CBSE Class 10 Mathematics Lines and Angles Notes Set 07

Basic Terms

A line segment has two end points; a ray has one end point; a line extends endlessly in both directions. Two rays with a common end point form an angle. Angles are classified as acute (less than 90°), right (90°), obtuse (between 90° and 180°), straight (180°) and reflex (between 180° and 360°).
Two angles are complementary if their sum is 90° and supplementary if their sum is 180°. The complement of x is 90° − x and its supplement is 180° − x.

Pairs of Angles

1. Linear pair: two adjacent angles whose non-common arms form a straight line; their sum is 180°.
2. Vertically opposite angles: when two lines intersect, the angles opposite each other are equal.
3. Angles around a point add up to 360°; angles on a straight line add up to 180°.
4. If a ray stands on a line, the sum of the two adjacent angles formed is 180°.

Parallel Lines and a Transversal

A line that cuts two or more lines at different points is a transversal. When a transversal cuts two parallel lines:
1. Corresponding angles are equal.
2. Alternate interior angles are equal.
3. Interior angles on the same side of the transversal are supplementary.
Conversely, if any one of these conditions holds, the two lines are parallel.
Lines parallel to the same line are parallel to each other.

Intercepts on Transversals

Three or more parallel lines cut any two transversals in the same ratio: if parallel lines meet one transversal at A, B, C and another at D, E, F, then AB : BC = DE : EF. If the intercepts on one transversal are equal, the intercepts on every other transversal are also equal.

Angle Sum Property

The three angles of a triangle add up to 180°, and an exterior angle of a triangle equals the sum of the two interior opposite angles. These facts are often used with parallel lines to find unknown angles.

Worked Examples

Illustration 1. Find an angle whose supplement is three times its complement.
Answer:
180 − x = 3(90 − x), so 180 − x = 270 − 3x and 2x = 90.
Answer: x = 45°.

Illustration 2. Lines AB and CD are parallel and a transversal makes alternate interior angles (2x + 10)° and (3x − 20)°. Find x.
Answer:
Alternate interior angles are equal: 2x + 10 = 3x − 20.
Answer: x = 30, and each angle is 70°.

Illustration 3. Parallel lines cut transversals so that the intercepts on one are 6 cm and 9 cm. If the first intercept on the other is 4 cm, find the second.
Answer:
Intercepts are proportional: 6 : 9 = 4 : y, so y = 6 cm.