### 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:**

- Angle formed in a pair of scissors

- Angle formed between the spokes of a wheel

- Angle formed between index and thumb finger

**Types of angles:**

Angles can be classified based on their measurements as:

- Acute angle

- Right angle

- Obtuse angle

- Straight angle

- Reflex angle

- 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:

- How are angles formed by intersecting lines related?
- Can we say that vertical angles are also adjacent angles? Explain.
- 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

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