#### Introduction:

## Use Congruent Triangles

**Example 1:** In the given figure, AB = BC and AD = CD. Show that BD bisects AC at right angles.

**Solution:**

From the given figure, ∆ABD ≅ ∆CBD

**Given:** AB = BC and AD = CD

**To prove** ∠BEA = ∠BEC = 90° and AE = EC.

**Proof:**

AB = BC (Given)

AD = CD (Given)

BD = BD (Common sides)

Therefore, ∆ABD ≅ ∆CBD (By SSS congruency)

∠ABD = ∠CBD (Corresponding angles)

Now, from ∆ABE and ∆CBE,

AB = BC (Given)

∠ABD = ∠CBD (Corresponding angles)

BE = BE (Common sides)

Therefore, ∆ABE≅ ∆CBE (By SAS congruency)

∠BEA = ∠BEC (Corresponding angles)

And ∠BEA +∠BEC = 180° (Linear pair)

2∠BEA = 180° (∠BEA = ∠BEC)

∠BEA = = 90° = ∠BEC

AE = EC (Corresponding sides)

Hence, BD AC.

**Example 2:** In a Δ ABC, if AB = AC and ∠ B = 70°, find ∠ A.

**Solution:**

**Given:** AB = AC and ∠B = 70°

∠ B = ∠ C [Angles opposite to equal sides are equal]

Therefore, ∠ B = ∠ C = 70°

Sum of angles in a triangle = 180°

∠ A + ∠ B + ∠ C = 180°

∠ A + 70° + 70° = 180°

∠ A = 180° – 140°

∠ A = 40°.

**Example 3:** In the given figure, PQ = PS and ∠QPR = ∠SPR. Prove that ∆ PQR ≅ ∆PSR, Use SAS congruence postulate.

**Solution:**

In ∆ PQR and ∆ PSR,

PQ = PS (Given)

∠QPR = ∠SPR (Given)

PR = PR (Common sides)

Therefore, ∆PQR ≅ ∆PSR (By SAS congruence).

**Example 4:** Identify the congruent triangle in the given figure.

**Solution:**

In ∆LMN,

65° + 45° + ∠L = 180°

110° + ∠L = 180°

∠L = 180° – 110°

Therefore, ∠L = 70°

Now in ∆XYZ and ∆LMN

∠X = ∠L (Given)

XY = LM (Given)

XZ = NL (Given)

Therefore, ∆XYZ ≅ ∆LMN by SAS congruence postulate.

**Example 5:**

Write a 2-column proof for the given figure.

**Given:** BD is an angle bisector of ∠CDA, ∠C ≅ ∠A

**To prove:** △CBD ≅ ∠ABD

**Solution:**

**How to prove construction:**

The following steps explain the construction of congruent triangles:

**Step 1:**

To copy∠ A, draw a segment with initial point D. Draw an arc with center A. Using the same radius, draw an arc with center D. Label points B, C, and E.

**Step 2:**

Draw an arc with radius BC and center E. Label the intersection F.

**Step 3:**

Draw .

**Example 6:**

Write a proof to verify that the construction for copying an angle is valid.

**Solution:**

Add and to the diagram. In the construction, , , , and are determined by the same compass. So, the required construction is and .

**Given:** ≅ , ≅ , ≅ .

**To prove:** D ≅ A

**Plan for Proof:**

Show that △CAB ≅ △FDE, so we can conclude that the corresponding parts ∠A and ∠D are congruent.

**Plan in action:**

#### Exercise:

- Write a plan for proof for the given information to prove that ∠1 ≅ ∠2.

2. Prove that ∠VYX ≅ ∠WYZ in the given figure.

3. Prove that (FL) ̅ ≅ (HN) ̅ in the given diagram.

4. Prove that △PUX ≅ △QSY in the given figure

5. Prove that ≅ in the given diagram.

6. Write a two-column proof from the given diagram.

7. Prove that ∠1 ≅ ∠2 from the given diagram with the given information.

Given: ≅ , ∠PMN ≅ ∠NKL

8. Prove that ∠1 ≅ ∠2 from the given diagram with the given information.

Given: (TS) ̅ ≅ (TV) ̅, (SR) ̅ ≅ (VW) ̅

9. Find the measure of each angle in the given triangle.

m∠A=x°;m∠B=(4x)°and m∠C=(5x)°.

10. Find the measure of each angle in the given triangle.

m∠A=x°;m∠B=(5x)°and m∠C=(x+19)°.

#### Concept Map:

#### What Have We Learned:

- Understand and apply SSS congruence postulate.
- Understand and apply SAS congruence postulate.
- Understand and apply AAS congruence postulate.
- Understand and apply construction proof.
- Solve problems on different congruence of triangles.
- Solve problems on different congruence postulates.

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