Angles in Polygons and Parallel Lines

Angles in Polygons

A polygon is a 2D shape with n straight sides.

Examples of Polygons:

• Triangle – 3 sides

• Quadrilateral – 4 sides

• Pentagon – 5 sides

• Hexagon – 6 sides

• Octagon – 8 sides

Regular Polygons:
A regular polygon has equal side lengths and equal angles.

• Example: An equilateral triangle (3 equal sides, 3 equal angles of $60 \degree$).

• Example: A square (4 equal sides, 4 equal angles of $90 \degree$)

Interior and Exterior Angles of a Polygon

Interior angles are inside the polygon at each vertex.

Exterior angles help form a straight line with the interior angles.

Interior and exterior angles at each vertex add up to $180 \degree$.

Sum of Interior Angles Formula:

$$\text{Sum of interior angles } = (n - 2) \times 180 \degree$$

where n is the number of sides.

Sum of Exterior Angles Formula:

$$\text{Sum of exterior angles } = 360 \degree$$

Finding the Interior and Exterior Angles of a Regular Polygon:

Find the sum of the interior angles:

Find each interior angle:

Find each exterior angle:

$$\frac{360 \degree}{n}$$

Example: Regular Pentagon (5 sides)

• Sum of interior angles: $$(5 - 2) \times 180 \degree = 540 \degree$$

• Each interior angle: $$540 \degree \div 5 = 108 \degree$$

• Each exterior angle: $$360 \degree \div 5 = 72 \degree$$

   

Final Answer: Interior angle = $108 \degree$, Exterior angle = $72 \degree$.

Angles in Parallel Lines

Parallel lines never meet and stay the same distance apart. When a transversal (crossing line) cuts through parallel lines, different angle relationships are formed.

Types of Angles in Parallel Lines include:

Corresponding Angles

• Corresponding Angles can be found by looking for an  F-Shape
• These angles are equal.

Alternate Angles

• Alternate Angles can be found by looking for an  Z-Shape
• These angles are equal.

Co-Interior (Supplementary) Angles

• Co-Interior (supplementary) Angles can be found by looking for an  C-Shape
• The angles add up to $180\degree$.

Example: Finding Missing Angles

Find angles p and q in diagram below. You must give reasons for your answers

1. For $p$ :

   • Use vertically opposite angles for p: $$p = 134 \degree$$

2. For $q$:

   1. Use corresponding angles (F-Shape) to find the angle on the same line as $q$ to be $134\degree$

   2. Now use angles on a line sum to $180\degree$ to find $q$ $\to$ $q = 180\degree - 134\degree$ $\therefore \quad q = 46\degree$

Final Answer:  $p = 134 \degree$,  $q = 46\degree$.