An essential concept in geometric discussions is the angle. In Grade 7, first learn how angles are represented and named, as well as their types. We will also organize positional relationships such as vertical angles, corresponding angles, and alternate angles.
Notation for Angles
A figure formed by two rays extending from one point is called an angle.
・Represent it with three points, placing the vertex in the middle: ∠AOB (or ∠BOA)
・It can also be represented using one symbol: ∠O, ∠a
The size of an angle is measured in degrees (°). It takes a value from 0° to 360°.
Types of Angles
- Acute angle: An angle greater than 0° and less than 90°.
- Right angle: An angle exactly equal to 90°.
- Obtuse angle: An angle greater than 90° and less than 180°.
- Straight angle: An angle exactly equal to 180° (forming a straight line).
Classify each angle as acute, right, obtuse, or straight.
(1) 35° (2) 90° (3) 120° (4) 180°
Solution:
(1) Acute (2) Right (3) Obtuse (4) Straight
Vertical, Corresponding, and Alternate Angles
Learn the positional relationships between angles formed when two lines intersect or when a transversal crosses two lines.
- Vertical angles: The two opposite angles formed when two lines intersect. They are always equal.
- Corresponding angles: Angles in the same position when a line crosses two lines.
- Alternate angles: Opposite angles arranged in a Z shape when a line crosses two lines.
Property of Parallel Lines (studied in greater detail in Grade 8, but useful to know in Grade 7):
Two lines are parallel ⇔ their corresponding angles are equal ⇔ their alternate angles are equal.
Of the four angles formed by two intersecting lines, one angle is 65°. Find the measures of the other three angles.
Solution:
Since vertical angles are equal, the opposite angle is 65°.
An adjacent angle is 180° − 65° ＝ 115°, and its vertical angle is also 115°.
Therefore, the four angles are 65°, 115°, 65°, and 115°.
- Types of angles: acute(＜90°), right(90°), obtuse(90°〜180°), straight(180°).
- Vertical angles are always equal.
- Two lines are parallel ⇔ their corresponding and alternate angles are equal.
- One angle + its adjacent angle = 180° (forming a straight line).
Practice Problems
- Is a 72° angle acute, right, obtuse, or straight?
- When two lines intersect and one angle is 38°, find the measure of its vertical angle.
- Lines ℓ // m are intersected by line n. If one of the angles formed by n and ℓ is 70°, find the measure of the corresponding angle formed by n and m.
Answers and Explanations
- Answer: Acute
Explanation: 72° is between 0° and 90°, so it is acute. - Answer: 38°
Explanation: Vertical angles are always equal. - Answer: 70°
Explanation: Corresponding angles are equal when lines are parallel.