Angle Geometry Problems

Why this matters

Surveyors, engineers and carpenters work out angles they cannot measure directly — the pitch of a roof, the angle of a support strut — from the ones they can. Each step uses a known fact, and writing down the reason is what turns a guess into a proof.

The three shapes to look for

Angle pair	Shape	Rule
Alternate	Z	equal
Corresponding	F	equal
Co-interior (allied)	C	add to $180^\circ$

These only work when the lines really are parallel — shown by matching arrows on the lines.

Setting out a solution

Give each step on its own line with its reason in brackets:

$\angle x = 70^\circ$ (corresponding angles) $\angle y = 180 - 70 = 110^\circ$ (angles on a straight line)

Examiners give marks for the reasons, not just the numbers.

Kites and other shapes

Sahira thinks ABCD is a kite because AB = AD. A kite needs two pairs of equal adjacent sides. To prove or disprove it, find the angles and see whether the shape is symmetrical about a diagonal — one pair of equal sides is not enough.

Key rules and formulas

Alternate angles are equal

Look for a Z shape between the parallel lines. The two angles inside the Z are equal.

Corresponding angles are equal

Look for an F shape. Angles in the same position at each crossing are equal.

Co-interior angles add to 180°

Look for a C (or U) shape. The two angles between the parallel lines on the same side of the crossing line add to

18 0^{\circ}

The basics still apply

Angles on a straight line add to

18 0^{\circ}

; angles at a point add to

36 0^{\circ}

; vertically opposite angles are equal; angles in a triangle add to

18 0^{\circ}

Worked examples

Worked example 1

Is it a right-angled triangle?

ABCD is a trapezium with AD parallel to BC. Triangle ABD is isosceles with AB = AD, and angle DBC

= 4 1^{\circ}

. Is ABD a right-angled triangle?

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Using algebra

A parallelogram has angles

2 y

and

8 y

next to each other. Find all four angles.