Congruent Triangles: Problems and Solutions

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:

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 △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.