Angle facts and parallel lines

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课程笔记

What is an angle?

An angle is formed when two lines meet at a point.
Each line is called a side of the angle; the point they share is the vertex.
The size of an angle is measured in degrees (°).
A full turn is 360°, so a half turn is 180° and a quarter turn is 90°.
Angles can be named using three letters with the vertex in the middle, e.g. angle BAC, or just by the vertex letter, e.g. angle A.

Types of angle

Acute: less than 90°.
Right: exactly 90°.
Obtuse: between 90° and 180°.
Straight: exactly 180°.
Reflex: between 180° and 360°.
Zero: 0° (not turned).

Measuring and drawing angles

Use a protractor to measure angles accurately.
Place the centre of the protractor on the vertex and line up the 0° mark with one side.
Read the scale that starts from 0° on the side you lined up.
To draw an angle, draw one side, place the protractor correctly, mark the required degree, then draw the second side.
Estimate first: acute angles are less than 90°, obtuse between 90° and 180°, reflex greater than 180°.

Angle facts: straight lines and points

Angles on a straight line add up to 180°.
Angles around a point add up to 360°.
Vertically opposite angles are equal.
These facts let you find missing angles without measuring.

Angles in triangles and quadrilaterals

The angles in a triangle add up to 180°.
The angles in a quadrilateral add up to 360°.
In an isosceles triangle, the angles opposite the equal sides are equal.
In an equilateral triangle, all angles are 60°.

Parallel lines and transversals

Parallel lines are always the same distance apart and never meet.
A transversal is a line that crosses two or more lines.
When a transversal crosses parallel lines, it creates equal and supplementary angle pairs.
Arrow markings (▸▸) show lines are parallel.

Alternate, corresponding and co-interior angles

Alternate angles are equal. They are on opposite sides of the transversal and between the parallel lines (forming a Z shape).
Corresponding angles are equal. They are in the same position at each intersection (forming an F shape).
Co-interior angles add up to 180°. They are on the same side of the transversal and between the parallel lines (forming a C or U shape).
These angle facts are true only when the lines are parallel.

Giving reasons

For each step in an angle calculation, write a reason.
Common reasons: angles on a straight line sum to 180°; angles around a point sum to 360°; vertically opposite angles are equal; angles in a triangle sum to 180°; alternate angles are equal; corresponding angles are equal; co-interior angles sum to 180°.
Use the correct notation: angle ABC or ∠ABC, and the degree symbol °.
Show all steps clearly so your method can be followed.

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练习题

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1.Which type of angle measures exactly 90°?
Easy
- AAcute
- BRight
- CObtuse
- DStraight
2.An angle that is greater than 90° but less than 180° is called:
Easy
- AAcute
- BRight
- CObtuse
- DReflex
3.What is the sum of angles on a straight line?
Easy
- A90°
- B180°
- C270°
- D360°
4.What is the sum of angles around a point?
Easy
- A90°
- B180°
- C270°
- D360°
5.In a triangle, the angles are 50° and 60°. What is the third angle?
Easy
- A60°
- B70°
- C80°
- D90°
6.Two angles are vertically opposite. If one angle is 40°, what is the other?
Easy
- A40°
- B50°
- C140°
- D180°
7.A quadrilateral has angles 80°, 100°, and 90°. What is the fourth angle?
Medium
- A80°
- B90°
- C100°
- D110°
8.Two parallel lines are cut by a transversal. Which pair of angles are equal?
Medium
- ACo-interior angles
- BAlternate angles
- CSupplementary angles
- DAdjacent angles on a straight line

复习笔记

## Angle Facts and Parallel Lines ### What is an angle? An angle is formed by two lines that meet at a point. Each line is a **side** of the angle; the shared point is the **vertex**. The size of an angle is its **measure**. Notation: \(\angle A\) (vertex alone) or \(\angle BAC\) (vertex in the middle). The measure is written \(m\angle A\). ### Types of angle - **Zero**: 0° - **Acute**: less than 90° - **Right**: exactly 90° (lines are perpendicular) - **Obtuse**: between 90° and 180° - **Straight**: exactly 180° - **Reflex**: between 180° and 360° A full turn is 360°. ### Angle facts - **Straight line**: angles add to 180°. - **Around a point**: angles add to 360°. - **Vertically opposite**: equal. - **Triangle**: angles add to 180°. - **Quadrilateral**: angles add to 360°. ### Parallel lines A transversal crossing parallel lines creates: - **Alternate angles** — equal (Z-shape). - **Corresponding angles** — equal (F-shape). - **Co-interior angles** — add to 180° (C-shape). ### Angle addition If D lies inside \(\angle BAC\), then \(m\angle BAC = m\angle BAD + m\angle DAC\). ### Exam tips - Give a **reason** for every step. - Use letters (a, b, c) or Greek letters (α, β, γ, θ) for unknown angles. - Check whether angles are on a line, around a point, or in a shape before calculating.

Angle: A figure formed by two lines meeting at a point; also the size of the opening between them.
Vertex: The point where the two sides of an angle meet.
Acute angle: An angle less than 90°.
Right angle: An angle of exactly 90°.
Obtuse angle: An angle between 90° and 180°.
Straight angle: An angle of exactly 180°.
Reflex angle: An angle between 180° and 360°.
Vertically opposite angles: Angles opposite each other when two lines cross; they are equal.
Alternate angles: Angles on opposite sides of a transversal between parallel lines; they are equal.
Corresponding angles: Angles in matching positions on parallel lines cut by a transversal; they are equal.
Co-interior angles: Angles on the same side of a transversal between parallel lines; they add to 180°.

闪卡

What is the vertex of an angle?

The vertex is the point where the two sides (lines or rays) of the angle meet.

How is an angle named using three points, and which point goes in the middle?

Use the angle symbol with three points, e.g. angle BAC, where the vertex is always the middle letter. So angle BAC has vertex A.

What is an acute angle?

An angle smaller than a right angle, so less than 90°.

What is a right angle, and what is the relationship between two lines that form one?

A right angle is 90°, which is 1/4 of a full turn. Two lines that form a right angle are perpendicular (also called normal or orthogonal).

What is an obtuse angle?

An angle larger than a right angle but smaller than a straight angle, so between 90° and 180°.

What is a straight angle?

An angle of 180°, which is 1/2 of a full turn.

What is a reflex angle?

An angle larger than a straight angle but less than a full turn, so between 180° and 360°.

What is a full angle (full turn) in degrees?

A full angle is 360°, which is 1 complete turn.

What is a zero angle?

An angle of 0°, where the two sides coincide (no turn).

What is the angle addition postulate?

If D lies inside angle BAC, then m angle BAC = m angle BAD + m angle DAC. In words, the whole angle equals the sum of the two parts.

What do angles on a straight line add up to?

Angles on a straight line add up to 180°.

What do angles around a point add up to?

Angles around a point add up to 360°.

What are vertically opposite angles, and what is true about them?

When two straight lines cross, the angles opposite each other are vertically opposite. They are equal.

What do the angles in a triangle add up to?

The angles in a triangle add up to 180°.

What do the angles in a quadrilateral add up to?

The angles in a quadrilateral add up to 360°.

What are alternate angles on parallel lines, and what is true about them?

Alternate angles are on opposite sides of the transversal and between the parallel lines, in a Z shape. They are equal.

What are corresponding angles on parallel lines, and what is true about them?

Corresponding angles are in matching positions at each intersection, in an F shape. They are equal.

What are co-interior angles on parallel lines, and what is true about them?

Co-interior angles are on the same side of the transversal between the parallel lines, in a C or U shape. They add up to 180°.

What is a transversal?

A line that crosses two or more other lines. On parallel lines it creates alternate, corresponding and co-interior angle pairs.

Why must you give a reason for each step of an angle calculation?

Each step must be justified by an angle fact, such as angles on a straight line sum to 180° or alternate angles are equal, so the reasoning is clear and correct.

What is the sum of angles on a straight line, and what is the sum of angles around a point?

Angles on a straight line sum to 180°; angles around a point sum to 360°.

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