cm (base).]

In ΔABC: AB = 7 cm, ∠B = 47° (angle at B, included between AB and BC), BC = 5 cm.

In ΔXYZ: XZ = 7 cm, ∠Z = 47° (angle at Z, included between XZ and ZY), ZY = 5 cm.

Matching equal parts: AB = XZ = 7 cm, ∠B = ∠Z = 47° (included angle), BC = ZY = 5 cm.

Two sides and the included angle are equal → SAS (Side Angle Side) condition is satisfied.

Correspondence: A↔X, B↔Z, C↔Y.

Therefore, ΔABC ≅ ΔXZY.

Answer: Yes, ΔABC ≅ ΔXZY by SAS (AB = XZ = 7 cm, ∠B = ∠Z = 47°, BC = ZY = 5 cm).

Angles of Isosceles and Equilateral Triangles

ΔAIR ≅ ΔFLY. Identify the corresponding vertices, sides and angles.

When ΔAIR ≅ ΔFLY, the vertices correspond in order: A↔F, I↔L, R↔Y.

Corresponding vertices: A and F, I and L, R and Y.

Corresponding sides:

AI = FL (sides between 1st and 2nd vertices),

IR = LY (sides between 2nd and 3rd vertices),

AR = FY (sides between 1st and 3rd vertices).

Corresponding angles:

∠A = ∠F, ∠I = ∠L, ∠R = ∠Y.

Answer: Corresponding vertices: A↔F, I↔L, R↔Y. Sides: AI=FL, IR=LY, AR=FY. Angles: ∠A=∠F, ∠I=∠L, ∠R=∠Y.

All 18 questions in Geometric Twins are solved step by step in the Solutions PDF above.

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Chapter 9 Solutions — Geometric Twins

Overview

What these solutions cover

Geometric Twins — Congruent Figures

Congruence of Triangles — SSS Condition

Congruence of Triangles — ASA and SAS Conditions

Angles of Isosceles and Equilateral Triangles

Worked examples

Suppose ΔHEN is congruent to ΔBIG. List all the other correct ways of expressing this congruence.

Identify whether the triangles below are congruent. What conditions did you use to establish their congruence? Express the congruence. [ΔABC: AB = 7 cm (left side), ∠B = 47°, BC = 5 cm (base). ΔXYZ: XZ = 7 cm (right side), ∠Z = 47°, YZ = 5 cm (base).]

ΔAIR ≅ ΔFLY. Identify the corresponding vertices, sides and angles.

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