Angles In Polygons And Parallel Lines

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Notes

Polygon Basics

A **polygon** is a 2D shape with×n×straight sides.
Triangle (3 sides), quadrilateral (4), pentagon (5), hexagon (6), octagon (8), decagon (10).
In a **regular polygon**, all sides are equal and all angles are equal.

Interior and Exterior Angles

**Interior angles** are inside the polygon at each corner.
**Exterior angle** at a corner is the angle needed to make a straight line with the interior angle.
Interior + exterior $= 180^{\circ}$ at each vertex.

Sum of Interior Angles

Sum of interior angles = **180° $\times (n$ – 2)**.
Triangle: 180°, quadrilateral: 360°, pentagon: 540°, hexagon: 720°, octagon: 1080°, decagon: 1440°.

Sum of Exterior Angles

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

Finding Angles in Regular Polygons

Interior angle $= sum$ of interior angles $\div n$ , or use formula **180(n–2)/n**.
Exterior angle = **360°/n**.
Alternatively, exterior angle $= 180^{\circ}$ – interior angle.

Finding Number of Sides

Given exterior angle×e×, number of sides $\times n\times =$ **360°/e**.
Given interior angle×i×, solve **i = 180(n–2)/n** for 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 are equal.

Problem-Solving Tips

Always give reasons: 'corresponding angles are equal', 'alternate angles are equal', 'co-interior angles sum to 180°'.
Use correct names (F, Z, C shapes) in reasoning to get full marks.

Interior and Exterior Angles

Corresponding, Alternate, Co-interior Angles

Practice questions

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1.What is the sum of the interior angles of a pentagon?
Easy
- A540°
- B360°
- C720°
- D180°
2.Work out the size of one interior angle of a regular 9-sided polygon.
Easy
- A140°
- B120°
- C160°
- D100°
3.Write down the mathematical name for an 8-sided polygon.
Easy
- AOctagon
- BHexagon
- CHeptagon
- DDecagon
4.In the diagram, two parallel lines are crossed by a transversal. Which angle is alternate to angle X?
Easy
- AAngle Y
- BAngle Z
- CAngle W
- DAngle V
5.A regular polygon has an exterior angle of 20°. Work out the number of sides.
Easy
- A18
- B20
- C16
- D12
6.Find the size of one interior angle of a regular octagon.
Medium
- A135°
- B120°
- C140°
- D130°
7.In the diagram, AB is parallel to CD. Angle x is 50°. What is the size of angle y?
Medium
- A50°
- B130°
- C40°
- D180°
8.The interior angle of a regular polygon is 162°. Calculate the number of sides.
Medium
- A20
- B18
- C16
- D24

Summary

## Angles in Polygons & Parallel Lines ### Polygons - **Polygon**: 2D shape with n straight sides. - **Regular polygon**: all sides equal, all angles equal. - **Interior angle**: inside at a corner; **exterior angle**: outside, forms straight line with interior. - **Sum of interior angles** = 180° × (n – 2) - **Sum of exterior angles** = 360° (always) - For a regular polygon: - Exterior angle = 360° / n - Interior angle = 180° – exterior angle, or (180°(n – 2))/n - Common polygons: - Triangle (3): sum 180°, int 60°, ext 120° - Square (4): sum 360°, int 90°, ext 90° - Pentagon (5): sum 540°, int 108°, ext 72° - Hexagon (6): sum 720°, int 120°, ext 60° - Octagon (8): sum 1080°, int 135°, ext 45° - Decagon (10): sum 1440°, int 144°, ext 36° - To find missing angle: sum known angles, subtract from total sum. - To find number of sides from interior angle: set angle = 180(n–2)/n and solve. ### Parallel Lines - **Corresponding angles** (F-shape): equal. - **Alternate angles** (Z-shape): equal. - **Co-interior angles** (C-shape): sum to 180°. - **Vertically opposite angles**: equal. - Always give reasons: "corresponding angles on parallel lines are equal", etc. ### Tips - Identify regular vs irregular first. - Use 180(n–2) to start missing angle problems. - For parallel lines, look for F, Z, C shapes.

Key terms

Polygon: A 2D shape with n straight sides.
Regular polygon: A polygon with all sides equal and all angles equal.
Interior angle: The angle inside a polygon at a corner.
Exterior angle: The angle between a side and an extended adjacent side; interior + exterior = 180°.
Sum of interior angles: 180° × (n – 2) for a polygon with n sides.
Sum of exterior angles: 360° for any polygon.
Corresponding angles: Angles in matching positions when a transversal crosses parallel lines; they are equal (F-shape).
Alternate angles: Angles on opposite sides of the transversal and inside the parallel lines; they are equal (Z-shape).
Co-interior angles: Angles inside the parallel lines on the same side of the transversal; they sum to 180° (C-shape).
Vertically opposite angles: Angles opposite each other when two lines cross; they are equal.

Slides

Polygons and Their Angles

A polygon is a 2D shape with n straight sides.
In a regular polygon, all sides are equal and all angles are equal.
Interior angles are inside the polygon; exterior angles are outside, and they add to 180° at each corner.

What is the sum of the interior angles of a pentagon?

360°
540°
720°
900°

Sum of Interior Angles

Sum of interior angles $= 180^{\circ} \times (n$ – 2).
For a triangle $(n=3)$ : 180°; quadrilateral $(n=4)$ : 360°; pentagon $(n=5)$ : 540°; hexagon $(n=6): 720^{\circ}$ .
This formula works for any polygon, regular or irregular.

The sum of the interior angles of an octagon $is ____$ degrees.

Exterior Angles of a Polygon

The sum of exterior angles of any polygon is always 360°.
For a regular polygon, exterior angle $= 360^{\circ} \div n$ .
Interior angle + exterior angle $= 180^{\circ}$ .

A regular polygon has an exterior angle of 45°. How many sides does it have?

Finding Interior Angle of a Regular Polygon

Method 1: Find sum of interior angles, then divide by n.
Method 2: Use formula: interior angle $= 180^{\circ} \times (n$ – $2) \div n$ .
Example: Regular pentagon: $180^{\circ} \times 3 \div 5 = 108^{\circ}$ .

One interior angle of a regular 10-sided polygon $is ____$ degrees.

Angles in Parallel Lines

Parallel lines are always the same distance apart and never meet.
Corresponding angles (F-shape) are equal.
Alternate angles (Z-shape) are equal.
Co-interior angles (C-shape) add up to 180°.

Which of these are true about parallel lines?

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

Identifying Angle Relationships

Look for F, Z, and C shapes to identify corresponding, alternate, and co-interior angles.
Vertically opposite angles are also equal.
Always give reasons using correct names (not F, Z, C) in exams.

In the diagram, two parallel lines are cut by a transversal. One angle is 70°. Which is true?

The corresponding angle is 110°.
The alternate angle is 70°.
The co-interior angle is 70°.
The vertically opposite angle is 110°.

Worked Example: Parallel Lines

Find angles a and b in a diagram with parallel lines.
$a = 64^{\circ} (vertically$ opposite angles are equal).
$b = 64^{\circ} (corresponding$ angles on parallel lines are equal).

If two parallel lines are cut by a transversal, co-interior angles sum $to ____$ degrees.

Combining Polygons and Parallel Lines

Problems often combine angles in polygons with parallel lines.
Use interior/exterior angle formulas and parallel line rules together.
Always show clear reasoning and give reasons for each step.

A regular polygon has interior angle 162°. How many sides does it have?

Flashcards

What is the sum of interior angles of a polygon with n sides?

$Sum = 180^{\circ} \times (n$ – 2).

What is the sum of exterior angles of any polygon?

360°.

How do you find the size of one interior angle of a regular n-sided polygon?

Use the formula: interior angle $= 180^{\circ} \times (n$ – 2) / n.

How do you find the size of one exterior angle of a regular n-sided polygon?

Exterior angle $= 360^{\circ} / n$ .

What is the relationship between an interior angle and its adjacent exterior angle?

They add up to 180°.

What is the mathematical name for an 8-sided polygon?

Octagon.

Work out the size of one interior angle of a regular 16-sided polygon.

Interior angle $= 180^{\circ} \times (16$ – $2) / 16 = 180^{\circ} \times 14 / 16 = 157.5^{\circ}$ .

Work out the size of one interior angle of a regular 9-sided polygon.

Interior angle $= 180^{\circ} \times (9$ – $2) / 9 = 180^{\circ} \times 7 / 9 = 140^{\circ}$ .

Work out the size of one interior angle of a regular octagon.

Interior angle $= 180^{\circ} \times (8$ – $2) / 8 = 180^{\circ} \times 6 / 8 = 135^{\circ}$ .

A regular polygon has an exterior angle of 20°. How many sides does it have?

Number of sides $= 360^{\circ} / 20^{\circ} = 18$ .

Each interior angle of a regular polygon is 162°. Calculate the number of sides.

Exterior angle $= 180^{\circ}$ – $162^{\circ} = 18^{\circ}$ . Number of sides $= 360^{\circ} / 18^{\circ} = 20$ .

What are corresponding angles?

Corresponding angles are equal and form an F-shape when a transversal crosses parallel lines.

What are alternate angles?

Alternate angles are equal and form a Z-shape when a transversal crosses parallel lines.

What are co-interior (supplementary) angles?

Co-interior angles add up to 180° and form a C-shape when a transversal crosses parallel lines.

In the diagram, two parallel lines are crossed by a transversal. Angle X is given. Which angle is alternate to angle X?

The angle that forms a Z-shape with X on the other parallel line.

In the diagram, two parallel lines are crossed by a transversal. Angle X is given. Which angle is corresponding to angle X?

The angle that forms an F-shape with X on the other parallel line.

Vertically opposite angles are ...

Equal.

Angles on a straight line sum to ...

180°.

Show that the sum of interior angles of a pentagon is 540°.

A pentagon can be divided into 3 triangles. Each triangle has 180°, so total $= 3 \times 180^{\circ} = 540^{\circ}$ .

Show that the sum of interior angles of a hexagon is 720°.

A hexagon can be divided into 4 triangles. Each triangle has 180°, so total $= 4 \times 180^{\circ} = 720^{\circ}$ .

Two regular octagons and a square meet at a point. Show that the three interior angles at that point add up to 360°.

Interior angle of regular octagon $= 135^{\circ}$ . Interior angle of square $= 90^{\circ}$ . $Sum = 135^{\circ} + 135^{\circ} + 90^{\circ} = 360^{\circ}$ .

In a regular polygon, the ratio of exterior angle to interior angle is 2:13. Work out the exterior angle.

Let exterior $= 2x$ , interior $= 13x$ . Then $2x + 13x = 180^{\circ}$ → $15x = 180^{\circ}$ → $x = 12^{\circ}$ . Exterior $= 24^{\circ}$ .

For the polygon in the previous card, work out the number of sides.

Number of sides $= 360^{\circ} /$ exterior angle $= 360^{\circ} / 24^{\circ} = 15$ .

What is the interior angle of a regular decagon (10 sides)?

Interior angle $= 180^{\circ} \times (10$ – $2) / 10 = 180^{\circ} \times 8 / 10 = 144^{\circ}$ .

What is the exterior angle of a regular decagon?

Exterior angle $= 360^{\circ} / 10 = 36^{\circ}$ .

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