In this chapter we will discuss questions related to exterior angle theorem of triangle.

All the questions are to the standard of grade 09.

**Question 01**

Observe the below image carefully and find value of all internal angle of triangle ABC.

**Solution****Consider the straight line EC**.

We know that in straight line, sum of all adjacent angle measure 180 degree.

∠EBA + ∠ABC = 180

136 + ∠ABC = 180

∠ABC = 180 – 136

∠ABC = 44 degree

Similarly **consider the adjacent angles in straight line BD**.

∠ACB + ∠ACD = 180

∠ACB + 104 = 180

∠ACB = 180 – 104

∠ACB = 76 degree

**Now consider triangle ABC.**

We know in triangle, sum of all internal angle measure 180 degree.

∠ABC + ∠ACB + ∠A = 180

44 + 76 + ∠A = 180

120 + ∠A = 180

∠A = 180 – 120

∠A = 60 degree

Hence, we calculated value of all internal angles of triangle.

**Question 02**

Observe the below image and calculate value of ∠ABC using exterior angle theorem.

**Solution**

Note that **∠EAF & ∠BAC are vertically opposite angles**.

∠EAF = ∠BAC = 45 degree

**Applying angle sum property in ∠ACD**.

∠ACD = ∠BAC + ∠ABC

105 = 45 + ∠ABC

∠ABC = 105 – 45

∠ABC = 60 degree

Hence, ∠ABC of triangle measures 60 degrees.

**Question 03**

Observe the below image and find the measure of ∠AEC using exterior angle theorem of triangle.

**Solution**

Note that AB & CD are parallel lines.

∠BAD and ∠ADC are alternate interior angles.

∠BAD = ∠ADC = 52 degree

**Now consider triangle EDC.**

Side DE of triangle EDC is extended outside to form exterior angle ∠AEC.

**Using exterior angle theorem, we can write;**

∠AEC = ∠ECD + ∠EDC

∠AEC = 40 + 52

∠AEC = 92 degree

Hence, value of ∠AEC is 92 degrees.

**Question 04**

Observe the below image and find the value of x using exterior angle theorem of triangle.

**Solution****EC is a straight line.**

We know that sum of alternate angle in straight line measures 180 degree.

∠BAE + ∠BAC = 180

120 + ∠BAC = 180

∠BAC = 180 – 120

∠BAC = 60 degree

Now applying **exterior angle theorem in ∠ACD.**

∠ACD = ∠x + ∠BAC

112 = ∠x + 60

∠x = 112 – 60

∠x = 52 degree

Hence, ∠x measures 52 degree.

**Question 05**

In the below figure ∠ACE = 90 degree and ∠A, ∠B & ∠C are in ratio 3 : 2 : 1. Using exterior angle theorem of triangle, find measure of ∠ECD.

**Solution**

Let ∠A = 3x

∠B = 2x

∠C = x

We know that **sum of all angle of triangle measure 180 degree.**

3x + 2x + x = 180

6x = 180

x = 30

Getting value of all the angles.

∠A = 3 (30) = 90

∠B = 2(30) = 60

∠C = 30 degree

**Now applying exterior angle theorem in ∠ACD**;

∠ACD = ∠A + ∠B

∠ACD = 90 + 60

∠ACD = 150 degree

**From above image we can write;**

∠ACD = ∠ACE + ∠ECD

150 = 90 + ∠ECD

∠ECD= 60 degree

Hence, ∠ECD measures 60 degree.

**Question 06**

In the below triangle ABC, ∠B = ∠C. The line AE bisect ∠DAC into two equal halves. Prove that line AE is parallel to BC.

**Solution**

Since AE bisect ∠DAC into two equal halves, we can write;

∠DAE = ∠ EAC

**Applying exterior angle theorem in triangle**, we can write;

∠DAC = ∠B + ∠C

∠DAE + ∠EAC = ∠C + ∠C

2∠EAC = 2∠C

∠EAC = ∠C

Both ∠EAC & ∠C are alternate interior angles.

Since both the angles are equal, it means that AE & BC are parallel lines intersected by transversal AC.

Hence Proved.

**Question 07**

In the below triangle ∠A = ∠B and ∠ACD = 100 degree. Calculate the value of ∠A of triangle.

**Solution**

Let ∠A = ∠B = x

Applying exterior angle theorem in ∠ACD;

∠ ACD = ∠A + ∠B

100 = x + x

2x = 100

x = 50 degree

Hence, ∠A measures 50 degrees.