Angles In Polygons And Parallel Lines

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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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Câu hỏi luyện tập

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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°

Thêm câu hỏi

9.In the diagram, two parallel lines are cut by a transversal. Angle a is 110°. What is the size of angle b, which is co-interior with a?
Easy
- A70°
- B110°
- C180°
- D90°
10.Find the sum of the interior angles of a decagon (10-sided polygon).
Easy
- A1440°
- B1800°
- C1080°
- D1260°
11.Two regular octagons and a square meet at a point. Show that the three interior angles at that point add up to 360°. What is the interior angle of a regular octagon?
Medium
- A135°
- B120°
- C140°
- D130°
12.ABC is a straight line, AD is parallel to BE, angle BAD = 34°, and AB = BD. Find angle p (angle ABD).
Hard
- A34°
- B68°
- C112°
- D146°
13.In the diagram, two parallel lines are cut by a transversal. One angle is 3x and another is 2x (co-interior). Find x.
Medium
- A36°
- B30°
- C40°
- D45°
14.The exterior angle of a regular polygon is 24°. What is the sum of its interior angles?
Hard
- A2340°
- B2160°
- C2520°
- D2700°
15.What is the sum of the interior angles of a hexagon?
Easy
- A720°
- B540°
- C900°
- D360°
16.A regular polygon has 12 sides. What is the size of each exterior angle?
Easy
- A30°
- B36°
- C45°
- D60°
17.In the diagram, two parallel lines are cut by a transversal. One angle is 70°. What is the size of the alternate angle?
Easy
- A70°
- B110°
- C180°
- D90°
18.A regular polygon has an interior angle of 150°. How many sides does it have?
Medium
- A12
- B10
- C15
- D20
19.In the diagram, AB is parallel to CD. Angle A = 55°. What is the size of angle B?
Easy
- A55°
- B125°
- C180°
- D35°
20.The exterior angle of a regular polygon is 18°. What is the sum of its interior angles?
Hard
- A3240°
- B2880°
- C3600°
- D3060°
21.What is the mathematical name for a 7-sided polygon?
Easy
- AHeptagon
- BHexagon
- COctagon
- DNonagon
22.Work out the size of one interior angle of a regular 24-sided polygon.
Medium
- A165°
- B150°
- C156°
- D160°
23.In the diagram, a transversal cuts two parallel lines. One co-interior angle is 110°. What is the other co-interior angle?
Easy
- A70°
- B110°
- C180°
- D90°
24.In the diagram, a regular polygon has 8 sides. What is the size of each exterior angle?
Easy
- A45°
- B40°
- C30°
- D36°
25.Find the size of one interior angle of a regular 20-sided polygon.
Medium
- A162°
- B170°
- C158°
- D165°
26.The ratio of exterior angle to interior angle of a regular polygon is 2:13. What is the exterior angle?
Hard
- A24°
- B20°
- C30°
- D36°
27.In the diagram, a triangle is drawn between two parallel lines. One angle is 40°, another is 60°. What is the third angle of the triangle?
Medium
- A80°
- B100°
- C70°
- D90°
28.What is the sum of the exterior angles of any polygon?
Easy
- A360°
- B180°
- C540°
- D720°
29.A regular polygon has an exterior angle of 30°. How many sides does it have?
Easy
- A12
- B10
- C8
- D6
30.What is the sum of the interior angles of a nonagon (9-sided polygon)?
Easy
- A1260°
- B1080°
- C1440°
- D900°
31.A regular pentagon and a regular hexagon share a side. Find the angle x between them.
Hard
- A132°
- B120°
- C108°
- D144°
32.What is the mathematical name for a 6-sided polygon?
Easy
- AHexagon
- BPentagon
- COctagon
- DHeptagon
33.A regular polygon has an interior angle of 144°. How many sides does it have?
Medium
- A10
- B12
- C8
- D9
34.In the diagram, triangle ABC is drawn between two parallel lines. Angle A = 50°, angle B = 70°. What is angle C?
Medium
- A60°
- B50°
- C70°
- D40°
35.The exterior angle of a regular polygon is 15°. What is the sum of its interior angles?
Hard
- A3960°
- B3600°
- C3240°
- D2880°
36.A regular octagon and a regular hexagon meet at a point. What is the angle between them?
Hard
- A105°
- B135°
- C120°
- D90°
37.What is the sum of the interior angles of a quadrilateral?
Easy
- A180°
- B270°
- C360°
- D540°
38.The sum of the exterior angles of any polygon is 360°.
Easy
True or false?
39.Match each angle relationship with the correct shape formed by parallel lines and a transversal.
Medium
- Corresponding angles
- Alternate angles
- Co-interior angles
- F-shape
- Z-shape
- C-shape
40.A regular polygon has an interior angle of 140°. How many sides does it have?
Medium
- A7
- B8
- C9
- D10
41.Which of the following statements about a regular pentagon are correct? (Select all that apply)
Medium
- AEach interior angle is 108°
- BEach exterior angle is 72°
- CSum of interior angles is 540°
- DSum of exterior angles is 360°
- EEach interior angle is 120°
42.Arrange the following regular polygons in increasing order of their interior angle size.
Medium
- Square
- Regular pentagon
- Regular hexagon
- Regular octagon
43.A regular polygon has an exterior angle that is one-fifth of its interior angle. How many sides does it have?
Hard
- A10
- B12
- C15
- D20
44.In the diagram, two parallel lines are cut by a transversal. One angle is 75°. What is the size of the alternate angle?
Easy
- A15°
- B75°
- C105°
- D180°
45.Which of the following statements about a regular hexagon are correct? (Select all that apply)
Medium
- AEach interior angle is 120°.
- BEach exterior angle is 60°.
- CSum of interior angles is 720°.
- DSum of exterior angles is 360°.
- EIt has 8 sides.
46.A regular polygon has an exterior angle of 36°. How many sides does it have?
Easy
47.Complete the sentence.
Easy
Angles that are on the same side of a transversal and inside the parallel lines are called ____ angles.
48.The interior angles of a pentagon are 100°, 110°, 120°, 130°, and x. Find x.
Medium
49.In the diagram, two parallel lines are cut by a transversal. One angle is 75°. What is the size of the alternate angle? Give a reason.
Medium
50.In the diagram, two parallel lines are cut by a transversal. One co-interior angle is 115°. Find the other co-interior angle.
Medium
51.A regular polygon has an interior angle of 156°. How many sides does it have?
Medium
52.What is the mathematical name for a 9-sided polygon?
Easy
53.The interior angles of a pentagon are 90°, 110°, 120°, 130°, and x. Find x.
Medium
54.A regular polygon has an exterior angle of 36°. How many sides does it have?
Easy
- A12
- B11
- C10
- D9
55.The interior angles of a pentagon are 100°, 110°, 120°, 130°, and x. Find x.
Medium
- A96°
- B80°
- C88°
- D72°
56.In the diagram, two parallel lines are cut by a transversal. One co-interior angle is 115°. Find the other co-interior angle.
Medium
- A58°
- B72°
- C79°
- D65°
57.A regular polygon has an interior angle of 156°. How many sides does it have?
Medium
- A15
- B17
- C16
- D14
58.The interior angles of a pentagon are 90°, 110°, 120°, 130°, and x. Find x.
Medium
- A99°
- B81°
- C90°
- D108°

Ghi chú ôn tập

## 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.

Thẻ ghi nhớ

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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