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
- Write a plan for proof for the given information to prove that ∠1 ≅ ∠2.
- Prove that ∠VYX ≅ ∠WYZ in the given figure.
- Prove that¯FL≅¯HN in the given diagram.
- Prove that¯FL≅¯HN in the given diagram.
- Prove that ¯AC≅¯GE in the given diagram.
- Write a two-column proof from the given diagram.
- Prove that ∠1 ≅ ∠2 from the given diagram with the given information.
Given: ¯MN≅¯KN,∠PMN≅∠NKL
- Prove that ∠1 ≅ ∠2 from the given diagram with the given information.
Given: TS≅¯TV,¯SR≅¯VW
- Find the measure of each angle in the given triangle.
m∠A=x°;m∠B=(4x)° and m∠C=(5x)°.
- 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
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