Angles In Polygons And Parallel Lines

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課程筆記

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° at each vertex.

Interior and exterior angles of a polygon

Sum of Interior Angles

Sum of interior angles = 180° × (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 ÷ n, or use formula 180(n–2)/n.
Exterior angle = 360°/n.
Alternatively, exterior angle = 180° – interior angle.

Finding Number of Sides

Given exterior angle×e×, number of sides ×n×= 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.

Corresponding angles (F-shape)

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.

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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.
Medium
- A140°
- B120°
- C160°
- D100°
3.Write down the mathematical name for an 8-sided polygon.
Easy
- AOctagon
- BHexagon
- CHeptagon
- DDecagon
4.A regular polygon has an exterior angle of 20°. Work out the number of sides.
Medium
- A18
- B20
- C16
- D12
5.Find the size of one interior angle of a regular octagon.
Easy
- A135°
- B120°
- C140°
- D130°
6.In the diagram, AB is parallel to CD. Angle x is 50°. What is the size of angle y?
Easy
- A50°
- B130°
- C40°
- D180°
7.The interior angle of a regular polygon is 162°. Calculate the number of sides.
Medium
- A20
- B18
- C16
- D24
8.A regular polygon has 15 sides. What is the size of each exterior angle?
Easy
- A24°
- B30°
- C20°
- D36°

複習筆記

## 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.

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.

字卡

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

Sum = 180° × (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° × (n – 2) / n.

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

Exterior angle = 360° / 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° × (16 – 2) / 16 = 180° × 14 / 16 = 157.5°.

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

Interior angle = 180° × (9 – 2) / 9 = 180° × 7 / 9 = 140°.

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

Interior angle = 180° × (8 – 2) / 8 = 180° × 6 / 8 = 135°.

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

Number of sides = 360° / 20° = 18.

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

Exterior angle = 180° – 162° = 18°. Number of sides = 360° / 18° = 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 × 180° = 540°.

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 × 180° = 720°.

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°. Interior angle of square = 90°. Sum = 135° + 135° + 90° = 360°.

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° → 15x = 180° → x = 12°. Exterior = 24°.

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

Number of sides = 360° / exterior angle = 360° / 24° = 15.

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

Interior angle = 180° × (10 – 2) / 10 = 180° × 8 / 10 = 144°.

What is the exterior angle of a regular decagon?

Exterior angle = 360° / 10 = 36°.

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