Part-II • Reprint 2026–27

Ganita Prakash • Part-II • Chapter 1

Geometric Twins

Discover how measurements can tell us when two figures are exactly alike. Explore congruent figures, triangle congruence criteria (SSS, SAS, ASA, AAS, RHS), the SSA ambiguous trap, and the deductive geometry of isosceles and equilateral triangles.

Start Exploring Test Your Geometry (40 Qs)

Estimated: 45 mins

✓ Superimposition✓ SSS, SAS, ASA, AAS, RHS✓ SSA Counterexample✓ Isosceles & 60° Equilateral✓ Expression Engineer

≅

Superimposition Simulator

Interactive

Rotate or flip the twin triangle. Notice that orientation does not change congruence!

Rotate: 0°

ΔABC ≅ ΔXYZ: Both triangles remain congruent under rotation and reflection!

Curriculum Progression

What Will You Learn in Part-II Chapter 1?

Chapter Progress17% Completed

1. Superimposition

Same shape and same size. Figures coincide regardless of rotation or reflection.

2. Congruence Criteria

Master SSS, SAS, ASA, AAS, and RHS. Understand why SSA fails.

3. Correspondence & CPCTC

Order of vertices matters! Match corresponding vertices, sides, and angles.

4. Isosceles & Equilateral

Deduce that angles opposite equal sides are equal, and equilateral angles are 60°.

Section 1.1 • Textbook pp. 1–3

1.1 Geometric Twins: The Signboard Puzzle

A checkmark symbol on a signboard needs to be recreated on another board. Can we take some measurements that allow us to exactly recreate the figure? Let the corner points be A, B, and C. Are the arm lengths AB and BC alone sufficient to exactly recreate the figure?

Interactive Arm Length & Angle Explorer

Arm AB: 4 cm

Arm BC: 8 cm

Angle ∠ABC: 80°

Question: Are arm lengths AB = 4 cm and BC = 8 cm alone enough to recreate the exact symbol?

What Are Congruent Figures?

Figures that are exact copies of each other, having the exact same shape and size, are said to be congruent (≅). Congruent figures can be superimposed (placed one over the other) so that they fit exactly with no overlap or gaps. Crucially, a figure can be rotated or flipped before superimposing!

Figure it Out: Which Pairs Appear Congruent?

What Measurements Fix a Figure?

Circle

A circle is completely determined by its Radius (r). Two circles with equal radii are always congruent.

Rectangle

Determined by two adjacent sides: Length & Breadth. All four angles are already 90°.

Triangle

Can 3 side lengths alone fix a triangle? Or do we need angles too? Let us find out in Section 1.2!

Official Textbook Exercises

Figure It Out Problems

Verified NCERT Ganita Prakash Part-II Solutions

Determine whether figures are congruent and identify the minimum measurements needed to recreate identical figures.

Q1. What does it mean for two plane figures to be congruent?

Hint: Think about shape, size, and superimposition.

Q3(a). What measurement(s) would you take to recreate a figure congruent to a given circle?

Hint: A circle's size is controlled by a single length from the center.

Q3(b). What measurement(s) would you take to recreate a figure congruent to a given rectangle?

Hint: Rectangles are determined by two perpendicular dimensions.

Q4. For a bent-arm symbol with arms AB and BC, are the lengths AB and BC alone sufficient to guarantee congruence?

Hint: Can the arms be opened wide or kept narrow?

Practice Engine

40 Chapter Practice Questions

Level 1: FoundationRef: p. 2

Two geometric figures that have the exact same shape and the exact same size are called:

Level 1: FoundationRef: p. 2

Can two figures be congruent if one is rotated or flipped compared to the other?

Level 1: FoundationRef: p. 6

What does the SSS congruence criterion state?

Level 1: FoundationRef: p. 7

If ΔPQR ≅ ΔXYZ, which vertex corresponds to Q?

Level 1: FoundationRef: p. 10

In the SAS congruence condition, what does 'A' represent?

Level 1: FoundationRef: p. 10

Does knowing that two triangles have the exact same three angles (AAA) guarantee that they are congruent?

Level 1: FoundationRef: p. 16

What does the hypotenuse of a right-angled triangle refer to?

Level 1: FoundationRef: p. 17

What does RHS congruence stand for?

Level 1: FoundationRef: p. 18

In an isosceles triangle with AB = AC, which two angles must be equal?

Level 1: FoundationRef: p. 19

What is the measure of each interior angle in an equilateral triangle (in degrees)?

Level 2: ApplicationRef: p. 7

If ΔABC ≅ ΔPQR, which side corresponds to side AC?

Level 2: ApplicationRef: p. 10

In ΔABC and ΔDEF, AB = DE = 5 cm, BC = EF = 7 cm. Which angle must be equal to prove congruence by SAS?

Final Assessment

20-Question Comprehensive Quiz

Question 1Congruence Definition

What must be true for two plane figures to be congruent?

Question 2Transformations

Which of the following actions can change whether two figures are congruent?

Question 3SSS Criterion

If two triangles have sides of lengths 4 cm, 6 cm, and 8 cm each, what can we conclude?

Question 4Vertex Correspondence

Given ΔABC ≅ ΔXYZ. Which of the following statements is FALSE?

Question 5SAS Criterion

In the SAS criterion, why is it critical that the angle is 'included'?

Question 6SSA Ambiguity

Under what condition does SSA fail to guarantee congruence?

Question 7ASA Criterion

In ΔABC and ΔDEF, ∠B = ∠E = 50°, ∠C = ∠F = 30°, and BC = EF = 5 cm. Why are they congruent?

Question 8AAS Criterion

How does AAS establish triangle congruence?

Question 9RHS Criterion

What parts must match to apply the RHS congruence criterion?

Question 10Isosceles Property

In any triangle, what is true about the angles opposite to equal sides?

Question 11Isosceles Calculation

An isosceles triangle has a vertex angle of 40°. What is the measure of each base angle?

Question 12Equilateral Property

Why must every angle of an equilateral triangle equal 60°?

Question 13AAA Counterexample

Two triangles have angles 30°, 60°, and 90°. Are they necessarily congruent?

Question 14Circle Central Angle

In a circle with centre A, radii AB and AC form ∠BAC = 120°. What is ∠B?

Question 15Kite Symmetry

In kite ABCD with AB = AD and CB = CD, which line is the line of symmetry?

Question 16Rectangle Diagonals

In rectangle ABCD, diagonal BD splits it into two triangles. Why is ΔABD ≅ ΔCDB?

Question 17Conditions Sorter

Which of the following is NOT a sufficient condition to guarantee triangle congruence?

Question 18Altitude in Isosceles

In an isosceles triangle ABC with AB = AC, the altitude AD to base BC divides the triangle into two triangles that are congruent by:

Question 19Truss Rigidity

Why do railway bridges (like Howrah Bridge) use triangular trusses?

Question 20Congruence of Circles

Two circles are congruent if and only if they have:

Pitfall Prevention

10 Common Mistakes & How to Avoid Them

Assuming Same Angles Means Congruent (AAA Trap)

✗ Misconception: Believing that if two triangles have the same three angles (e.g., 30°, 70°, 80°), they must be congruent.

✓ Correct: Equal angles only guarantee that figures have the same SHAPE (similarity). They can be different sizes (scaled up or down).

Example & Correction:To prove congruence, at least ONE side length must be known and matched.

Treating SSA as a General Congruence Test

✗ Misconception: Using two sides and a non-included angle to claim two triangles are congruent.

✓ Correct: SSA is the ambiguous case. The swinging side can cut the ray at two different points, creating an acute and an obtuse triangle.

Example & Correction:Only use SAS (included angle) or RHS (for right-angled triangles). Never use SSA generally.

Ignoring Vertex Order in Congruence Statements

✗ Misconception: Writing ΔBAC ≅ ΔXYZ when A corresponds to X and B corresponds to Y.

✓ Correct: The order of vertices in the naming statement must strictly reflect the corresponding parts: A ↔ X, B ↔ Y, C ↔ Z requires ΔABC ≅ ΔXYZ.

Example & Correction:Write ΔBAC ≅ ΔYXZ so the 1st, 2nd, and 3rd letters match their corresponding vertices.

Confusing Included Angle (SAS) with Any Angle

✗ Misconception: Calling any combination of two sides and an angle 'SAS'.

✓ Correct: The angle MUST be formed by the two known sides (the included angle).

Example & Correction:The included angle between AB and BC is ∠B (the vertex shared by both sides).

Believing Congruent Figures Must Have the Same Orientation

✗ Misconception: Thinking two triangles are not congruent if one is rotated or flipped upside down.

✓ Correct: Congruence is an intrinsic property of shape and size. Translation, rotation, and reflection do not alter congruence.

Example & Correction:Superimposition allows rotating and flipping the cutout to test for exact fit.

Confusing ASA and AAS

✗ Misconception: Using the terms ASA and AAS interchangeably without checking side position.

✓ Correct: In ASA, the side is between the two known angles. In AAS, the side is adjacent to one angle and opposite to the other.

Example & Correction:Both guarantee congruence, but ASA uses the included side while AAS uses a non-included side.

Applying RHS When Hypotenuse is Not Equal

✗ Misconception: Applying RHS to any right-angled triangle with two equal sides, even if the hypotenuse is not one of them.

✓ Correct: RHS strictly requires: (1) 90° angle, (2) HYPOTENUSE must be equal, and (3) one leg must be equal.

Example & Correction:If the hypotenuse is not one of the matched sides, the criterion is SAS (with the 90° as included angle).

Mismatches in Isosceles Base Angles

✗ Misconception: Assuming the two base angles of an isosceles triangle can be obtuse (> 90°).

✓ Correct: Since base angles are equal, two obtuse angles would sum to > 180°, which violates the 180° triangle sum theorem.

Example & Correction:The obtuse angle MUST be the vertex angle. The base angles are acute: (180° - 100°) / 2 = 40° each.

Forgetting That Radii of a Circle Are Equal

✗ Misconception: Failing to recognize that triangles formed by two radii and a chord inside a circle are always isosceles.

✓ Correct: All radii from the centre of a circle to any point on its circumference are equal in length (OA = OB).

Example & Correction:ΔOAB is automatically isosceles with OA = OB, so ∠OAB = ∠OBA.

Thinking Equal Perimeters or Areas Mean Congruence

✗ Misconception: Assuming that if two rectangles or triangles have the same area, they must be congruent.

✓ Correct: Figures can have the same area but completely different shapes and side lengths (e.g. 2 × 8 vs 4 × 4 rectangle).

Example & Correction:They are not congruent because their corresponding side lengths are not equal.

Terminology & Rules

Key Terms & Math Toolbox

Congruence (≅)

A ≅ B

The geometric relation between two figures that have identical shape and identical size, coinciding completely upon superimposition.

Superimposition

Overlaying

The mental or physical process of placing one figure directly over another (possibly with translation, rotation, or reflection) to test if they coincide.

Corresponding Parts (CPCTC)

A ↔ X, AB ↔ XY

The matching vertices, sides, and angles in two congruent figures that lie over each other when the figures are superimposed.

SSS Criterion

Side-Side-Side

If three sides of one triangle are equal to the three corresponding sides of another triangle, the triangles are congruent.

SAS Criterion

Side-Angle-Side

If two sides and the included angle of one triangle are equal to two sides and the included angle of another, the triangles are congruent.

SSA (Ambiguous Case)

Side-Side-Angle

Given two sides and a non-included angle, a unique triangle is NOT always guaranteed because the swinging side can intersect a ray twice.

ASA Criterion

Angle-Side-Angle

If two angles and the included side of one triangle are equal to two angles and the included side of another, the triangles are congruent.

AAS Criterion

Angle-Angle-Side

If two angles and a non-included side of one triangle are equal to two angles and the corresponding non-included side of another, the triangles are congruent.

RHS Criterion

Right-Hypotenuse-Side

If the hypotenuse and one leg of a right-angled triangle are equal to the hypotenuse and one leg of another right-angled triangle, the triangles are congruent.

Isosceles Triangle Theorem

AB = AC ⇔ ∠B = ∠C

In any triangle with two equal sides, the angles opposite those equal sides are equal in measure.

Quick Reference

Frequently Asked Questions

What are congruent figures?

Congruent figures are geometric figures that have the exact same shape and the exact same size. If you trace one figure and place it over the other, it will fit exactly over it with all edges and corners matching.

What does superimposition mean?

Superimposition is the act of placing one figure on top of another to check whether they coincide. While superimposing, you are allowed to slide (translate), turn (rotate), or flip (reflect) the figure.

What is SSS congruence?

SSS stands for Side-Side-Side. It states that if all three side lengths of one triangle are equal to the three corresponding side lengths of another triangle, the two triangles are guaranteed to be congruent.

What is SAS congruence and what does 'included angle' mean?

SAS stands for Side-Angle-Side. It requires two sides and the angle between them (the included angle) to match. For example, between side AB and side BC, the included angle is ∠B. If the angle is not between the two known sides, SAS cannot be used.

Why does SSA not always guarantee congruence?

SSA gives two sides and a non-included angle. When you draw the known angle and first side, the second side swings like a compass arc. This arc can cut the ray at two distinct points, creating two completely different (non-congruent) triangles: one acute and one obtuse.

What is the difference between ASA and AAS?

In ASA (Angle-Side-Angle), the equal side is the included side located between the two known angles. In AAS (Angle-Angle-Side), the equal side is located outside the two known angles. However, because the three angles of any triangle always add up to 180°, AAS can always be rewritten as ASA.

What is the RHS congruence criterion?

RHS applies exclusively to right-angled triangles. It requires: (1) both triangles have a Right angle (90°), (2) their Hypotenuses are equal, and (3) one pair of corresponding Sides (legs) are equal.

Why does the order of letters matter in ΔABC ≅ ΔXYZ?

Writing ΔABC ≅ ΔXYZ communicates the exact correspondence of vertices: vertex A matches X, vertex B matches Y, and vertex C matches Z. If you write ΔACB ≅ ΔXYZ, it would mean side AC matches XY, which is mathematically false if those sides are not equal.

Why are angles opposite to equal sides in an isosceles triangle equal?

If you draw the perpendicular altitude from the vertex to the base, it splits the isosceles triangle into two right-angled triangles. By the RHS criterion, these two smaller triangles are congruent! By CPCTC (corresponding parts of congruent triangles), the base angles are equal.

Why is every angle in an equilateral triangle equal to 60°?

Since all three sides of an equilateral triangle are equal, the isosceles theorem applies to every pair of sides. This means all three angles must be equal: ∠A = ∠B = ∠C. Since the angle sum of a triangle is always 180°, 180° ÷ 3 = 60°.

Why doesn't AAA guarantee congruence?

AAA means all three angles are equal. This guarantees that the triangles have the same shape, but NOT the same size. For example, a small equilateral triangle with sides 1 cm and a huge equilateral triangle with sides 100 cm both have angles 60°, 60°, 60°, but they are not congruent.

Where are congruent triangles used in real life?

Triangular trusses are used in bridges (like Kolkata's Howrah Bridge), roofs, and cranes because triangles are rigid by SSS—they cannot deform without changing side lengths. Repeating congruent triangles also create aesthetically pleasing architectural designs like the Louvre Pyramid in Paris and geodesic domes.

Your Chapter Notes & Observations

Write down your reflections, correspondence rules, and key criteria. Saved automatically in your browser.

Content aligned with Ganita Prakash, Grade 7, Part-II, Chapter 1: Geometric Twins.

← Part I Chapter 8: Working with Fractions Part II Chapter 2: Operations with Integers