Angles in Triangles and Polygons | JHS Mathematics

Angles in Triangles

Sum of Interior Angles

The sum of the interior angles in any triangle is always 180°. This property allows us to find unknown angles when two angles are known.

Example 1: Find the missing angle in a triangle with angles 45° and 60°.

\[ \text{Let the missing angle be } x \] \[ 45° + 60° + x = 180° \] \[ 105° + x = 180° \] \[ x = 180° - 105° = 75° \]

Types of Triangles by Angles:

Acute: All angles < 90°
Right: One angle = 90°
Obtuse: One angle > 90°

Special Cases:

Equilateral triangle: All angles = 60°
Isosceles triangle: Two equal angles

Exterior Angle Theorem

The exterior angle of a triangle is equal to the sum of the two opposite interior angles.

Example 2: Find angle x if angles A and B are 50° and 70° respectively.

\[ x = 50° + 70° = 120° \]

Angles in Polygons

Sum of Interior Angles

The sum of the interior angles of an n-sided polygon is given by: \[ S = (n - 2) \times 180° \]

Example 3: Find the sum of interior angles of a pentagon (5 sides).

\[ S = (5 - 2) \times 180° = 3 \times 180° = 540° \]

Derivation:

Any n-sided polygon can be divided into (n-2) triangles. Since each triangle sums to 180°, the total is (n-2)×180°.

Finding Individual Angles

For regular polygons (where all sides and angles are equal), each interior angle is: \[ \text{Each angle} = \frac{(n - 2) \times 180°}{n} \]

Example 4: Find one interior angle of a regular hexagon.

\[ \text{Each angle} = \frac{(6 - 2) \times 180°}{6} = \frac{720°}{6} = 120° \]

Regular Polygons

Properties

Regular polygons have equal sides and equal angles. This makes calculating their angles simpler.

Example 5: Find the value of x in this regular hexagon.

First calculate the sum of interior angles:

\[ S = (6 - 2) \times 180° = 720° \]

Since all angles are equal:

\[ x = \frac{720°}{6} = 120° \]

Exterior Angles:

The sum of exterior angles for any polygon is always 360°. For regular polygons:

\[ \text{Each exterior angle} = \frac{360°}{n} \]

Practice Exercise

Question 1

Find the missing angle in a triangle with angles 35° and 75°.

Sum of angles = 180°

\[ 35° + 75° + x = 180° \] \[ 110° + x = 180° \] \[ x = 180° - 110° = 70° \]

Question 2

Calculate the sum of interior angles of a nonagon (9-sided polygon).

\[ S = (n - 2) \times 180° \] \[ S = (9 - 2) \times 180° = 7 \times 180° = 1260° \]

Question 3

Find one interior angle of a regular octagon.

First find sum of angles:

\[ S = (8 - 2) \times 180° = 1080° \]

Then divide by number of angles:

\[ \text{Each angle} = \frac{1080°}{8} = 135° \]

Question 4

In triangle ABC, angle A = 40° and angle B = angle C. Find angle B.

Let angle B = angle C = x

\[ 40° + x + x = 180° \] \[ 2x = 140° \] \[ x = 70° \]

Question 5

Find the value of x in the hexagon with angles: 110°, 120°, 130°, x, 115°, 125°.

First find sum of angles:

\[ S = (6 - 2) \times 180° = 720° \]

Then set up equation:

\[ 110° + 120° + 130° + x + 115° + 125° = 720° \] \[ 600° + x = 720° \] \[ x = 120° \]

Ready for more challenges?

Full Practice Set

Problem Solving

Complex Problems

Let's apply our knowledge to solve more complex problems involving multiple concepts.

Example 6: Find angles x and y in the following diagram (assuming a regular pentagon and an equilateral triangle).

First calculate interior angle of pentagon:

\[ \text{Each angle} = \frac{(5 - 2) \times 180°}{5} = 108° \]

For the equilateral triangle, all angles = 60°

At point where they meet:

\[ x = 360° - 108° - 60° = 192° \]

For angle y, consider the straight line:

\[ y = 180° - 60° = 120° \]

                                
                                     Common Mistakes
                                
                                Forgetting that the sum of angles in a triangle is always 180°
Using the wrong formula for polygon angle sums
Assuming all polygons are regular when they're not
Miscounting the number of sides in a polygon

Math Challenge

Timed Test in 7 Levels

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