# Questions on exterior angle of triangle

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.

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.

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