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# Congruent Triangles

Sep 10, 2022

## Key Concepts

• Identify congruent parts

## Congruent Triangles

### 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.
1. Symmetric Property – For any angles of A and B, if ∠A≅∠B, then ∠B≅∠A. Order of congruence is the same.
1. 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)

mA = mD, m B = mE (Congruent angles)

mA + m+ mC = 180° (Triangle sum theorem)

mD + m+ mF = 180°

mA + m+ mC = mD + mE + mF (Substitution property)

mD + m+ mC = mD + mE + mF (Substitution property)

m= mF (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)

mACD  + mCAN  + mCDN = 180° (Triangle sum theorem)

mACD  = 180° – 45° – 30° = 105°.

mACD  = mBDC = 105°. (Congruent angles)

### Properties of congruent triangles:

1. Reflexive property of congruent triangles

For any triangle ABC, ∆ABC ≌ ∆ABC.

1. Symmetric property of congruent triangles:

If ∆ABC ≌ ∆DEF, then ∆DEF ≌ ∆ABC.

1. 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:
1. Find the value of x in the given figure.
1. Find the values of x and y in the diagram.
1.  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.

1. Find  from the given figure.
1. Find the values of x and y in the diagram.
1. Find the value of x in the figure.
1. Write the congruence statements for the given figure.
1. Identify the congruent corresponding parts.
1. 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.

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