Triangles by Sides and by Angles

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.