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 :

m∠1 + m∠2 + m∠3 = 180 -----(1)

So, m∠3 + m∠4 = 180 -----(2)

(2) = (1)

m∠1 + m∠2 + m∠3 = m∠3 + m∠4

m∠1 + m∠2 = m∠4

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.

21^º + 34^º = x^º

Step 3 :

55^º = x^º

So, the value of x is 55^º.

Problem 2 :

Solve for x and find m ∠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 + 51^º = 100^º

Step 3 :

Solve the equation for x.

(2x + 3)^º + 51^º = 100^º

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 48^º.

Problem 3 :

Solve for x and find m ∠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.

60^º + 2x^º = 94^º

Subtract 60^º from both sides.

60^º - 60^º + 2x^º = 94^º - 60^º

2x^º = 34

Divide both sides by 2.

2x^º/2 = 34/2

x^º = 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 34^º.

Problem 4 :

Solve for x.

Solution :

Step 1 :

m ∠DCA = m ∠CAB + m ∠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 = 180^º

Step 2 :

Substitute the given angle measures.

50^º + 62^º + m ∠C = 180^º

112 + m ∠C = 180^º

Subtract 112 from both sides.

m ∠C = 180 - 112

x = 68

Step 3 :

In a triangle ∠DEF

m ∠D + m ∠E + m ∠F = 180^º

Step 4 :

Substitute the given angle measures.

53^º + m ∠E + 80^º = 180^º

133 + m ∠E = 180^º

m ∠E = 180 - 133

m ∠E = 47

Step 5 :

∠EOC = 180 – (∠ABC + ∠DEF)

= 180 - 115

= 65

So, the value of x is 65.

Problem 6 :

Solve for x, find m ∠ 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.

x^º + 90^º = 122^º

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 = 56^º

∠CDA = x^º

∠ABC + ∠ACB + ∠BAC = 180

2∠ABC + 56 = 180

2∠ABC = 180 - 56

2∠ABC = 124

∠ABC = 124/2

∠ABC = 62

∠ACD = 56 + 62 ==> 118

∠CAD = ∠CDA = x

x + x + 118 = 180

2x + 118 = 180

2x = 62

x = 31

Determine the unknown angle measures in the figure.

Problem 8 :

Describe and correct the error in finding m∠1.

Solution :

Sum of interior angles of triangle = 180 degree

115 + 39 + m∠1 = 180

154 + m∠1 = 180

m∠1 = 180 - 154

m∠1 = 26

Problem 9 :

Describe and correct the error in finding m∠1.

Solution :

Exterior angle = sum of remote interior angles

m∠1 = 80 + 50

m∠1 = 130

So, the measure of exterior angle is 130 degree.

Problem 10 :

The measure of one of the base angles of an isosceles triangle is 42°. The measure of an exterior angle at the vertex of the triangle is

1) 42°        2) 84°         3) 96°       4) 138°

Solution :

Since it is isosceles triangle, the same side will make equal angle measures. Let 42 degree the equal angle measure, then the third angle measure will be x

42 + 42 + x = 180

84 + x = 180

x = 180 - 84

= 96

Option 3) is correct.