Angles In Polygons And Parallel Lines

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課程筆記

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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練習題

免費預覽——45 題中的 8 題。註冊即可查看全部。

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

複習筆記

## 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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