Solve Problem Using Angle Relationship

Key Concepts

• What is an angle.
• Solve problems involving adjacent and vertical angles.
• Solve problems involving complementary and supplementary angles.
• Find the measure of an unknown angle.

8.4 Solve problems using angle relationship 

Angle: 

The measure of rotation of a ray from its initial position to terminal position about a fixed point ‘O’ is defined as an angle. 

(OR) 

In geometry, an angle can be defined as the figure formed by two rays meeting at a commonend point. The angle is represented by the symbol “

∠

”. The angle above is written as

∠∠AOB. 

Parts of an angle: 

Arms: The two rays that are joining to form an angle are called arms of an angle. Here, OA and OB are arms of

∠

AOB. 

Vertex: The common endpoint at which the two rays meet to form an angle is called the vertex. Here, point O is the vertex of

∠AOB. 

Examples of angles: 

1. Angle formed in a pair of scissors 

2. Angle formed between the spokes of a wheel 

3. Angle formed between index and thumb finger 

Types of angles: 

Angles can be classified based on their measurements as: 

1. Acute angle 

2. Right angle 

3. Obtuse angle  

4. Straight angle  

5. Reflex angle  

6. Complete angle 

Adjacent angles: 

Two angles are said to be adjacent angles if they share a common vertex and side. Adjacent angles can be a complementary angle or a supplementary angle when they share the common vertex and side. 

Here, we observe that

∠AOB and∠BOC have a common arm, i.e., OB and a common vertex O.  

Therefore,∠AOB and∠BOC can be called adjacent angles. 

Vertical angles: 

When a pair of lines intersect each other at a common point, then four angles are formed. Two non-adjacent angles that are opposite and equal are called vertical angles or vertically opposite angles. 

Example: 

Complementary angles: 

Two angles are said to be complementary if they add up to 90⁰. Complementary angles can be adjacent or non-adjacent angles. 

Supplementary angles: 

Two angles are said to be supplementary if they add up to 180⁰. Supplementary angles can be adjacent or non-adjacent angles. They form a straight line if they are adjacent angles. 

8.4.1 Solve problems involving adjacent and vertical angles 

Example 1: From the figure given below, find the value of ‘x’ and the measure of

∠AOC and∠BOD.  

Solution: We know that vertical angles have equal measures.

∠AOC and∠BOD are the vertical angles. 

Step 1: Since

∠AOC and∠BOD is equal, we write m

∠AOC = m

∠BOD 2x + 96 = 5x – 9 

2x + 96 + 9 = 5x – 9 + 9 (∵Add 9 on both the sides) 

2x + 105 = 5x 

2x + 105 – 2x = 5x – 2x (∵Subtract 2x from both sides) 

105 = 3x  

105/3=3×3 (∵Divide by 3 on both sides of the equation) 

35 = x 

Therefore, value of x = 35. 

m ∠AOC = 2x + 96 m ∠BOD = 5x – 9 

                = 2(35) + 96    = 5(35) – 9 

  = 70 + 96    = 175 – 9 

  = 166    = 166  

Example 2:

∠MNQ and ∠PNR are vertical angles. Find the value of ‘x’. 

Solution: We know that vertical angles have equal measures.

∠MNQ and∠PNR are the vertical angles. 

Step 1: Since

∠MNQ and ∠PNR is equal, we write m

∠MNQ = m ∠PNR 

3x – 6 = 114 

3x – 6 + 6 = 114 + 6 (∵Add 9 on both the sides) 

3x = 120 

3x / 3 = 120 / 3 (∵Divide by 3 on both sides of the equation) 

x = 40 

Therefore, value of x = 40. 

m ∠MNQ = 3x – 6  

                 = 3(40) – 6  

   = 120 – 6     

   = 114   

8.4.2 Solve problems involving complementary and supplementary angles 

Example 1: Determine the value of ‘x’ in the given figure. 

Solution:  

We know that sum of two adjacent angles is complementary by observing the figure. 

Step 1: m∠AOB + m∠BOC = 90⁰ 

x + 62 = 90⁰ 

x = 90⁰ – 62 

x = 28⁰ 

Example 2: Determine the value of ‘x’ if the following two angles are supplementary. 

Solution:  

We know that sum of two angles is supplementary. 

Step 1:

x / 2+ x /3 = 180 

3 × x+2 ×x /  6 = 180 

3x+2x / 6= 180 

5x / 6 = 180 

5x X 6 / 6 = 180 × 6 (∵Multiply by 6 on both the sides) 

5x = 1080 (∵Divide by 5 on both the sides) 

5x / 5 =10805

x = 216⁰ 

8.4.3 Find the measure of an unknown angle 

Example 1: If three lines AB, CD and EF intersect each other at a common point ‘O’ such that AB is perpendicular to CD, determine the value of y and

∠AOE. 

Solution: We know that AB is perpendicular to CD. So, m

∠AOC = 90⁰. 

Step 1: Use vertical angles to find the value of ‘y’. 

m∠COE = m∠DOF 

5y + 16 = 3y + 20 

5y + 16 – 3y = 3y + 20 – 3y (∵Subtract 3y on both the sides) 

2y + 16 = 20 

2y + 16 – 16 = 20 –16 (∵Subtract 16 on both the sides) 

2y = 4 

2y / 2 = 4 / 2 (∵Divide by 2 on both the sides) 

y = 2 

∠COE = 5y + 16 

            = 5(2) + 16 

            = 10 + 16 

∠COE = 26 

Step 2: We observe that

∠AOE and∠COE are complementary. Since AB and CD are perpendicular to each other. 

∠AOC = 90⁰ 

∠AOE +∠COE = 90⁰ 

∠AOE + 26⁰ = 90⁰ 

∠AOE = 90⁰ – 26⁰ 

∠AOE = 64⁰ 

Exercise:

1. How are angles formed by intersecting lines related?
2. Can we say that vertical angles are also adjacent angles? Explain.
3. List two pairs of adjacent angles and vertical angles from the figure given below.

• In the given figure, AB, CD and EF intersect at O. Find the values of x, y and z if it is being given that x:y:z =2:3:5

• Two angles are equal and complementary to each other. Determine the measure of the two angles.
• Determine the value of ‘x’ in the figure given below.

• Determine the complement angle of 35⁰.
• Two angles are equal and supplementary to each other. Determine the measure of the two angles.
• Determine the value of a from the figure given below.

• Determine the supplement angle of 123⁰.

What have we learned?

• What an angle is?
• Solving problems involving adjacent and vertical angles.
• Solving problems involving complementary and supplementary angles.
• Determining the measure of an unknown angle