Meaning of an Angle

An angle is the amount of turning or rotation between two straight lines that meet at a point. The point where the two lines meet is called the vertex, and the lines themselves are called the arms or sides of the angle.

In simple words, an angle measures how much one line turns away from another around a point.

Arm 1
   \
    \
     \
      \
       \
        \  ← Angle
         \
          \
           B-----------C
                ↑
             Vertex
  

Here, ∠ABC is the angle, B is the vertex, and BA and BC are the arms.

An angle is formed when two straight lines meet at a point or when one straight line turns about a fixed point. It is measured in degrees (°).

Example of Turning

If a door turns about its hinge, the hinge is the vertex, the sides of the door are the arms, and the space opened by the door is the angle.

Unit of Measurement

The standard unit for measuring an angle is the degree (°). There are:

• 360° in a complete revolution


• 180° in a straight line


• 90° in a right angle

Diagram of a Circle and Line:

Full circle = 360°
Half circle = 180°
Quarter circle = 90°
  

Instrument for Measuring Angles

A protractor is used to measure or draw angles in degrees. It is a semicircular or circular instrument marked from 0° to 180° or 0° to 360°.

Parts of an Angle

• Vertex — The common point where the two lines meet.


• Arms or Sides — The two straight lines that form the angle.


• Angle Space — The opening or region between the two arms.

      Arm 1
       \
        \
         \
          \
           Vertex (O)
            \
             \
              Arm 2
  

Angles are classified according to their sizes.

(a) Acute Angle

Definition: An acute angle is an angle less than 90°. It is smaller than a right angle.

Examples: 25°, 40°, 60°

\
 \
  \__
  

Real-life examples: The tip of a slice of pizza; the corner of a road turning sharply.

(b) Right Angle

Definition: A right angle is an angle exactly equal to 90°. It forms a perfect corner — like the corner of a square or a book.

|__
  

Real-life examples: The corner of a wall; the corner of a sheet of paper.

(c) Obtuse Angle

Definition: An obtuse angle is an angle that is greater than 90° but less than 180°. It is wider than a right angle.

Examples: 120°, 135°

\
 \
  \
   \____
  

Real-life examples: The angle formed when you open a door wide; the slant of a hill.

(d) Straight Angle

Definition: A straight angle is exactly 180°. It looks like a straight line. It is equal to two right angles combined.

-------------------------
  

Real-life examples: The top edge of a ruler; the horizon line where the sky meets the earth.

(e) Reflex Angle

Definition: A reflex angle is an angle greater than 180° but less than 360°. It looks like a large opening or turning beyond a straight line.

Examples: 220°, 300°

\
 \
  \_________
          \
           \
            \
  

Real-life examples: The turning of a clock hand from 12 to 8; the wide opening of a car door.

(f) Complete Angle (Full Angle)

Definition: A complete angle or full rotation is exactly 360°. It represents a full turn around a point.

     ___
   /     \
  |       |
   \ ___ /
  

Real-life examples: The second hand of a clock making a full circle; the wheel of a car turning one complete round.

(a) Complementary Angles

Two angles are complementary if their sum is 90°.

Example: 30° and 60° are complementary (30° + 60° = 90°).

|__
 \
  \
  

(b) Supplementary Angles

Two angles are supplementary if their sum is 180°.

Example: 70° and 110° are supplementary (70° + 110° = 180°).

(c) Vertically Opposite Angles

When two straight lines cross each other, the opposite angles are equal.

     \
      \
       X
      / \
     /   \
  

If the lines intersect, the pairs of opposite angles formed are equal.

(d) Adjacent Angles

Two angles that share a common vertex and one common arm, but do not overlap, are called adjacent angles.

(e) Angles on a Straight Line

Angles that lie on a straight line add up to 180°.

A------B------C
  

∠ABD + ∠DBC = 180°

(f) Angles at a Point

The sum of all angles around a point = 360°.

      |
   ---O---
      |
  

When two parallel lines are cut by another line (called a transversal), certain angles are formed:

   A---------B
     \     /
      \   /
       \ /
        X
       / \
      /   \
   C---------D
  

Here, AB ∥ CD and the slanting line is the transversal.

Types of Angles Formed

• Corresponding Angles – These are equal.


• Alternate Angles – These are also equal.


• Co-interior (Allied) Angles – These add up to 180°.

Bisection and Construction of Angles

Angles can be constructed (drawn) and bisected (divided into two equal parts) using:

• Ruler


• Compass


• Protractor

To Construct 60°

Steps:

1. Draw a straight line AB.


2. With compass on A, draw an arc that cuts AB at C.


3. With compass on C, draw another arc to meet the first at D.


4. Draw a line from A through D. ∠DAB = 60°.

To Bisect an Angle

Steps:

1. Draw ∠ABC.


2. With compass on vertex B, draw an arc to cut both arms at points P and Q.


3. With compass on P and Q, draw arcs to intersect at R.


4. Draw a line from B to R. This line divides ∠ABC into two equal parts.

PROPERTIES OF ANGLES

Property	Description	Total
Angles on a straight line	Sum of angles = 180°	180°
Angles at a point	Sum of angles = 360°	360°
Angles in a triangle	Sum of angles = 180°	180°
Angles in a quadrilateral	Sum of angles = 360°	360°
Vertically opposite angles	Equal	—
Alternate and corresponding angles	Equal when lines are parallel	—

REAL-LIFE USES OF ANGLES

• Construction and architecture (for corners, slopes, and measurements)


• Navigation (bearings and directions)


• Sports (aiming, passing, and shooting)


• Art and design (patterns and geometry)


• Engineering (machine parts and joints)

QUICK FACTS TO REMEMBER

• Angle is measured in degrees (°).


• Right angle = 90°, Straight angle = 180°, Full angle = 360°.


• Complementary → 90°, Supplementary → 180°.


• Angles at a point = 360°.


• Vertically opposite angles are equal.


• Alternate and corresponding angles are equal when lines are parallel.

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Meaning of a Polygon

A polygon is a closed plane figure made up of three or more straight sides joined end to end.

Each polygon is named according to the number of sides it has.

Examples of Polygons

Name of Polygon	Number of Sides	Example
Triangle	3	ΔABC
Quadrilateral	4	Square, Rectangle
Pentagon	5	Regular pentagon
Hexagon	6	Regular hexagon
Heptagon	7	Regular heptagon
Octagon	8	Regular octagon
Nonagon	9	—
Decagon	10	—

Diagram Examples

Triangle (3 sides):

   /\
  /__\

Quadrilateral (4 sides):

 ______
|      |
|______|

Pentagon (5 sides):

   /\
  /  \
 /____\
 \    /
  \__/

Types of Polygons

Regular Polygon: All sides and all angles are equal (e.g., square, regular hexagon).

Irregular Polygon: Sides and angles are not all equal (e.g., a rectangle or any uneven shape).

Interior and Exterior Angles

Interior angles are inside the polygon.

Exterior angles are outside the polygon, formed by extending one side.

Diagram (for one vertex of a polygon):

     /
    /|
   /_|
  (inside)

The inner part (inside) = interior angle

The outer part (outside) = exterior angle

Derivation of the Formula for Sum of Interior Angles

A polygon can be divided into triangles by drawing diagonals from one vertex.

Since the sum of angles in one triangle = 180°, and the polygon can be divided into (n − 2) triangles,

Sum of interior angles = (n − 2) × 180°

where n = number of sides.

Quadrilateral (4 sides):

A------B
|     /|
|   /  |
| /    |
C------D

From vertex A, one diagonal (AC) divides the quadrilateral into 2 triangles.

Hence, Sum of angles = (4 − 2) × 180° = 2 × 180° = 360°.

For a Pentagon (5 sides):

   A
  / \
 B---E
 |   /
 |  /
 C-D

From vertex A, diagonals divide the shape into 3 triangles.

Therefore: Sum of angles = (5 − 2) × 180° = 3 × 180° = 540°.

Formula for One Interior Angle (Regular Polygon)

If the polygon is regular (all angles equal):

Each interior angle = ((n − 2) × 180°) ÷ n

Worked Examples

Example 1: Find the sum of the interior angles of a hexagon.

Solution: n = 6
Sum = (6 − 2) × 180° = 4 × 180° = 720°

Example 2: Find each interior angle of a regular pentagon.
Solution: n = 5
Each = [(5 − 2) × 180°] ÷ 5 = (3 × 180°) ÷ 5 = 540° ÷ 5 = 108°

Example 3: Find the sum of the interior angles of a decagon.

Solution: n = 10
Sum = (10 − 2) × 180° = 8 × 180° = 1440°

Relationship Between Interior and Exterior Angles

At each vertex of a polygon:

Interior angle + Exterior angle = 180°

Sum of All Exterior Angles

No matter how many sides a polygon has, the sum of all exterior angles is always 360°.

Sum of exterior angles = 360°

To Find Each Exterior Angle (Regular Polygon)

Each exterior angle = 360° ÷ n

Worked Examples

Example 4: Find the size of each exterior angle of a regular hexagon.

Solution: n = 6

Each = 360° ÷ 6 = 60°

Example 5: Find the size of each interior angle of a regular hexagon.

Solution: Each interior = 180° − 60° = 120°

Example 6: Find the number of sides of a regular polygon whose exterior angle is 40°.
Solution: Each exterior = 360° ÷ n

So, 40° = 360° ÷ n

n = 360° ÷ 40° = 9 sides (nonagon)

Summary Table

Polygon	Number of Sides (n)	Sum of Interior Angles (n−2)×180°	Each Interior Angle (Regular)	Each Exterior Angle (Regular)
Triangle	3	180°	60°	120°
Quadrilateral	4	360°	90°	90°
Pentagon	5	540°	108°	72°
Hexagon	6	720°	120°	60°
Heptagon	7	900°	128.6°	51.4°
Octagon	8	1080°	135°	45°
Nonagon	9	1260°	140°	40°
Decagon	10	1440°	144°	36°

Quick Facts to Remember

1. Sum of interior angles of a polygon = (n − 2) × 180°


2. Each interior angle (regular polygon) = [(n − 2) × 180°] ÷ n


3. Sum of exterior angles of any polygon = 360°


4. Each exterior angle (regular polygon) = 360° ÷ n


5. Interior angle + Exterior angle = 180°


6. Triangles = 180°, Quadrilaterals = 360°, Pentagons = 540°, etc.

Real-Life Applications

1. Used in architecture and engineering (designing corners and sides of structures).


2. Used in graphic design (drawing polygons and patterns).


3. Used in navigation and robotics (calculating turns).


4. Used in road design (curves and intersections).

Practice Exercises

1. Find the sum of the interior angles of a 12-sided polygon.


2. Find each interior angle of a regular octagon.


3. Find the number of sides of a polygon if the sum of interior angles is 1260°.


4. If each exterior angle of a regular polygon is 30°, find the number of sides.


5. Find each exterior and interior angle of a regular decagon.

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Meaning of Angle of Elevation

The angle of elevation is the angle formed when you look upward from a horizontal line to an object that is above the level of your eyes.

It helps us to measure how high an object appears when we look up at it.

Simple meaning:
When you lift your head or eyes to look at something above you (like a bird, a kite, or the top of a building), the angle your eyes make with the ground is called the angle of elevation.

(Angle of Elevation)

         Object (Top of a Building)
                    *
                   /|
                  / |
                 /θ |
                /___|
     Observer———Ground Level

θ = Angle of Elevation

In this diagram:

1. The observer is the person looking up.


2. The horizontal line is the ground.


3. The line of sight is the slant line from the observer’s eyes to the object.


4. The angle θ between the line of sight and the ground is called the angle of elevation.

Important Terms

Term	Meaning
Line of Sight	The straight line from the observer’s eye to the object.
Horizontal Line	The line parallel to the ground from the observer’s eyes.
Vertical Line	The line representing the height of the object (upward).

Relationship with Right-Angled Triangle

Whenever you look up to an object, the ground, your line of sight, and the height of the object form a right-angled triangle.

In that triangle:

1. The height of the object = Opposite side


2. The distance from the observer to the object = Adjacent side


3. The line of sight = Hypotenuse

Real-Life Examples of Angle of Elevation

1. Looking up at a kite in the sky.


2. Looking at the top of a tree or tower.


3. A pilot looking up at another aircraft above.


4. A person at the foot of a hill looking at the top.


5. Engineers calculating slope of roads or bridges.

Formula (Using Trigonometric Ratio — Understanding Only)

In the right-angled triangle:

tan(θ) = Opposite side (height) ÷ Adjacent side (distance)

This formula helps to find:

1. The height of the object


2. The distance from the object


3. The angle of elevation, if other values are known

Summary of Key Formulas

1. tan(θ) = Opposite ÷ Adjacent


2. sin(θ) = Opposite ÷ Hypotenuse


3. cos(θ) = Adjacent ÷ Hypotenuse


4. Height = Distance × tan(θ)


5. Distance = Height ÷ tan(θ)

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EXAMPLES ON ANGLE OF ELEVATION

Example 1: A Boy Looking at a Tree

A boy is standing 20 m away from a tree. The top of the tree is seen at an angle of elevation of 30°. Find the height of the tree.

        Top of Tree
             *
            /|
           / | h
          /  |
         /   |
        *----*
     Boy     20 m
Angle = 30°

Step 1: Write what we are given.

Distance from boy to tree = 20 m

Angle of elevation = 30°

We are asked to find the height of the tree (h).

Step 2: Recall the trigonometric ratio that connects opposite and adjacent.

Use tan(θ) = opposite ÷ adjacent

Step 3: Substitute the values.

tan(30°) = h ÷ 20

Step 4: From your calculator, tan(30°) = 0.577.

So, 0.577 = h ÷ 20

Step 5: Multiply both sides by 20.

h = 20 × 0.577 = 11.54

Therefore, the height of the tree is about 11.5 m.

---

Example 2: A Man Looking at a Flagpole

A man 1.7 m tall stands 30 m away from a flagpole. The angle of elevation to the top of the pole is 40°. Find the total height of the flagpole.

     Top of Flagpole
           *
          /|
         / | (h - 1.7)
        /  |
       /   |
      *----*
   Man     30 m
Angle = 40°

Step 1: Write what we know.

Distance = 30 m

Angle = 40°

Man’s eye level = 1.7 m

We are to find the total height of the pole = h.

Step 2: Use the tangent ratio:

tan(40°) = (h - 1.7) ÷ 30

Step 3: From calculator, tan(40°) = 0.8391

Step 4: Substitute.

0.8391 = (h - 1.7) ÷ 30

Step 5: Multiply both sides by 30.

h - 1.7 = 30 × 0.8391 = 25.17

Step 6: Add the man’s height.

h = 25.17 + 1.7 = 26.87

The flagpole is about 26.9 m tall.

---

Example 3: Girl and Building

A girl stands 10 m away from a building. The top of the building is observed at an angle of 60°. Find the height of the building.

       Top of Building
            *
           /|
          / | h
         /  |
        /   |
       *----*
     Girl   10 m
Angle = 60°

Step 1: Given:

Distance = 10 m

Angle = 60°

Find h.

Step 2: Formula:

tan(60°) = h ÷ 10

Step 3: From calculator, tan(60°) = 1.732

Step 4: Substitute and solve.

1.732 = h ÷ 10

h = 10 × 1.732 = 17.32

The building is 17.3 m high.

Example 4: Streetlight Pole

From a point 15 m away from a streetlight, the angle of elevation to the top is 45°. Find the height of the pole.

       Top of Pole
           *
          /|
         / | h
        /  |
       *---*
      15 m
Angle = 45°

Step 1: Given: Angle = 45°, Distance = 15 m

Step 2: Formula:

tan(45°) = h ÷ 15

Step 3: tan(45°) = 1

Step 4: Substitute and solve.
1 = h ÷ 15 → h = 15 × 1 = 15

The pole is 15 m high.

---

Example 5: Tree Across a River

A man on one side of a river sees the top of a tree on the other bank at an angle of elevation of 35°. The river is 25 m wide. Find the height of the tree.

        Top of Tree
             *
            /|
           / | h
          /  |
         /   |
        *----*
     Man     25 m
Angle = 35°

Step 1: Given: Angle = 35°, Distance = 25 m

Step 2: Formula:

tan(35°) = h ÷ 25

Step 3: tan(35°) = 0.7002

Step 4: Solve for h.

h = 25 × 0.7002 = 17.5

The tree is 17.5 m tall.

---

Example 6: Helicopter Above a Building

A helicopter is flying above a building 12 m tall. From the top of the building, the angle of elevation to the helicopter is 50°. If the horizontal distance is 40 m, find how high the helicopter is from the ground.

      Helicopter
           *
          /|
         / | (h - 12)
        /  |
   12m *---*
Building   40 m
Angle = 50°

Step 1: Given:

Height of building = 12 m

Angle = 50°

Distance = 40 m

Find total height h.

Step 2: Use formula:

tan(50°) = (h - 12) ÷ 40

Step 3: tan(50°) = 1.1918

Step 4: Substitute:

1.1918 = (h - 12) ÷ 40

h - 12 = 1.1918 × 40 = 47.67

h = 47.67 + 12 = 59.67

The helicopter is about 59.7 m above the ground.

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Example 7: Flagpole and Distance

A flag on a 9 m pole is seen from a point on the ground at an angle of 45°. Find how far the person is standing from the pole.

       Top of Flag
           *
          /|
         / | 9 m
        /  |
       *---*
         d
Angle = 45°

Step 1: Given: Angle = 45°, Opposite = 9 m
Find the distance (d).

Step 2: Formula:

tan(45°) = 9 ÷ d

Step 3: Since tan(45°) = 1,

1 = 9 ÷ d → d = 9

The person is 9 m away.

---

Example 8: Ladder Against Wall

A ladder leans against a wall making an angle of 60° with the ground. If the base of the ladder is 4 m away from the wall, how high up does it reach?

       Wall
        |\
        | \
      h |  \ Ladder
        |   \
        |____\
          4 m
Angle = 60°

Step 1: Given: Angle = 60°, Distance = 4 m

Step 2: Formula:

tan(60°) = h ÷ 4

Step 3: tan(60°) = 1.732

Step 4: Substitute and solve.

1.732 = h ÷ 4 → h = 4 × 1.732 = 6.93

The ladder reaches 6.93 m high on the wall.

---

Example 9: Tower Observation

A boy standing 50 m away observes the top of a tower at an angle of 28°. Find the height of the tower.

        Top of Tower
             *
            /|
           / | h
          /  |
         /   |
        *----*
      Boy    50 m
Angle = 28°

Step 1: Given: Angle = 28°, Distance = 50 m

Step 2: Formula:

tan(28°) = h ÷ 50

Step 3: tan(28°) = 0.5317

Step 4: Solve for h.

h = 50 × 0.5317 = 26.6

The tower is 26.6 m tall.

---

Definition:

The angle of depression is the angle formed between the horizontal line of sight and the line of sight downwards to an object below the observer’s eye level.

In simple words, when you look down at an object, the angle your eyes move downward from the horizontal is called the angle of depression.

          Eye level (Horizontal line)
               -----------------
               \   θ  /
                \   /
                 \ /
                  *
                 Car

Here:

1. The horizontal line represents your eye level.


2. The line going down to the car is the line of sight.


3. The angle θ between them is the angle of depression.

Key Points:

1. The angle of depression is always measured downward from the horizontal.


2. The angle of elevation (from the object looking up) is equal to the angle of depression (from the observer looking down).


3. These problems are usually solved using trigonometric ratios such as tan, sin, and cos.


4. Most questions involve right-angled triangles.

Trigonometric Formula:

We mostly use the tangent formula:

tan(θ) = opposite / adjacent

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EXAMPLES

Example 1: Girl Looking Down at a Dog

A girl standing on a 10 m high balcony looks down at a dog. The dog is 15 m away from the base of the building. Find the angle of depression.

     Girl
      *
      |\
   10m| \
      |  \  Line of sight
      |   \
      *----*
           15 m
           Dog

Step 1: Given
Height = 10 m

Distance = 15 m

Find angle θ.

Step 2: Formula

tan(θ) = opposite / adjacent = 10 / 15

Step 3: Calculate
tan(θ) = 0.6667

Step 4: Find θ

θ = tan⁻¹(0.6667) = 33.7°

Angle of depression = 33.7°

---

Example 2: Man Looking Down at a Boat

       Man
        *
        |\
     40m| \
        |  \25°
        |   \
        *----*
            x (distance)

Step 1: Given

Height = 40 m

Angle = 25°

Find x.

Step 2: Formula

tan(25°) = 40 / x

Step 3: Rearrange

x = 40 / tan(25°)

Step 4: Compute

tan(25°) = 0.4663

x = 40 / 0.4663 = 85.8

Distance = 85.8 m

---

Example 3: Building and Car

A man standing on top of a 20 m tall building sees a car on the road at an angle of depression of 45°. Find the distance between the car and the building.

Step 1: Given

Height = 20 m

Angle = 45°

Step 2: Formula

tan(45°) = 20 / x

Step 3: Since tan(45°) = 1:

1 = 20 / x → x = 20

Distance = 20 m

---

Example 4: Tower Observation

A boy at the top of a tower 50 m high observes a car moving away. When the angle of depression is 60°, find the distance of the car from the base of the tower.

Step 1: Given

Height = 50 m

Angle = 60°

Step 2: Formula

tan(60°) = 50 / x

Step 3: tan(60°) = 1.732

1.732 = 50 / x → x = 50 / 1.732 = 28.9

Distance = 28.9 m

---

Example 5: Plane Descending

An airplane is flying at an altitude of 500 m. The pilot sees the airport runway at an angle of depression of 10°. Find the horizontal distance to the runway.

Step 1: Given

Height = 500 m

Angle = 10°

Step 2: Formula

tan(10°) = 500 / x

Step 3: tan(10°) = 0.1763

x = 500 / 0.1763 = 2836.6

The runway is about 2837 m away.

---

Example 6: Top of Hill

A person standing on top of a 60 m hill sees a car at an angle of depression of 35°. Find how far the car is from the hill.

Step 1: tan(35°) = 60 / x

Step 2: x = 60 / tan(35°) = 60 / 0.7002 = 85.7

Distance = 85.7 m

---

Example 7: Bridge and Boat

A man stands on a bridge 30 m above the river. He looks down at a boat at an angle of depression of 20°. Find how far the boat is from the base of the bridge.

tan(20°) = 30 / x

x = 30 / 0.3640 = 82.4

Boat is 82.4 m away.

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Example 8: Kite Observation

A kite is flying at a height of 25 m. The string makes an angle of depression of 50° from the boy’s hand. Find how far the kite is horizontally from the boy.

tan(50°) = 25 / x

x = 25 / 1.1918 = 21.0

Horizontal distance = 21.0 m

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Example 9: Cliff and Ship

A ship is seen at an angle of depression of 15° from the top of a cliff that is 70 m high. Find the distance of the ship from the base of the cliff.

tan(15°) = 70 / x

x = 70 / 0.2679 = 261.2

Ship is 261.2 m from the cliff.

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Summary:

1. Angle of elevation → looking upward.


2. Angle of depression → looking downward.


3. Both angles are equal when measured from parallel horizontal lines.


4. Use tan(θ) = opposite / adjacent for most problems.


5. Always draw and label a diagram before solving.