Angles In Polygons And Parallel Lines

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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.

Interior and exterior angles of a polygon

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').

Corresponding angles (F-shape)

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°.

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Câu hỏi luyện tập

Xem trước miễn phí — 8 trên 45 câu hỏi. Đăng ký để xem tất cả.

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?
Medium
- A12
- B10
- C15
- D8
3.Angles on a straight line add up to:
Easy
- A180°
- B360°
- C90°
- D270°
4.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°
5.A regular polygon has 72 sides. Find the size of an interior angle.
Medium
- A175°
- B170°
- C180°
- D165°
6.In a regular polygon, the interior angle is 11 times the exterior angle. Find the number of sides.
Hard
- A24
- B22
- C20
- D30
7.The diagram shows a regular pentagon. Find the size of angle x.
Medium
- A36°
- B72°
- C108°
- D54°
8.Vertically opposite angles are:
Easy
- Aequal
- Bsupplementary
- Ccomplementary
- Dnone of the above

Thêm câu hỏi

9.The exterior angle of a regular polygon is 45°. What is the name of the polygon?
Easy
- Aoctagon
- Bhexagon
- Cpentagon
- Ddecagon
10.What is the sum of the exterior angles of any polygon?
Easy
- A360°
- B180°
- C540°
- D720°
11.A regular octagon (interior angle 135°) and a regular hexagon (interior angle 120°) meet at a common vertex, with their other two sides lying along a straight line. Find the size of angle x, the remaining angle at that vertex.
Hard
- A105°
- B120°
- C135°
- D150°
12.What is the sum of the interior angles of a hexagon?
Easy
- A720°
- B540°
- C360°
- D900°
13.A regular polygon has an interior angle of 156°. How many sides does it have?
Medium
- A15
- B12
- C18
- D20
14.In a regular polygon, each exterior angle is 30°. Find the number of sides.
Easy
- A12
- B10
- C8
- D6
15.The diagram shows two parallel lines and a transversal. One angle is 125°. What is the size of the co-interior angle?
Easy
- A55°
- B125°
- C65°
- D35°
16.The interior angle of a regular polygon is 150°. How many sides does the polygon have?
Medium
- A12
- B10
- C15
- D8
17.Find the number of sides of a regular polygon with interior angle 162°.
Medium
- A20
- B18
- C24
- D16
18.A regular polygon has an interior angle of 150°. How many sides does it have?
Medium
- A10
- B12
- C15
- D20
19.A regular polygon has an exterior angle of 45°. What is the name of the polygon?
Easy
- AHexagon
- BOctagon
- CNonagon
- DDecagon
20.Angles around a point add up to:
Easy
- A360°
- B180°
- C90°
- D270°
21.What is the sum of the interior angles of a quadrilateral?
Easy
- A360°
- B180°
- C540°
- D720°
22.Find the size of one interior angle of a regular polygon with 40 sides.
Medium
- A171°
- B162°
- C180°
- D140°
23.Angles that form a straight line add up to:
Easy
- A180°
- B360°
- C90°
- D270°
24.In the diagram, two parallel lines are cut by a transversal, forming a Z-shape at the two marked angles. One angle is 60°. Find the value of x, the alternate angle.
Medium
- A50°
- B60°
- C70°
- D80°
25.Triangle ABC is isosceles with AB = AC. Angle ABC = 32°. Find angle BAC.
Medium
- A116°
- B148°
- C64°
- D32°
26.The diagram shows two parallel lines PAQ and SBCT. AB = AC and angle QAC = 43°. Find the value of x.
Hard
- A43°
- B86°
- C137°
- D47°
27.In a regular polygon, the interior angle is 11 times the exterior angle. Work out the number of sides of this polygon.
Hard
- A20
- B22
- C24
- D30
28.Vertically opposite angles are always equal.
Easy
True or false?
29.The sum of the interior angles of a quadrilateral is:
Easy
- A180°
- B270°
- C360°
- D540°
30.In the diagram, two parallel lines are cut by a transversal. One angle is 130°. Which statement is true?
Medium
- AThe corresponding angle is 50°
- BThe alternate angle is 130°
- CThe co-interior angle is 130°
- DThe vertically opposite angle is 50°
31.Which of the following are true about a regular hexagon? (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 6 lines of symmetry
32.Match each polygon to the sum of its interior angles.
Easy
- Triangle
- Quadrilateral
- Pentagon
- Hexagon
- 180°
- 360°
- 540°
- 720°
33.Co-interior angles are always equal.
Easy
True or false?
34.Arrange the steps in the correct order to find the number of sides of a regular polygon given that each interior angle is 156°.
Medium
- Set up equation: 180(n-2)/n = 156
- Multiply both sides by n: 180(n-2) = 156n
- Expand: 180n - 360 = 156n
- Subtract 156n: 24n - 360 = 0
- Add 360: 24n = 360
- Divide by 24: n = 15
35.Which of the following statements about angles in parallel lines are true? (Select all that apply)
Medium
- ACorresponding angles are equal
- BAlternate angles add up to 180°
- CCo-interior angles are equal
- DVertically opposite angles are equal
- ECo-interior angles add up to 180°
36.The sum of the interior angles of a pentagon is 540°.
Easy
True or false?
37.A regular polygon has an exterior angle of 40°. How many sides does it have?
Easy
38.Complete the sentence: The sum of the exterior angles of any polygon is always ___.
Easy
The sum of the exterior angles of any polygon is always ____.
39.What is the name of the angle pair that forms a 'Z' shape when a transversal crosses parallel lines?
Easy
40.A regular polygon has an interior angle of 162°. Find the number of sides.
Medium
41.Complete the sentence: In a regular polygon, the interior angle and the exterior angle add up to ___.
Easy
In a regular polygon, the interior angle and the exterior angle add up to ____.
42.A regular polygon has an exterior angle of 24°. How many sides does it have?
Easy
43.A regular polygon has an exterior angle of 40°. How many sides does it have?
Medium
- A10
- B11
- C8
- D9
44.A regular polygon has an interior angle of 162°. Find the number of sides.
Medium
- A20
- B22
- C18
- D24
45.A regular polygon has an exterior angle of 24°. How many sides does it have?
Medium
- A14
- B16
- C17
- D15

Ghi chú ôn tập

## 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**

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.

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