Angles & angle relationships

What you will learn

name the five types of angle,
use the angle sums on a line, at a point, and in a triangle,
use the properties of vertically opposite, co-interior, alternate, and corresponding angles,
find unknown angles in geometric diagrams.

The five named angle types (each arc shows the angle).

Angles on a straight line sum to 180 deg

a + b = 180^\circ.

Angles at a point sum to 360 deg

a + b + c + \ldots = 360^\circ.

Vertically opposite angles are equal

When two lines cross, the pair of “opposite” angles are equal.

a = c, \qquad b = d.

Interior angles of a triangle sum to 180 deg

\alpha + \beta + \gamma = 180^\circ.

Worked example 1 Unknown angle on a line

Two angles on a line are $x$ and $125^\circ$ . Find $x$ .

x + 125 = 180 \;\Longrightarrow\; x = 55^\circ.

Worked example 2 Angles at a point

At a point, four angles are $90^\circ$ , $x$ , $120^\circ$ , and $40^\circ$ . Find $x$ .

90 + x + 120 + 40 = 360 \;\Longrightarrow\; x = 110^\circ.

Worked example 3 Triangle angle

In a triangle, two angles are $42^\circ$ and $73^\circ$ . Find the third.

180 - (42 + 73) = 180 - 115 = 65^\circ.

When a straight line (the transversal) crosses two parallel lines, three special pairs of equal or supplementary angles appear.

Corresponding angles are equal

Same corner position on each parallel line. Look for an “F” shape.

Alternate (Z) angles are equal

On opposite sides of the transversal, inside the parallels. Look for a “Z” shape.

Co-interior (C) angles are supplementary (sum to 180 deg)

On the same side of the transversal, inside the parallels. Look for a “C” shape.

a + b = 180^\circ.

The three angle patterns on parallel lines. F = corresponding (equal), Z = alternate (equal), C = co-interior (sum to 180°).

Worked example 4 Alternate angles

Two parallel lines are crossed by a transversal. One of the alternate angles is $68^\circ$ . What is the size of the other alternate angle?

Alternate angles are equal, so the other is $68^\circ$ .

Worked example 5 Co-interior

On a diagram of two parallel lines cut by a transversal, one co-interior angle is $115^\circ$ . Find the other.

115 + x = 180 \;\Longrightarrow\; x = 65^\circ.

Fluency

Classify as acute, right, obtuse, straight, or reflex: $55^\circ$ .
Classify: $90^\circ$ .
Classify: $142^\circ$ .
Classify: $210^\circ$ .
Classify: $180^\circ$ .
Two angles on a line are $x$ and $78^\circ$ . Find $x$ .
Two angles on a line are $x$ and $119^\circ$ . Find $x$ .
Two angles at a point are $130^\circ$ and $x$ . (These are the only two.) Find $x$ .
Three angles at a point are $90^\circ$ , $150^\circ$ , $x$ . Find $x$ .
Two lines cross. One of the angles is $63^\circ$ . Find the other three.
In a triangle, two angles are $40^\circ$ and $60^\circ$ . Find the third.
In an isosceles triangle, the apex angle is $80^\circ$ . Find each base angle.
In a right-angled triangle, one of the non-right angles is $35^\circ$ . Find the other.
A transversal cuts two parallel lines. Corresponding angles of $x$ and $73^\circ$ . Find $x$ .
A transversal cuts two parallel lines. Alternate angles of $x$ and $112^\circ$ . Find $x$ .
A transversal cuts two parallel lines. Co-interior angles of $x$ and $108^\circ$ . Find $x$ .

Reasoning

Three angles on a straight line are $x$ , $2x$ and $60^\circ$ . Find $x$ .
At a point the angles are $x$ , $x + 40^\circ$ and $150^\circ$ , with no other angles. Find $x$ .
In a triangle the angles are in the ratio $1 : 2 : 3$ . Find each angle.
In a triangle the angles are $2x$ , $3x$ and $4x$ . Find $x$ and each angle.
An exterior angle of a triangle is $120^\circ$ . The two interior angles not adjacent to it sum to what?
An isosceles triangle has a base angle of $72^\circ$ . Find the apex angle.
Two parallel lines are cut by a transversal. One co-interior angle is $3x$ and the other is $5x$ . Find $x$ .
A right-angled triangle has angles $90^\circ$ , $x$ and $2x + 15^\circ$ . Find $x$ .
Three angles around a point are $x$ , $110^\circ$ and $2x$ . Find $x$ .
Two parallel lines have transversal angles of $2x + 20^\circ$ and $x + 70^\circ$ as corresponding angles. Find $x$ .

Reasoning

A student claims “vertically opposite angles add to $180^\circ$ ”. Is this always true? If not, when is it wrong?
Emma says “the three angles in a triangle always sum to $180^\circ$ , so any three angles that add to $180^\circ$ form a triangle”. Is Emma correct? Justify with an example.
Tom says a co-interior angle pair must be equal. What is Tom mixing up? Give the correct relationship.
Is it possible for a triangle to have two right angles? Explain.

Problem solving

A clock shows $3{:}00$ . What is the angle between the hands?
A clock shows $6{:}00$ . What is the angle between the hands?
A staircase makes a $35^\circ$ angle with the floor. What angle does it make with the wall (assumed vertical)?
A sign is tilted $15^\circ$ from vertical. What angle does it make with the horizontal ground?
A road crosses two parallel train tracks. One of the acute angles at the crossing is $48^\circ$ . What are the sizes of the other three angles at each crossing?
A triangular piece of land has one angle of $90^\circ$ and another of $53^\circ$ . What is the third angle?