Home → Curriculum Hub → Maths Lessons → Angle Properties → Problem Solving with Angle Properties
Ready-to-teach lesson. Includes a video tutorial, printable worksheets and KS3 / Foundation GCSE exam-style practice with worked solutions.
Students learn problem solving with angle properties by combining several angle facts in one diagram. The lesson covers straight lines, angles around a point, triangles, quadrilaterals and isosceles triangles.
Most students arrive able to state each angle fact on its own. They can find one missing angle in a triangle or on a straight line. However, exam diagrams often ask for three or four steps in order.
This lesson treats every diagram as a chain of clues. Students who justify each step move on to angles in parallel lines and angles of polygons without relearning the topic.
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Watch this tutorial first to see how a diagram is worked through one angle at a time. Download the handout to work along with the video, then try the worksheet below.
Begin by retrieving the facts students are about to combine. Every one of them is already familiar.
The difficulty is not recall. It is choosing which fact to use, and in what order.
Ask students to state each fact and point to an example of it on the board. A fact they cannot locate in a diagram is a fact they will not use.
Six questions with full solutions, for classwork, homework or independent revision. Questions 1 to 3 mix quadrilaterals and triangles, so students choose the property before they calculate.
Questions 4 to 6 combine isosceles triangles with straight lines. Each needs at least two properties, and the final answer depends on getting the first step right.
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The starter gives two angles and asks for a third. The 104° angle sits outside the triangle. The 72° angle sits below the base line.
That placement is deliberate. Students who reach straight for the angle sum find they have nothing to add up.
Angles on a straight line turn 104° into 76°. Vertically opposite angles turn 72° into an interior angle. The angle sum then gives m = 32°.
Prompts / Questions to consider
The teaching example has four unknowns and one pair of tick marks. Work through it in the order the diagram allows, not left to right.
Start with the tick marks. They make the right-hand triangle isosceles, so the 66° angle and a are base angles. That gives a = 66°, then b = 180° − 66° − 66° = 48°.
The remaining two angles use different properties again. c and 66° are angles on a straight line, so c = 114°. d is vertically opposite 34°, so d = 34°.
Ask students to write the property beside each answer as they go. That habit is what the exam questions reward.
Take answers on mini-whiteboards after a and b. It shows quickly who read the tick marks and who guessed.
Prompts / Questions to consider
The plenary moves from arithmetic to algebra. Pose both diagrams as they are, with no extra scaffolding.
In the first diagram the three angles of the large triangle are 40°, 32° + r and 2r. The angle sum gives 40 + 32 + r + 2r = 180, so 3r = 108 and r = 36°.
The challenge is harder, and both marked angles sit outside the triangle. The tick marks make two sides equal, so two interior angles are equal.
Each base angle is 180° − (t + 30°), which is 150° − t. The exterior angle t equals the two remote interior angles added together, so t = 2(150 − t).
Solving gives 3t = 300 and t = 100°. The interior angles are 80°, 50° and 50°.
Prompts / Questions to consider
More able: ask for a second route through the teaching example, then move to the challenge diagram where the answer comes from an equation rather than a sum.
Less able: give the diagram with the first angle already labelled, and keep the list of angle facts on the desk. The learning point is choosing a property, not recalling it.
| Phase | Focus | Time |
|---|---|---|
| Starter | Two properties before the angle sum: m = 32° | 5 to 10 min |
| Development | Four unknowns in one diagram, each with a reason | 15 to 20 min |
| Check | Mini-whiteboards: a = 66°, b = 48° | 10 min |
| Main / stretch | Independent practice on the worksheet | 15 to 20 min |
| Plenary | Forming equations: r = 36°, then t = 100° | 5 to 10 min |
Use this checklist after the lesson or as a revision self-check.
Try these KS3 and Foundation GCSE-style questions on problem solving with angle properties, then reveal the solutions to check your working. Select the image to view it full screen.
The triangle sits on a straight line. Work out angles a, b and c, and give a reason for each one.
a and 135° are angles on a straight line, so a = 180° − 135° = 45°.
The angles in the triangle add to 180°, so b = 180° − 108° − 45° = 27°.
b and c are angles on a straight line, so c = 180° − 27° = 153°.
Check the answer with the exterior angle property: 108° + 27° = 135°.
The tick marks show two equal sides. Work out angles p and q.
q and 110° are angles on a straight line, so q = 180° − 110° = 70°.
The tick marks make the triangle isosceles, so the angle at the top vertex is also 70°.
The angles in the triangle add to 180°, so p = 180° − 70° − 70° = 40°.
The angle marked 298° is a reflex angle. Work out angle a.
Angles around a point add to 360°, so the angle at the apex is 360° − 298° = 62°.
In the right-hand triangle the angles add to 180°, so the third angle is 180° − 62° − 42° = 76°.
That 76° angle is on a straight line with the angle inside the left-hand triangle, so that angle is 180° − 76° = 104°.
The tick marks make the left-hand triangle isosceles, so a = (180° − 104°) ÷ 2 = 38°.
The diagram has rotational symmetry of order three. Work out angle x.
The matching arcs show both base angles are equal, so each one is 72°.
The angles in the triangle add to 180°, so the angle at the centre is 180° − 72° − 72° = 36°.
Three identical triangles meet at the centre, so they use 3 × 36° = 108°.
Angles around a point add to 360°, and the three gaps are equal, so x = (360° − 108°) ÷ 3 = 84°.
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Finding a missing angle by applying two or more angle facts in sequence. Each step uses a named property, such as angles on a straight line, and each answer feeds the next step.
Angles on a straight line add to 180°. Angles around a point add to 360°. Angles in a triangle add to 180°. Angles in a quadrilateral add to 360°. Vertically opposite angles are equal, and the base angles of an isosceles triangle are equal.
They label every angle they can find from a single property, without planning a full route first. The next step usually becomes clear once two or three angles are written on the diagram.
Because the question asks for a reason. A correct number with no property named will not gain the reasoning mark on most mark schemes.
Angles in parallel lines, then angles of polygons. Both add new properties to the same chain of reasoning rather than replacing it.
The natural next lesson is angles in parallel lines, which adds alternate and co-interior angles to the same chain of reasoning. Angles of polygons follow. For the full topic hub, visit All Angle Properties Lessons.
If you want to try one thing tomorrow, put the starter diagram on the board and ask for the property before the number. How many students reach for the angle sum first tells you where to spend the lesson.
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Students learn to solve geometric problems using properties of triangles, straight lines, and angle facts like vertically opposite and corresponding angles. They apply reasoning to find unknown values.
As they progress, students tackle multi-step problems with isosceles triangles and reflex angles, strengthening their ability to justify relationships and structure solutions logically.
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