Angles on a Line and at a Point

Angles on a straight line sum to 180°. Angles around a full point sum to 360°. Vertically opposite angles are always equal.

Angles on a Straight Line

When angles are formed on one side of a straight line, they always add up to 180°.

$\alpha + \beta + \gamma = 180°$

Angles at a Point

All angles around a single point add up to 360°.

$a + b + c + d = 360°$

Vertically Opposite Angles

When two straight lines cross, the angles directly across from each other are equal.

Vertically Opposite Angles

\alpha = \gamma \quad \text{and} \quad \beta = \delta

Angles at a crossing of two lines come in equal pairs.

Note Vertically opposite angles are sometimes called "vertical angles." They are always equal, regardless of the angle values.

Angles Formed by a Transversal

When a line (transversal) crosses two parallel lines:

Corresponding angles are equal (same position at each intersection)
Alternate interior angles are equal (between the lines, opposite sides)
Co-interior angles are supplementary (sum to 180°)

Find the missing angles

Two lines cross. One angle is 65°.

The vertically opposite angle is also 65°.

Its neighbour on the straight line is $180° - 65° =$ 115°.

The fourth angle is also 115° (vertically opposite to 115°).

Real world: road junction

At a crossroads two roads meet in a straight line. A side road branches off at 65° on one side, leaving $180° - 65° = 115°$ on the other — angles on a straight line always add to 180°. Town planners use this to design safe sight lines.

Real world: scissor blades

When scissors are partly open, the two blades and the straight handle form supplementary angles. If one blade opens to 40° above the line, the other side is $180° - 40° = 140°$ .

Real world: compass bearings

A hiker walks on a bearing of 120° from North. The angle on the other side of the North line is $360° - 120° = 240°$ . Navigators use angles around a point (summing to 360°) constantly.

Practice Problems

Three angles on a straight line are x, 2x, and 3x. Find x.

Show answer

x + 2x + 3x = 180° → 6x = 180° → x = 30°.

Two lines intersect. One angle is 42°. List all four angles.

Show answer

42°, 138°, 42°, 138°.