Applications of Congruent Triangles

Introduction 

Applications of 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: 

Now, from ∆ABE and ∆CBE, 

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° 

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

Solution: 

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

Solution: 

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 DF

.  

Example 6: 

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

Solution: 

To prove:  

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. 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¯FL≅¯HN  in the given diagram.

5. Prove that ¯AC≅¯GE  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: ¯MN≅¯KN,∠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)°.

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.

Concept Map