Angles in Geometry

A lesson on measuring and classifying angles, using angle properties on lines and in triangles, and solving for unknown angles.

Measuring Angles

Angles are measured in degrees (°) using a protractor. A full turn = 360°.

Acute (< 90°), right (= 90°), obtuse (90°–180°).

Angle Types

Name	Measure
Zero	0°
Acute	0° – 90°
Right	90°
Obtuse	90° – 180°
Straight	180°
Reflex	180° – 360°
Full turn	360°

Key Angle Rules

Angles on a Straight Line

\alpha + \beta = 180°

Angles on one side of a straight line sum to 180°.

Angles at a Point

\alpha + \beta + \gamma + \ldots = 360°

All angles around a point sum to 360°.

Vertically Opposite

\alpha = \gamma

Angles directly across from each other at a crossing are equal.

Angles in a Triangle

The three interior angles of any triangle always add up to 180°:

$\alpha + \beta + \gamma = 180°$

Angles in Quadrilaterals

The four interior angles of any quadrilateral sum to 360°.

1
Identify what you know
Write down given angle values.
2
Choose the right rule
Straight line, point, vertically opposite, or triangle?
3
Set up an equation
Use the sum rule to write an algebraic equation.
4
Solve for the unknown
Simplify and find the missing angle.

A triangle has angles 48° and 65°. Find the third angle.

$\alpha + \beta + \gamma = 180°$

$48 + 65 + \gamma = 180°$

$\gamma = 180° - 113° = 67°$

Real world: roof pitch in architecture

A builder designs a roof with one side rising at 35° and the other at 35°, meeting at the ridge. The angle between the two faces is $180° - 35° - 35° = 110°$ . Knowing supplementary angles helps architects ensure the structure is stable and symmetric.

Real world: road junction signs

A crossroads where one road meets another at 70° means the opposite angle (vertically opposite) is also 70°, and the adjacent angles are each $180° - 70° = 110°$ . Traffic engineers use these rules to plan safe junction layouts.

Real world: sports pitch markings

A football penalty area has corners marked at exactly 90°. Ground staff use the rule that the angles of a rectangle sum to 360° ( $4 \times 90°$ ) to verify that the lines are truly perpendicular and the area is square.

Tip When you see two lines crossing, always look for vertically opposite angles first — they give you a free value instantly.

Practice Problems

Two angles on a straight line are 3x and 2x. Find x and both angles.

Show answer

5x = 180° → x = 36°. Angles: 108° and 72°.

In a quadrilateral three angles are 85°, 100°, and 95°. Find the fourth.

Show answer

85 + 100 + 95 + d = 360° → d = 80°.