CBSE Class 8 Mathematics Lines and Angles Notes Set 08

Basic Terms and Types of Angles

A ray has one end point; two rays with the same end point form an angle, and that common point is the vertex. Points on one line are collinear; three or more lines through one point are concurrent.
Acute angle: between 0° and 90°. Right angle: exactly 90°. Obtuse angle: between 90° and 180°. Straight angle: 180°. Reflex angle: more than 180° and less than 360°.

Pairs of Angles

Complementary angles add up to 90°; supplementary angles add up to 180°.
Adjacent angles share a vertex and an arm, with their other arms on opposite sides of it. If the non-common arms are opposite rays the pair is a linear pair, and a linear pair adds up to 180°.
When two lines intersect, the angles facing each other are vertically opposite angles and they are equal. Angles around a point add up to 360°.

Parallel Lines and a Transversal

A transversal is a line that cuts two or more lines at different points. If the two lines are parallel then:
corresponding angles are equal, alternate interior angles are equal, and interior angles on the same side of the transversal add up to 180°.
The converse also holds: if any one of these conditions is true, the two lines are parallel. Lines parallel to the same line are parallel to each other.
Useful trick: when a bent path joins two parallel lines, draw a line through the bend parallel to them and add the alternate angles on both sides.

Angles of a Triangle and a Polygon

The three angles of a triangle add up to 180°. An exterior angle of a triangle equals the sum of the two interior opposite angles.
The interior angles of an n-sided polygon add up to (n − 2) × 180°. The exterior angles of a convex polygon add up to 360°, so each exterior angle of a regular polygon is 360° ÷ n.

Worked Examples

Example 1: Find the angle which is 24° more than its complement. Let the angle be x, then x = (90° − x) + 24°, so 2x = 114° and x = 57°.
Example 2: Two parallel lines are cut by a transversal and two interior angles on the same side are (2x + 10)° and (3x − 20)°. They add up to 180°, so 5x − 10 = 180 and x = 38. The angles are 86° and 94°.
Example 3: The exterior angle of a regular polygon is 30°. Number of sides = 360° ÷ 30° = 12, and the sum of the interior angles is 10 × 180° = 1800°.