## Key Concepts

- Identify congruent parts

## Congruent Triangles

### Introduction

### What are congruent figures?

Two figures are said to be congruent if they have the same corresponding side lengths and the corresponding angles.

### What are similar figures?

Any two figures have the same shape, but their size is not the same.

### Identify congruent parts

#### Congruence Statement:

From the above figures, the corresponding sides and the corresponding angles of both the triangles are equal.

## Properties of congruence:

The following properties define an equivalence relation.

**Reflexive Property**– For all angles of A, ∠A≅∠A. An angle is congruent to itself.

**Symmetric Property**– For any angles of A and B, if ∠A≅∠B, then ∠B≅∠A. Order of congruence is the same.

**Transitive Property**– For any angles A, B, and C, if ∠A≅∠B and ∠B≅∠C, then ∠A≅∠C. If two angles are congruent to a third angle, then the first two angles are also congruent.

**Example 1:** Identify the pairs of congruent corresponding parts in the figure.

**Solution:** From the given figure,

∆JKL ≌ ∆TSR

Corresponding angles: ∠J ≌ ∠T, ∠K ≌ ∠S, ∠L ≌ ∠R

**Example 2:** Find the values of *x* and *y* in the diagram using properties of congruent figures if DEFG ≌ SPQR.

**Solution: **

Given that DEFG ≌ SPQR,

We know that FG ≌ QR.

⇒ FG = QR

12=2x−4

16=2x

8=x

Since ∠ F ≌ ∠Q.

m∠F= m∠Q

68°=(6y+x)°

68=6y+8

6y=68−8

y=606=10

### Third angles theorem:

If two angles of one triangle are congruent to two angles of another triangle, then the third angle is also congruent.

**Given:**

∠ A ≅ ∠D,

∠B ≅ ∠E.

**To Prove:**

∠ C ≅ ∠F.

**Proof: **

If ∠ A ≅ ∠D, and ∠B ≅ ∠E (given)

*m*∠ *A = m*∠*D, m*∠* B = m*∠*E* (Congruent angles)

*m*∠*A *+* m*∠*B *+* m*∠*C = *180° (Triangle sum theorem)

*m*∠*D *+* m*∠*E *+* m*∠*F = *180°

*m*∠*A *+* m*∠*B *+* m*∠*C = m*∠*D *+* m*∠*E *+* m*∠*F* (Substitution property)

*m*∠*D *+* m*∠*E *+* m*∠*C = m*∠*D *+* m*∠*E *+* m*∠*F* (Substitution property)

*m*∠*C *=* m*∠*F* (Subtraction property of equality)

∠C ≅ ∠F. (Congruent angles)

**Example 3:** Find

m∠BDC

in the given figure.

**Solution: **

From the given figure, we have

∠A ≅ ∠B and ∠ADC ≅∠BCD

⇒∠ACD ≅ ∠BDC (Third angles theorem)

*m*∠*ACD *+* m*∠*CAN *+* m*∠*CDN = *180° (Triangle sum theorem)

*m*∠*ACD *=* *180° – 45° – 30° = 105°.

*m*∠*ACD *=* m*∠*BDC *= 105°. (Congruent angles)

### Properties of congruent triangles:

**Reflexive property of congruent triangles:**

For any triangle ABC, ∆ABC ≌ ∆ABC.

**Symmetric property of congruent triangles:**

If ∆ABC ≌ ∆DEF, then ∆DEF ≌ ∆ABC.

**Transitive property of congruent triangles:**

If ∆ABC ≌ ∆DEF and ∆DEF ≌ ∆JKL, then ∆ABC ≌ ∆JKL.

**Example 4:** Prove that the triangles are congruent.

**Solution:**

**Given: **AD ≌ CB, DC ≌ BA, ∠ ACD ≅ ∠ CAB, ∠ CAD ≅ ∠ ACB** **

**To prove:** ∆ACD ≅ ∆CAB

proof:

a. Use the reflexive property to show that AC ≌ AC

b. Use the third angles theorem to show that ∠ B ≅ ∠ D

**Plan in action:**

## Exercise

- Identify all pairs of congruent corresponding parts in the following figure:

- Find the value of
*x*in the given figure.

- Find the values of
*x*and*y*in the diagram.

- Write the proof with the help of a diagram.

Given: WX ⊥ VZ at Y, Y is the midpoint of WX, VW ≅ VX, and VZ bisects ∠WVX.

Proof: ∆VWY ≅ ∆VXY.

- Find from the given figure.

- Find the values of
*x*and*y*in the diagram.

- Find the value of
*x*in the figure.

- Write the congruence statements for the given figure.

- Identify the congruent corresponding parts.

- Find in the figure.

### What have we learned

- Understand congruent of two figures.
- Understand the congruence statements.
- Find corresponding angles and corresponding sides.
- Identify congruent parts for the given triangles.
- Find the values using properties of congruent figures.
- Understand third angles theorem.
- Understand properties of congruent triangles.
- Solve problems on congruent figures.

### Summary

#### Properties of congruence:

The following properties define an equivalence relation.

**1.Reflexive property** – For all angles of A, ∠A≅∠A. An angle is congruent to itself.

**2. Symmetric property** – For any angles of A and B, if ∠A≅∠B, then ∠B≅∠A. Order of congruence is the same.

**3.Transitive property** – For any angles A, B, and C, if ∠A≅∠B and ∠B≅∠C, then ∠A≅∠C. If two angles are congruent to a third angle, then the first two angles are also congruent.

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