## Key Concepts

- Use congruent triangles

**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 =

180°2180°2

= 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, and use the SAS congruence postulate.

**Solution:**

In QPR 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 starting at point D. Draw an arc with the 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

DF−→−DF→

.

**Example 6:**

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

**Solution:**

Add

BC−BC-

and

EF−EF- to the diagram. In the construction,

AB−AB-,

DE−DE-,

AC−AC-, and

DF−DF- are determined by the same compass. So, the required construction is

BC−BC- and

EF−EF-.

**Given:**

AB−AB-

≅

DE−DE-,

AC−AC- ≅

DF−DF-,

BC−BC- ≅

EF−EF-.

**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

1. Prove that FL O HN in the given diagram.

2. Prove that APUX – AQSY in the given figure.

3. Prove that AC = GE in the given diagram.

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

5. Prove that 21 22 from the given diagram with the given information. Given: MNKN, ZPMN

6. Prove that Z1 Z2 from the given diagram with the given information. Given: TS TV, SR_VW 1 RA

7. Find the measure of each angle in the given triangle. m2A=xo;m_B=(4x)”and m_C=(5x)”.

8 Find the measure of each angle in the given triangle. m2A=xo;mB=(5x)’and m2C=(x+19)o.

**Concept Map**

## What have we learned:

- Understand and apply the SSS congruence postulate.
- Understand and apply SAS congruence postulate.
- Understand and apply the 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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