Introduction
Applications of Congruent Triangles
Example 1: In the given figure, AB = BC and AD = CD. Show that BD bisects AC at right angles.
![Applications of Congruent Triangles](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1634.png)
Solution:
From the given figure, ∆ABD ≅ ∆CBD
Given: AB = BC and AD = CD
To prove: ∠BEA = ∠BEC = 90° and AE = EC.
Proof:
![Proof:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1635.png)
Now, from ∆ABE and ∆CBE,
![Proof:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1613.png)
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°
![Solution:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1636.png)
Example 3: In the given figure, PQ = PS and ∠QPR = ∠SPR. Prove that ∆ PQR ≅ ∆PSR, Use SAS congruence postulate.
![Example 3](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1612.png)
Solution:
![Solution:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1615.png)
Example 4: Identify the congruent triangle in the given figure.
![Example 4](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1614.png)
Solution:
![Solution:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1617.png)
Example 5:
Write a 2-column proof for the given figure.
![Example 5:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1618.png)
Given: BD is an angle bisector of ∠CDA, ∠C ≅ ∠A
To prove: △CBD ≅ ∠ABD
Solution:
![Solution](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1616.png)
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 1](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1619.png)
![Step 1](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1620.png)
Step 2:
Draw an arc with radius BC and center E. Label the intersection F.
![Step 2](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1621.png)
![Step 2](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1622.png)
Step 3:
Draw DF
.
![Draw DF](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1623.png)
![Draw DF](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1624.png)
Example 6:
Write a proof to verify that the construction for copying an angle is valid.
Solution:
![](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/Screenshot-2360.png)
To prove:
![To prove:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1625.png)
Plan for Proof:
Show that △CAB ≅ △FDE, so we can conclude that the corresponding parts ∠A and ∠D are congruent.
Plan in action:
![Plan for Proof:](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1626-1024x323.png)
Exercise
- Write a plan for proof for the given information to prove that ∠1 ≅ ∠2.
![Write a plan for proof for the given information to prove that ∠1 ≅ ∠2.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1637.png)
- Prove that ∠VYX ≅ ∠WYZ in the given figure.
![Prove that ∠VYX ≅ ∠WYZ in the given figure.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1638.png)
- Prove that¯FL≅¯HN in the given diagram.
![Prove that¯FL≅¯HN in the given diagram.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1639.png)
- Prove that¯FL≅¯HN in the given diagram.
![Prove that¯FL≅¯HN in the given diagram.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1640.png)
- Prove that ¯AC≅¯GE in the given diagram.
![Prove that ¯AC≅¯GE in the given diagram.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1641.png)
- Write a two-column proof from the given diagram.
![Write a two-column proof from the given diagram.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1642.png)
- 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.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1643.png)
- Prove that ∠1 ≅ ∠2 from the given diagram with the given information.
Given: TS≅¯TV,¯SR≅¯VW
![Prove that ∠1 ≅ ∠2 from the given diagram with the given information.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1644.png)
- Find the measure of each angle in the given triangle.
![m∠A=x°;m∠B=(4x)° and m∠C=(5x)°.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1645.png)
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)°.](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1646.png)
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
![Concept Map](https://www.turito.com/learn-internal/wp-content/uploads/2022/09/image-1633-1024x482.png)
![applications of congruent triangles](/_next/image?url=https%3A%2F%2Fwww.turito.com%2Flearn-internal%2Fwp-content%2Fuploads%2F2022%2F09%2Fapplications-of-congruent-triangles.png&w=1920&q=50)
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