Class 7MathematicsChapter 9NCERT • Ganita Prakash

Geometric Twins

Understand congruent figures and the measurements that guarantee two triangles have exactly the same shape and size.

Chapter Snapshot

Congruent figures have the same shape and size.
Congruent figures fit exactly when superimposed.
Triangle congruence can be proved using SSS, SAS, ASA, AAS or RHS.
Order of corresponding vertices matters.
Two sides and a non-included angle do not always guarantee congruence.
Angles opposite equal sides of an isosceles triangle are equal.

What You Will Learn

Understand the meaning of congruent figures and triangles
Match corresponding vertices, sides and angles
Use SSS and SAS congruence criteria
Use ASA and AAS congruence criteria
Use RHS congruence for right triangles
Identify conditions that do not guarantee congruence
Use corresponding parts of congruent triangles in reasoning
Apply congruence to properties of isosceles and equilateral triangles

Exam PriorityBoardsVery HighFoundationHigh

Chapter Overview

Two figures may look similar, but congruence means something stronger: they must have exactly the same shape and size.

1. Congruent Triangles

If two triangles are congruent, all corresponding sides and corresponding angles are equal.

For example:

\triangle ABC\cong\triangle XYZ

means the correspondence is:

$A\leftrightarrow X$ ;
$B\leftrightarrow Y$ ;
$C\leftrightarrow Z$ .

2. SSS Congruence

SSS

AB=XY,\quad BC=YZ,\quad AC=XZ

3. SAS Congruence

The included angle is the angle between the two known sides.

4. ASA and AAS Congruence

AAS also guarantees congruence when two angles and a corresponding non-included side are equal.

Triangle Congruence Conditions

Condition	Information Needed
SSS	Three corresponding sides
SAS	Two sides and included angle
ASA	Two angles and included side
AAS	Two angles and a corresponding side
RHS	Right angle, hypotenuse and one side

5. RHS Congruence

The hypotenuse is the side opposite the right angle.

6. Corresponding Parts

Once congruence is established, corresponding sides and angles are equal.

This is useful because we may prove triangles congruent using only a few measurements and then deduce the remaining measurements.

Use Congruence

Question

ΔABC ≅ ΔPQR. If AB = 6 cm and ∠C = 48°, find PQ and ∠R.

Solution

A\leftrightarrow P,\quad B\leftrightarrow Q,\quad C\leftrightarrow R

7. Isosceles Triangles

An isosceles triangle has two equal sides.

A key result is:

Isosceles Triangle Property

AB=AC

Angles in an Isosceles Triangle

Question

AB = AC and ∠A = 80°. Find ∠B and ∠C.

Solution

\angle B=\angle C

8. Equilateral Triangle

All sides of an equilateral triangle are equal, so all three angles are equal.

Since their sum is $180^\circ$ :

3\times\text{angle}=180^\circ

Therefore each angle is:

60^\circ

9. Correspondence Must Be Written Carefully

If two triangles are congruent, the order of the letters tells which vertices correspond.

For example:

\triangle ABC\cong\triangle PQR

means:

$A\leftrightarrow P$
$B\leftrightarrow Q$
$C\leftrightarrow R$

Therefore:

AB=PQ,\quad BC=QR,\quad AC=PR

and corresponding angles are equal.

10. Why Some Conditions Are Enough

A triangle is rigid. Certain sets of measurements determine only one triangle up to rotation or reflection.

SSS: three sides fix the triangle.
SAS: two sides and the included angle fix the triangle.
ASA/AAS: two angles determine the third angle, and one side fixes the size.
RHS: in right triangles, the hypotenuse and one other side are sufficient.

11. Conditions That Do Not Guarantee Congruence

AAA does not guarantee congruence. Two triangles can have the same three angles but different sizes.

SSA is also not generally sufficient because the same measurements can sometimes form more than one triangle.

Congruence Tests

Information	Guarantees Congruence?
SSS	Yes
SAS	Yes
ASA	Yes
AAS	Yes
RHS for right triangles	Yes
AAA	No
SSA in general	No

12. Using Congruence in Proofs

A typical proof has three stages:

state the known equal sides or angles;
name the congruence criterion;
use corresponding parts to conclude the required equality.

Using Congruence

Question

In isosceles triangle ABC, AB = AC. AD bisects angle A and meets BC at D. Show that BD = DC.

Solution

ABD

13. Isosceles and Equilateral Triangles

Congruence gives a powerful way to understand special triangles. In an isosceles triangle, the angles opposite equal sides are equal. In an equilateral triangle all sides are equal, so all three angles are equal. Since their sum is $180^\circ$ , each is $60^\circ$ .

14. Worked Congruence Notes

Congruent Does Not Mean “Looks the Same”

Two figures are congruent only if one can be placed exactly over the other after moving, rotating or reflecting it. Orientation does not matter; size and shape do.

A small triangle and a large triangle may have the same shape, but they are not congruent.

Matching the Correct Parts

Suppose:

\triangle ABC\cong\triangle DEF

Then the order gives:

A\leftrightarrow D,\quad B\leftrightarrow E,\quad C\leftrightarrow F

Hence:

AB=DE,\ BC=EF,\ AC=DF

and:

\angle A=\angle D,\ \angle B=\angle E,\ \angle C=\angle F

Choosing the Correct Criterion

If three side equalities are given, think SSS. If two sides and the angle between them are given, think SAS. If two angles and a side are given, think ASA or AAS. For right triangles, if the hypotenuse and one corresponding side are equal, think RHS.

Identify the Criterion

Given $AB=PQ$ , $AC=PR$ and $\angle A=\angle P$ , the equal angle lies between the two equal sides. Therefore:

\triangle ABC\cong\triangle PQR

by SAS, provided the correspondence of the remaining vertices is correct.

Why AAA Is Different

If all corresponding angles are equal, triangles have the same shape but can be scaled. So AAA establishes similarity of shape, not necessarily equality of size.

CPCT After Congruence

Once congruence has been proved, all corresponding parts are equal. This is often the final step of a geometry proof.

For example, if two triangles are proved congruent and side $BC$ corresponds to side $EF$ , then:

BC=EF

Isosceles Triangle Reasoning

Congruence can show that the base angles of an isosceles triangle are equal by splitting the triangle into two matching triangles. Similar reasoning can establish that an angle bisector from the vertex may also act as a median and altitude in an isosceles triangle.

Quick Check

If triangle ABC is congruent to triangle RST, which angle corresponds to angle C?

Show answer

Angle T.

15. More Congruence Practice Notes

SSS Check

Question

Triangle ABC has sides 4 cm, 5 cm and 7 cm. Triangle PQR has sides 7 cm, 4 cm and 5 cm. Are they congruent?

Solution

Yes. The three side lengths match pairwise, so the triangles are congruent by SSS . The drawing orientation does not matter.

Reflection Does Not Break Congruence

A triangle and its mirror image are congruent because reflection changes orientation but not distances or angles. Congruence allows translation, rotation and reflection.

A Reliable Proof Format

Write:

the three equal facts;
the congruence criterion;
the congruence statement in correct corresponding order;
the required conclusion using corresponding parts.

Quick Check

If two right triangles have equal hypotenuse and one equal corresponding side, which test applies?

Show answer

RHS congruence.

Important Terms

Key Terms

CongruenceCorresponding PartsSSSSASASAAASRHSHypotenuse

Common Mistakes

Exam Focus

Important Exam Topics

meaning of congruence;
correct vertex correspondence;
SSS, SAS, ASA, AAS and RHS;
identifying which condition applies;
isosceles triangle angle property;
equilateral triangle angles.

Quick Revision

Congruent figures have the same shape and size.
Rotation or reflection does not change congruence.
SSS uses three sides.
SAS uses two sides and included angle.
ASA/AAS use two angles and a side.
RHS is for right triangles.
AAA does not guarantee congruence.
Two sides and a non-included angle may not guarantee congruence.
Angles opposite equal sides are equal.
Every angle of an equilateral triangle is 60°.

Practice Questions

Q1MCQEasyClass 7

Which condition uses three corresponding sides? A. SAS B. SSS C. ASA D. RHS

View Solution

Answer: B. SSS

Q2Short AnswerEasyClass 7

\triangle ABC\cong\triangle PQR

View Solution

\angle Q

Q3NumericalModerateClass 7

AB=AC

View Solution

180^\circ-40^\circ=140^\circ

Q4Short AnswerModerateFoundation

Why does AAA not prove congruence?

View Solution

Triangles can have the same three angle measures but different side lengths, so they may have the same shape but different sizes.

Additional Practice

Q5Short AnswerEasyClass 7

What does CPCT mean in congruent triangles?

View Solution

Corresponding parts of congruent triangles are equal.

Q6Short AnswerModerateClass 7

Do three equal angles guarantee two triangles are congruent? Explain.

View Solution

No. AAA fixes shape but not size; triangles can have the same angles and different side lengths.

Q7Short AnswerModerateClass 7

\triangle ABC\cong\triangle PQR

View Solution

QR

Q8Short AnswerHardFoundation

Two right triangles have equal hypotenuses and one corresponding leg equal. Which criterion proves congruence?

View Solution

RHS congruence.

Chapter Summary

Congruence means exact equality of shape and size.
Corresponding parts must be matched in the correct order.
SSS, SAS, ASA, AAS and RHS guarantee triangle congruence.
Congruence helps prove further side and angle relationships.
Isosceles and equilateral triangle properties follow naturally from congruence.

Class 7MathematicsChapter 9NCERT • Ganita Prakash

Geometric Twins

Understand congruent figures and the measurements that guarantee two triangles have exactly the same shape and size.

Chapter Snapshot

Congruent figures have the same shape and size.
Congruent figures fit exactly when superimposed.
Triangle congruence can be proved using SSS, SAS, ASA, AAS or RHS.
Order of corresponding vertices matters.
Two sides and a non-included angle do not always guarantee congruence.
Angles opposite equal sides of an isosceles triangle are equal.

What You Will Learn

Understand the meaning of congruent figures and triangles
Match corresponding vertices, sides and angles
Use SSS and SAS congruence criteria
Use ASA and AAS congruence criteria
Use RHS congruence for right triangles
Identify conditions that do not guarantee congruence
Use corresponding parts of congruent triangles in reasoning
Apply congruence to properties of isosceles and equilateral triangles

Exam PriorityBoardsVery HighFoundationHigh

Chapter Overview

Two figures may look similar, but congruence means something stronger: they must have exactly the same shape and size.

1. Congruent Triangles

If two triangles are congruent, all corresponding sides and corresponding angles are equal.

For example:

\triangle ABC\cong\triangle XYZ

means the correspondence is:

$A\leftrightarrow X$ ;
$B\leftrightarrow Y$ ;
$C\leftrightarrow Z$ .

2. SSS Congruence

SSS

AB=XY,\quad BC=YZ,\quad AC=XZ

3. SAS Congruence

The included angle is the angle between the two known sides.

4. ASA and AAS Congruence

AAS also guarantees congruence when two angles and a corresponding non-included side are equal.

Triangle Congruence Conditions

Condition	Information Needed
SSS	Three corresponding sides
SAS	Two sides and included angle
ASA	Two angles and included side
AAS	Two angles and a corresponding side
RHS	Right angle, hypotenuse and one side

5. RHS Congruence

The hypotenuse is the side opposite the right angle.

6. Corresponding Parts

Once congruence is established, corresponding sides and angles are equal.

This is useful because we may prove triangles congruent using only a few measurements and then deduce the remaining measurements.

Use Congruence

Question

ΔABC ≅ ΔPQR. If AB = 6 cm and ∠C = 48°, find PQ and ∠R.

Solution

A\leftrightarrow P,\quad B\leftrightarrow Q,\quad C\leftrightarrow R

7. Isosceles Triangles

An isosceles triangle has two equal sides.

A key result is:

Isosceles Triangle Property

AB=AC

Angles in an Isosceles Triangle

Question

AB = AC and ∠A = 80°. Find ∠B and ∠C.

Solution

\angle B=\angle C

8. Equilateral Triangle

All sides of an equilateral triangle are equal, so all three angles are equal.

Since their sum is $180^\circ$ :

3\times\text{angle}=180^\circ

Therefore each angle is:

60^\circ

9. Correspondence Must Be Written Carefully

If two triangles are congruent, the order of the letters tells which vertices correspond.

For example:

\triangle ABC\cong\triangle PQR

means:

$A\leftrightarrow P$
$B\leftrightarrow Q$
$C\leftrightarrow R$

Therefore:

AB=PQ,\quad BC=QR,\quad AC=PR

and corresponding angles are equal.

10. Why Some Conditions Are Enough

A triangle is rigid. Certain sets of measurements determine only one triangle up to rotation or reflection.

SSS: three sides fix the triangle.
SAS: two sides and the included angle fix the triangle.
ASA/AAS: two angles determine the third angle, and one side fixes the size.
RHS: in right triangles, the hypotenuse and one other side are sufficient.

11. Conditions That Do Not Guarantee Congruence

AAA does not guarantee congruence. Two triangles can have the same three angles but different sizes.

SSA is also not generally sufficient because the same measurements can sometimes form more than one triangle.

Congruence Tests

Information	Guarantees Congruence?
SSS	Yes
SAS	Yes
ASA	Yes
AAS	Yes
RHS for right triangles	Yes
AAA	No
SSA in general	No

12. Using Congruence in Proofs

A typical proof has three stages:

state the known equal sides or angles;
name the congruence criterion;
use corresponding parts to conclude the required equality.

Using Congruence

Question

In isosceles triangle ABC, AB = AC. AD bisects angle A and meets BC at D. Show that BD = DC.

Solution

ABD

13. Isosceles and Equilateral Triangles

14. Worked Congruence Notes

Congruent Does Not Mean “Looks the Same”

Two figures are congruent only if one can be placed exactly over the other after moving, rotating or reflecting it. Orientation does not matter; size and shape do.

A small triangle and a large triangle may have the same shape, but they are not congruent.

Matching the Correct Parts

Suppose:

\triangle ABC\cong\triangle DEF

Then the order gives:

A\leftrightarrow D,\quad B\leftrightarrow E,\quad C\leftrightarrow F

Hence:

AB=DE,\ BC=EF,\ AC=DF

and:

\angle A=\angle D,\ \angle B=\angle E,\ \angle C=\angle F

Choosing the Correct Criterion

Identify the Criterion

Given $AB=PQ$ , $AC=PR$ and $\angle A=\angle P$ , the equal angle lies between the two equal sides. Therefore:

\triangle ABC\cong\triangle PQR

by SAS, provided the correspondence of the remaining vertices is correct.

Why AAA Is Different

If all corresponding angles are equal, triangles have the same shape but can be scaled. So AAA establishes similarity of shape, not necessarily equality of size.

CPCT After Congruence

Once congruence has been proved, all corresponding parts are equal. This is often the final step of a geometry proof.

For example, if two triangles are proved congruent and side $BC$ corresponds to side $EF$ , then:

BC=EF

Isosceles Triangle Reasoning

Quick Check

If triangle ABC is congruent to triangle RST, which angle corresponds to angle C?

Show answer

Angle T.

15. More Congruence Practice Notes

SSS Check

Question

Triangle ABC has sides 4 cm, 5 cm and 7 cm. Triangle PQR has sides 7 cm, 4 cm and 5 cm. Are they congruent?

Solution

Yes. The three side lengths match pairwise, so the triangles are congruent by SSS . The drawing orientation does not matter.

Reflection Does Not Break Congruence

A triangle and its mirror image are congruent because reflection changes orientation but not distances or angles. Congruence allows translation, rotation and reflection.

A Reliable Proof Format

Write:

the three equal facts;
the congruence criterion;
the congruence statement in correct corresponding order;
the required conclusion using corresponding parts.

Quick Check

If two right triangles have equal hypotenuse and one equal corresponding side, which test applies?

Show answer

RHS congruence.

Important Terms

Key Terms

CongruenceCorresponding PartsSSSSASASAAASRHSHypotenuse

Common Mistakes

Exam Focus

Important Exam Topics

meaning of congruence;
correct vertex correspondence;
SSS, SAS, ASA, AAS and RHS;
identifying which condition applies;
isosceles triangle angle property;
equilateral triangle angles.

Quick Revision

Congruent figures have the same shape and size.
Rotation or reflection does not change congruence.
SSS uses three sides.
SAS uses two sides and included angle.
ASA/AAS use two angles and a side.
RHS is for right triangles.
AAA does not guarantee congruence.
Two sides and a non-included angle may not guarantee congruence.
Angles opposite equal sides are equal.
Every angle of an equilateral triangle is 60°.

Practice Questions

Q1MCQEasyClass 7

Which condition uses three corresponding sides? A. SAS B. SSS C. ASA D. RHS

View Solution

Answer: B. SSS

Q2Short AnswerEasyClass 7

\triangle ABC\cong\triangle PQR

View Solution

\angle Q

Q3NumericalModerateClass 7

AB=AC

View Solution

180^\circ-40^\circ=140^\circ

Q4Short AnswerModerateFoundation

Why does AAA not prove congruence?

View Solution

Triangles can have the same three angle measures but different side lengths, so they may have the same shape but different sizes.

Additional Practice

Q5Short AnswerEasyClass 7

What does CPCT mean in congruent triangles?

View Solution

Corresponding parts of congruent triangles are equal.

Q6Short AnswerModerateClass 7

Do three equal angles guarantee two triangles are congruent? Explain.

View Solution

No. AAA fixes shape but not size; triangles can have the same angles and different side lengths.

Q7Short AnswerModerateClass 7

\triangle ABC\cong\triangle PQR

View Solution

QR

Q8Short AnswerHardFoundation

Two right triangles have equal hypotenuses and one corresponding leg equal. Which criterion proves congruence?

View Solution

RHS congruence.

Chapter Summary

Congruence means exact equality of shape and size.
Corresponding parts must be matched in the correct order.
SSS, SAS, ASA, AAS and RHS guarantee triangle congruence.
Congruence helps prove further side and angle relationships.
Isosceles and equilateral triangle properties follow naturally from congruence.