Angles

1Summary 重點摘要

Right angle 90°, straight angle 180°, full turn 360°; angles on a line sum to 180°, vertically opposite angles are equal — these rules solve for angles.

2Vocabulary cards Vocabulary

3Step by step Steps

1. 1. What is an angle?An angle is formed by two rays starting from the same point, and it measures how far the two rays open apart. When line OA and line OB meet at O, angle AOB is formed, with O called the vertex and the two rays called the arms. Angles appear everywhere: between the hands of a clock, in a pair of open scissors, in an opened book. When naming an angle, the vertex letter must sit in the middle, as in angle AOB; if a single letter is used, it must be the capital vertex letter, such as angle O, otherwise angles sharing a vertex cannot be told apart.
2. 2. Parts of an angleAn angle consists of two arms and a vertex: the arms are rays from the same point, and the vertex is where they meet. In angle AOB, OA and OB are the arms and O is the vertex. To measure or draw an angle, place the protractor's centre on the vertex, align its zero line with one arm, and read the scale where the other arm points. Common errors are placing the centre away from the vertex, or reading the wrong ring of the scale, turning 40° into 140°; estimate first whether the angle is acute or obtuse and the mistake shows at once.
3. 3. Unit: degreeAngles are measured in degrees (°), with a full turn divided into 360 equal parts of one degree each. A full circle is therefore 360°, half a circle is 180°, and a quarter circle is 90°. The clock is a ready example: the minute hand turns 360° in 60 minutes, which is 6° per minute, and moving from the 12 to the 3 covers a quarter turn of 90°. Memorise these basic values, because adjacent angles on a line and angles at a point later all build on 180° and 360°; a wrong base value corrupts every calculation that follows.
4. 4. Right angleA right angle equals 90° and is the most important benchmark among angles. All four corners of a square or rectangle are right angles, and so are the angles between two perpendicular lines. Diagrams mark a right angle with a small square symbol, and the symbol alone tells you the angle is 90°. To test whether an angle is right, stack a set square's right angle onto it; larger is obtuse and smaller is acute. Never judge by eye alone: without the symbol or a statement in the question, confirm with a protractor.
5. 5. Acute angleAn acute angle is smaller than 90°; 30°, 45° and 60° are the common examples, and the smaller angles on a set square — 45° and 30°, 60° — are all acute. An acute angle opens less than a right angle and looks sharp. To classify one, compare against a right angle: align one arm with the set square, and if the other arm falls inside, the angle is acute. Mind the boundary: an acute angle is strictly greater than 0° and strictly less than 90°, so counting 90° itself as acute is a common error, since 90° is a right angle.
6. 6. Obtuse angleAn obtuse angle is greater than 90° and less than 180°; 120°, 150° and 170° are typical values. An obtuse angle opens wider than a right angle but has not reached a straight angle. Everyday examples include a laptop screen opened beyond a quarter turn, or the angle between the clock hands at 10:10. To classify any angle as acute, right or obtuse, first compare it with 90°. Both endpoints are excluded: 90° is right and 180° is straight, so an obtuse angle lies strictly between them, and filing 180° under obtuse is another common mistake.
7. 7. Straight angleA straight angle equals 180°, its two arms lying on the same straight line and pointing in opposite directions, so it looks exactly like a line. It equals two right angles: 2 × 90° = 180°. Straight angles underpin adjacent angles on a line: all angles formed at any point on a straight line sum to 180°, for example 118° and 62° fitting together into one straight angle. A common error is dismissing a straight angle as 'no angle at all'; it is a genuine angle whose arms form a line, and pointing out that two angles make a straight angle is a standard argument in geometry.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Angle: two lines meet
• Has vertex and two arms
• Unit: degree (°)
• Right = 90°, straight = 180°
• Acute < 90°, obtuse 90°-180°