6.6 Geometrical Problems

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: multi-step angle problems: combining facts, giving reasons, using algebra and “show that” questions. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Choose and combine angle facts to solve multi-step problems
• Give a correct reason for every step of your working
• Form and solve equations from angle facts
• Answer “show that” questions with a clear chain of reasoning

Key words

• Reason: The angle fact that explains a step, written in full, e.g. “angles in a triangle add up to 180°”.
• Show that: You are given the answer. You must show every step that leads to it.
• Hence: Use the answer you have just found to do the next part.
• Not drawn accurately: Do not measure. Work the angles out using facts.

A plan for angle problems

1. Copy the given angles onto the diagram.

2. Look for: straight lines (180°), points (360°), vertically opposite angles, triangles (180°), quadrilaterals (360°), isosceles triangles (dashes), parallel lines (arrows), regular polygons.

3. Work out any angle you can, write it on the diagram and give the reason.

4. Keep going until you reach the angle you need.

• Straight line; Reason to write Angles on a straight line add up to 180°.
• Point; Reason to write Angles around a point add up to 360°.
• Crossing lines; Reason to write Vertically opposite angles are equal.
• Triangle; Reason to write Angles in a triangle add up to 180°.
• Isosceles triangle; Reason to write Base angles of an isosceles triangle are equal.
• Quadrilateral; Reason to write Angles in a quadrilateral add up to 360°.
• Parallel lines; Reason to write Alternate / corresponding angles are equal; co-interior angles add up to 180°.

Multi-step problems with reasons

Worked example: three steps, three reasons

Angle ACB = 180 − 112 = 68° (angles on a straight line add up to 180°)
Angle ABC = 68° (base angles of an isosceles triangle are equal)
Angle BAC = 180 − 68 − 68 = 44° (angles in a triangle add up to 180°)

Answer: angle BAC = 44°

Algebra in angle problems

Worked example: form and solve an equation

Angles 2x + 15° and 3x − 5° lie on a straight line. Find each angle.
2x + 15 + 3x − 5 = 180 (angles on a straight line add up to 180°)
5x + 10 = 180, 5x = 170, x = 34. Angles: 83° and 97°.

Answer: 83° and 97°

Shapes put together

Worked example: square and equilateral triangle

Angle ABE = 90 + 60 = 150° (angle of a square + angle of an equilateral triangle)
AB = BE (both equal to the side of the square), so triangle ABE is isosceles.
Angle BAE = (180 − 150) ÷ 2 = 15°

Answer: 15°

“Show that” questions

You are told the answer, so a correct answer alone gets no marks. Show every step with a reason, and finish by stating what you have shown.

Worked example: show that a triangle is isosceles

A triangle has angles of 40° and 70°. Show that it is isosceles.
Third angle = 180 − 40 − 70 = 70° (angles in a triangle add up to 180°).
Two angles are 70°, so two sides are equal.

Answer: the triangle is isosceles

Common mistakes

• Measuring a diagram that is “not drawn accurately”.
• Giving short reasons such as “Z angles” or “isosceles”. Write the full fact.
• Using a fact that is not shown: only use parallel-line facts if the lines have arrows, and isosceles facts if sides are marked equal.
• In a “show that”, writing only the given answer without the working.

Revision questions

Try these, then check your answers below each one.

Question 1. Triangle ABC has AB = AC. BC is extended to D so that BCD is a straight line. Angle ACD = 124°. Work out angle BAC.

Answer: 68 °

Working: ACB = 180 - 124 = 56°. ABC = 56° (base angles). BAC = 180 - 56 - 56 = 68°.

Question 2. Lines AB and CD are parallel. B and C are joined. Angle ABC and angle BCD are on opposite sides of BC, between the parallel lines. Which is the correct reason why they are equal? (Choose from: Alternate angles are equal., Corresponding angles are equal., Co-interior angles add up to 180°., Vertically opposite angles are equal..)

Answer: Alternate angles are equal.

Working: They form a Z shape, so they are alternate angles, which are equal.

Question 3. ABCDEF is a regular hexagon. A square ABGH is drawn on the outside of side AB. Triangle AFH is formed by joining F to H. Work out angle AFH.

Answer: 15 °

Working: FAH = 360 - 120 - 90 = 150°. AF = AH, so AFH = (180 - 150) ÷ 2 = 15°.

Related lessons

• Previous lesson: 6.5 More Exterior and Interior Angles
• Next lesson: 7.1 Mean and Range
• In this chapter: 6.1 Properties of Shapes
• In this chapter: 6.2 Angles in Parallel Lines
• In this chapter: 6.3 Angles in Triangles

Continue in the full course

This lesson is section 6.6 (EGCMF0606) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.