Congruent Triangles

Key Concepts

• Identify congruent parts

Congruent Triangles

Introduction

What are congruent figures? 

Two figures are said to be congruent if they have the same corresponding side lengths and the corresponding angles. 

What are similar figures? 

Any two figures have the same shape, but their size is not the same. 

Identify congruent parts 

Congruence Statement: 

From the above figures, the corresponding sides and the corresponding angles of both the triangles are equal. 

Properties of congruence: 

The following properties define an equivalence relation. 

1. Reflexive Property – For all angles of A, ∠A≅∠A. An angle is congruent to itself. 

2. Symmetric Property – For any angles of A and B, if ∠A≅∠B, then ∠B≅∠A. Order of congruence is the same. 

3. Transitive Property – For any angles A, B, and C, if ∠A≅∠B and ∠B≅∠C, then ∠A≅∠C. If two angles are congruent to a third angle, then the first two angles are also congruent. 

Example 1: Identify the pairs of congruent corresponding parts in the figure. 

Solution: From the given figure, 

∆JKL ≌ ∆TSR 

Corresponding angles: ∠J ≌ ∠T, ∠K ≌ ∠S, ∠L ≌ ∠R 

Example 2: Find the values of x and y in the diagram using properties of congruent figures if DEFG ≌ SPQR. 

Solution:  

Given that DEFG ≌ SPQR, 

We know that FG ≌ QR. 

⇒ FG = QR 

12=2x−4

16=2x

8=x

Since ∠ F ≌ ∠Q. 

m∠F= m∠Q

68°=(6y+x)°

68=6y+8

6y=68−8

y=606=10

Third angles theorem: 

If two angles of one triangle are congruent to two angles of another triangle, then the third angle is also congruent. 

Given:

∠ A ≅ ∠D,

∠B ≅ ∠E. 

To Prove:

∠ C ≅ ∠F. 

Proof:  

If ∠ A ≅ ∠D, and ∠B ≅ ∠E (given) 

m∠ A = m∠D, m∠ B = m∠E (Congruent angles) 

m∠A + m∠B  + m∠C = 180° (Triangle sum theorem) 

m∠D + m∠E  + m∠F = 180° 

m∠A + m∠B  + m∠C = m∠D + m∠E + m∠F (Substitution property) 

m∠D + m∠E  + m∠C = m∠D + m∠E + m∠F (Substitution property) 

m∠C  = m∠F (Subtraction property of equality) 

∠C ≅ ∠F. (Congruent angles) 

Example 3: Find

m∠BDC

in the given figure. 

Solution:  

From the given figure, we have 

∠A ≅ ∠B and ∠ADC ≅∠BCD 

⇒∠ACD ≅ ∠BDC (Third angles theorem) 

m∠ACD  + m∠CAN  + m∠CDN = 180° (Triangle sum theorem) 

m∠ACD  = 180° – 45° – 30° = 105°. 

m∠ACD  = m∠BDC = 105°. (Congruent angles) 

Properties of congruent triangles: 

1. Reflexive property of congruent triangles: 

For any triangle ABC, ∆ABC ≌ ∆ABC. 

2. Symmetric property of congruent triangles: 

If ∆ABC ≌ ∆DEF, then ∆DEF ≌ ∆ABC. 

3. Transitive property of congruent triangles: 

If ∆ABC ≌ ∆DEF and ∆DEF ≌ ∆JKL, then ∆ABC ≌ ∆JKL. 

Example 4: Prove that the triangles are congruent. 

Solution: 

Given: AD ≌ CB, DC ≌ BA, ∠ ACD ≅ ∠ CAB, ∠ CAD ≅ ∠ ACB    

To prove: ∆ACD ≅ ∆CAB    

proof:    

a. Use the reflexive property to show that AC ≌ AC 

b. Use the third angles theorem to show that ∠ B ≅ ∠ D 

Plan in action: 

Exercise

1. Identify all pairs of congruent corresponding parts in the following figure:

2. Find the value of x in the given figure.

3. Find the values of x and y in the diagram.

4.  Write the proof with the help of a diagram.

Given: WX ⊥ VZ at Y, Y is the midpoint of WX, VW ≅ VX, and VZ bisects ∠WVX.

Proof: ∆VWY ≅ ∆VXY.

5. Find  from the given figure.

6. Find the values of x and y in the diagram.

7. Find the value of x in the figure.

8. Write the congruence statements for the given figure.

9. Identify the congruent corresponding parts.

10. Find  in the figure.

What have we learned

• Understand congruent of two figures.
• Understand the congruence statements.
• Find corresponding angles and corresponding sides.
• Identify congruent parts for the given triangles.
• Find the values using properties of congruent figures.
• Understand third angles theorem.
• Understand properties of congruent triangles.
• Solve problems on congruent figures.

Summary

Properties of congruence:

The following properties define an equivalence relation.

1.Reflexive property – For all angles of A, ∠A≅∠A. An angle is congruent to itself.

2. Symmetric property – For any angles of A and B, if ∠A≅∠B, then ∠B≅∠A. Order of congruence is the same.

3.Transitive property – For any angles A, B, and C, if ∠A≅∠B and ∠B≅∠C, then ∠A≅∠C. If two angles are congruent to a third angle, then the first two angles are also congruent.