# EXTERIOR ANGLE THEOREM OF TRIANGLE

An interior angle of a triangle is formed by two sides of the triangle. An exterior angle is formed by one side of the triangle and the extension of an adjacent side.

Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle.

Extend the base of the triangle and label the exterior angle as 4.

The Triangle Sum Theorem states :

m1 + m2 + m3 = 180 -----(1)

So, m3 + m4 = 180 -----(2)

(2) = (1)

m1 + m2 + m3 = m3 + m4

m1 + m2 = m4

Problem 1 :

Solve for x.

Solution :

Step 1 :

Write the Exterior Angle Theorem as it applies to this triangle.

m A + m B = m ACD

Step 2 :

Substitute the given angle measures.

21º + 34º = xº

Step 3 :

55º = xº

So, the value of x is 55º.

Problem 2 :

Solve for x and find ∠A.

Solution :

Step 1 :

Write the Exterior Angle Theorem as it applies to this triangle.

m A + m B = m C

Step 2 :

Substitute the given angle measures.

(2x + 3)º + 51º = 100º

Step 3 :

Solve the equation for x.

(2x + 3)º + 51º = 100º

2x + 3 + 51 = 100

2x + 54 = 100

Subtract 54 from both sides.

2x + 54 – 54 = 100 – 54

2x = 46

Divide both sides by 2.

2x/2 = 46/2

x = 23

Step 4 :

Use the value of x to find m A.

m A = 2x + 3

= 2(23) + 2

= 46 + 2

m A= 48

So, the value of m A is 48º.

Problem 3 :

Solve for x and find ∠B.

Solution :

Step 1 :

Write the Exterior Angle Theorem as it applies to this triangle.

m A + m B = m C

Step 2 :

Substitute the given angle measures.

60º + 2xº = 94º

Subtract 60º from both sides.

60º - 60º + 2xº = 94º - 60º

2xº = 34

Divide both sides by 2.

2xº/2 = 34/2

xº = 17

Step 3 :

Use the value of x to find m B.

m B = 2xº

= 2(17)

= 34

So, the value of m B is 34º.

Problem 4 :

Solve for x.

Solution :

Step 1 :

DCA CAB + ∠ABC

Step 2 :

Substitute the given angle measures.

6x - 7 = 2x + 103 - x

6x - 7 = x + 103

6x - x = 103 + 7

5x = 110

x = 110/5

x = 22

Problem 5 :

Solve for x.

Solution :

Step 1 :

Write the Exterior Angle Theorem as it applies to this triangle.

In a triangle ABC

m A + m B + m ∠C = 180º

Step 2 :

Substitute the given angle measures.

50º + 62º∠C = 180º

112 + ∠C = 180º

Subtract 112 from both sides.

∠C = 180 - 112

x = 68

Step 3 :

In a triangle DEF

m D + m E + m F = 180º

Step 4 :

Substitute the given angle measures.

53º+ 80º = 180º

133 + E = 180º

E = 180 - 133

= 47

Step 5 :

∠EOC = 180 – (ABC + DEF)

= 180 - 115

= 65

So, the value of x is 65.

Problem 6 :

Solve for x, find ∠ A.

Solution :

Step 1 :

Write the Exterior Angle Theorem as it applies to this triangle.

m A + m B = m C

Step 2 :

Substitute the given angle measures.

xº + 90º = 122º

Step 3 :

x + 90 = 122

Subtract 90 from both sides.

x + 90 – 90 = 122 - 90

x = 32

Step 4 :

Use the value of x to find m A.

m A = xº

= 32

So, the value of m A is 32.

Problem 7 :

Solve for x.

Solution :

In triangle ABC,

ABC = ACB

BAC = 56º

CDA = xº

ABC + ACB + ∠BAC = 180

2ABC + 56 = 180

2ABC = 180 - 56

2ABC = 124

ABC = 124/2

ABC = 62

ACD = 56 + 62 ==> 118

x + x + 118 = 180

2x + 118 = 180

2x = 62

x = 31

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