Angles In Polygons And Parallel Lines

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Notes

Basic Angle Properties

Angles around a point sum to **360°**.
Angles on a straight line sum to **180°**.
**Vertically opposite angles** are equal when two lines intersect.
Interior angles of a triangle sum to **180°**.
Interior angles of a quadrilateral sum to **360°**.
In an isosceles triangle, base angles are equal.
In an equilateral triangle, each angle is **60°**.

Angles in Polygons

A polygon with $n$ sides has interior angle $sum =$ $180^\circ \times (n-2)$ .
Exterior angles of any polygon sum to **360°**.
For a regular polygon: interior angle = $\frac{180(n-2)}{n}$ , exterior angle = $\frac{360}{n}$ .
Interior and exterior angles at a vertex sum to **180°**.
To find missing angles in a polygon, subtract known angles from the sum.
If given interior angle of a regular polygon, solve $\frac{180(n-2)}{n} = \text{angle}$ to find $n$ .

Angles in Parallel Lines

**Corresponding angles** (F-shape) are equal.
**Alternate angles** (Z-shape) are equal.
**Co-interior angles** (C-shape) sum to **180°**.
Vertically opposite angles also apply in parallel line problems.
Always give reasons (e.g., 'corresponding angles on parallel lines are equal').

Common Regular Polygons

Equilateral triangle (3 sides): interior 60°, exterior 120°.
Square (4 sides): interior 90°, exterior 90°.
Regular pentagon (5 sides): interior 108°, exterior 72°.
Regular hexagon (6 sides): interior 120°, exterior 60°.
Regular octagon (8 sides): interior 135°, exterior 45°.
Regular decagon (10 sides): interior 144°, exterior 36°.

Basic Angle Properties

Angles in Polygons

Angles in Parallel Lines

Regular Polygon Table

Practice questions

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1.What is the sum of the interior angles of a pentagon?
Easy
- A540°
- B360°
- C720°
- D180°
2.In a regular polygon, each interior angle is 150°. How many sides does the polygon have?
Easy
- A12
- B10
- C15
- D8
3.Angles on a straight line add up to:
Easy
- A180°
- B360°
- C90°
- D270°
4.The interior angle of a regular polygon is 156°. Work out the number of sides n.
Medium
- A15
- B12
- C18
- D20
5.In the diagram, two parallel lines are cut by a transversal. One angle is 70°. What is the size of the alternate angle?
Medium
- A70°
- B110°
- C180°
- D90°
6.A regular polygon has 72 sides. Find the size of an interior angle.
Medium
- A175°
- B170°
- C180°
- D165°
7.In a regular polygon, the interior angle is 11 times the exterior angle. Find the number of sides.
Medium
- A24
- B22
- C20
- D30
8.A hexagon has five angles each measuring 115°. Calculate the size of the sixth angle.
Hard
- A105°
- B115°
- C125°
- D95°

Summary

## Angles in Polygons & Parallel Lines ### Basic Angle Properties - Angles around a point sum to **360°**. - Angles on a straight line sum to **180°**. - Vertically opposite angles are **equal**. - Triangle interior angles sum to **180°**. - Quadrilateral interior angles sum to **360°**. - Isosceles triangle: two equal angles opposite equal sides. ### Angles in Polygons - **Interior angle** + **exterior angle** = **180°**. - Sum of interior angles = **180° × (n – 2)**. - Sum of exterior angles = **360°** (always). - For a **regular polygon**: - Interior angle = **180(n – 2)/n**. - Exterior angle = **360/n**. - To find number of sides from interior angle: set angle = 180(n – 2)/n and solve. ### Angles in Parallel Lines - **Corresponding angles** (F-shape) are **equal**. - **Alternate angles** (Z-shape) are **equal**. - **Co-interior angles** (C-shape) sum to **180°**. - Always give reasons: "corresponding angles are equal", "alternate angles are equal", "co-interior angles sum to 180°". ### Key Formulas - Sum of interior angles: **180(n – 2)** - Exterior angle (regular): **360/n** - Interior angle (regular): **180(n – 2)/n**

Key terms

Interior angle: Angle inside a polygon at a corner.
Exterior angle: Angle between a side and an extended adjacent side; interior + exterior = 180°.
Regular polygon: Polygon with all sides equal and all angles equal.
Sum of interior angles: 180° × (n – 2) for an n-sided polygon.
Sum of exterior angles: 360° for any polygon.
Corresponding angles: Equal angles formed when a transversal cuts parallel lines (F-shape).
Alternate angles: Equal angles formed when a transversal cuts parallel lines (Z-shape).
Co-interior angles: Angles that sum to 180° when a transversal cuts parallel lines (C-shape).
Vertically opposite angles: Equal angles opposite each other when two lines intersect.

Slides

Basic Angle Properties

Angles around a point add up to 360°.
Angles on a straight line add up to 180°.
Vertically opposite angles are equal.

Two lines intersect. One angle is 50°. What is the vertically opposite angle?

40°
50°
130°
180°

Angles in Triangles and Quadrilaterals

Interior angles of a triangle sum to 180°.
Interior angles of a quadrilateral sum to 360°.
In an isosceles triangle, base angles are equal.

The sum of interior angles in a triangle $is ____$ degrees.

Angles in Polygons

Sum of interior angles of an n-sided polygon $= 180^{\circ} \times (n$ – 2).
Exterior angles of any polygon sum to 360°.
In a regular polygon, all interior angles are equal and all exterior angles are equal.

What is the sum of interior angles of a hexagon (6 sides)?

540°
720°
900°
1080°

Interior and Exterior Angles in Regular Polygons

Interior angle $= 180^{\circ} \times (n$ – 2) / n.
Exterior angle $= 360^{\circ} / n$ .
Interior + Exterior $= 180^{\circ}$ .

A regular polygon has interior angle 150°. Its number of sides $is ____$ .

Angles in Parallel Lines

Corresponding angles (F-shape) are equal.
Alternate angles (Z-shape) are equal.
Co-interior angles (C-shape) sum to 180°.

Which of these are true about parallel lines?

Corresponding angles are equal.
Alternate angles sum to 180°.
Co-interior angles are equal.
Vertically opposite angles are equal.

Finding Missing Angles in Parallel Lines

Look for F, Z, or C shapes to identify angle relationships.
Use vertically opposite angles and angles on a straight line.
Always give reasons for each step in your working.

Two parallel lines are cut by a transversal. One alternate angle is 70°. What is the other alternate angle?

70°
110°
180°
Not enough info

Problem Solving with Polygons and Parallel Lines

Combine angle properties from polygons and parallel lines.
Use the sum of interior angles to find missing angles in polygons.
Apply parallel line rules to find unknown angles in diagrams.

A regular polygon has exterior angle 45°. It $has ____$ sides.

Exam Tips

Identify whether a polygon is regular or irregular.
Use 180(n-2) to find sum of interior angles.
Give full reasons for each angle you find.

The interior angle of a regular polygon is 108°. How many sides does it have?

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