Revision Notes: Angles - With their Types and Properties | Mathematics Class 6 ICSE PDF Download

Lines and Angles

Angle and Points

• An Angle is a figure formed by two rays with a common endpoint, called the vertex.
  
• Angles can have points in the interior, in the exterior or on the angle.
  

Points A, B and C are on the angle. D is the interior and E is in the exterior. 
B is the vertex.

Types of Angles

Angles are classified by their measures in degrees. The common types are:

• Zero angle: An angle of 0°. Both rays coincide.
• Acute angle: An angle greater than 0° and less than 90° (0° < angle < 90°).
• Right angle: An angle of exactly 90°. It forms a square corner.
• Obtuse angle: An angle greater than 90° and less than 180° (90° < angle < 180°).
• Straight angle: An angle of exactly 180°. Its sides form a straight line.
• Reflex angle: An angle greater than 180° and less than 360° (180° < angle < 360°).
• Full angle (complete angle): An angle of exactly 360°, a full turn.

To measure angles we use a protractor. Angles with equal measures are called congruent angles.

Pair of Angles

There are several important relationships between two angles. These pairs are used frequently in geometric reasoning.

• Adjacent angles: Two angles are adjacent if they have a common vertex, a common side, and their interiors do not overlap.
• Linear pair: Two adjacent angles whose non‑common sides form a straight line. The sum of the measures of a linear pair is 180°.
• Vertically opposite angles: When two lines intersect, they form two pairs of vertically opposite angles. Vertically opposite angles are equal.
• Complementary angles: Two angles whose measures add up to 90°.
• Supplementary angles: Two angles whose measures add up to 180°.
• Congruent angles: Angles that have equal measures.

Examples of facts you should remember:

• If ∠P and ∠Q are a linear pair, then ∠P + ∠Q = 180°.
• If two lines intersect and one angle is 50°, then the vertically opposite angle is also 50° and the adjacent angles are 130° (since 50° + 130° = 180°).

Angles Made by a Transversal

Transversal: A line that intersects two or more lines in the same plane at distinct points is called a transversal.

When a transversal cuts two lines, several named angle pairs are formed. If the two lines are parallel, these pairs have special relationships.

• Corresponding angles: Angles that are in the same relative position at each intersection. If the two lines are parallel then corresponding angles are equal.
• Alternate interior angles: Angles on opposite sides of the transversal and inside the two lines. If the two lines are parallel then alternate interior angles are equal.
• Alternate exterior angles: Angles on opposite sides of the transversal and outside the two lines. If the two lines are parallel then alternate exterior angles are equal.
• Co‑interior (consecutive interior or same‑side interior) angles: Two interior angles on the same side of the transversal. If the two lines are parallel then co‑interior angles are supplementary; their sum is 180°.

Important consequences when two lines are parallel and cut by a transversal:

• If one corresponding angle equals 70°, every corresponding angle equals 70°.
• If one alternate interior angle equals 110°, the other alternate interior angle equals 110°.
• If one co‑interior pair is given as x and y then x + y = 180°.

Worked Examples

Example 1. Two lines are intersected by a transversal. If one of the corresponding angles is 65°, find the alternate interior angle equal to it.

Sol.

The corresponding angle given is 65°.

Corresponding angles are equal when the two lines are parallel.

Therefore the alternate interior angle equal to it is 65°.

Example 2. Two parallel lines are cut by a transversal. One interior angle on the same side of the transversal measures 120°. Find the measure of its co‑interior partner.

Sol.

Co‑interior angles are supplementary when the lines are parallel.

So their measures add to 180°.

Required angle = 180° − 120°.

Required angle = 60°.

How to Identify Angle Types Quickly

• Look at the vertex and the two rays forming the angle to name it (e.g., ∠XYZ).
• If the angle measures are not given, use relationships such as vertically opposite, complementary, supplementary, corresponding or alternate to find unknown measures.
• Use a protractor to measure an angle directly when needed.