Angles: Lines, Points and Triangles
Basic angle facts are the rules that let missing angles be found from a diagram without measuring: angles on a straight line add up to 180 degrees, angles around a point add up to 360 degrees, vertically opposite angles, formed where two lines cross, are equal, and the three angles inside any triangle add up to 180 degrees. An isosceles triangle also has two equal base angles, and the exterior angle of a triangle equals the sum of the two interior angles that are not next to it.
Method
- Identify which angle fact applies: angles on a straight line (180), angles around a point (360), vertically opposite angles (equal), or angles in a triangle (180).
- Mark on the diagram any angles you can already work out, including ones found using symmetry, such as the equal base angles of an isosceles triangle.
- Set up an equation using the relevant angle fact, letting the unknown angle be x if it is not given directly.
- Solve the equation to find the missing angle.
- Check that your answer makes sense for the type of angle shown (for example, an angle marked as obtuse should be between 90 and 180 degrees).
- State your answer in degrees, with a reason based on the angle fact you used, such as 'angles on a straight line sum to 180 degrees'.
Worked example
Three angles lie on a straight line: 55 degrees, 2x degrees and (3x + 5) degrees. Find x, and find the size of each of the two unknown angles.
- Angles on a straight line sum to 180 degrees: 55 + 2x + (3x + 5) = 180.
- Simplify the left-hand side: 55 + 2x + 3x + 5 = 5x + 60, so 5x + 60 = 180.
- Subtract 60 from both sides: 5x = 120.
- Divide both sides by 5: x = 24.
- The angles are 2x = 48 degrees and 3x + 5 = 3(24) + 5 = 77 degrees. Check: 55 + 48 + 77 = 180.
Practice questions
Try each question, then tap to reveal the answer.
Q1Two angles on a straight line are 110 degrees and x. Find x.Show answer
Answer: x = 180 - 110 = 70 degrees
Q2Three angles around a point are 90 degrees, 130 degrees and x. Find x.Show answer
Answer: x = 360 - 90 - 130 = 140 degrees
Q3Two angles are vertically opposite. One is 65 degrees. Find the other.Show answer
Answer: 65 degrees, since vertically opposite angles are equal
Q4A triangle has angles 50 degrees, 70 degrees and x. Find x.Show answer
Answer: x = 180 - 50 - 70 = 60 degrees
Q5An isosceles triangle has an angle of 40 degrees at its apex. Find the size of each base angle.Show answer
Answer: Each base angle = (180 - 40) / 2 = 70 degrees
Q6Three angles on a straight line are 3x, 2x and x. Find x.Show answer
Answer: 6x = 180, so x = 30 degrees
Q7Four angles around a point are 2x, 3x, x and 4x. Find x.Show answer
Answer: 10x = 360, so x = 36 degrees
Q8A triangle has an exterior angle of 130 degrees at one vertex, where the two interior angles at the other two vertices are equal to each other. Find each of those interior angles.Show answer
Answer: The exterior angle equals the sum of the two opposite interior angles, so each equals 130 / 2 = 65 degrees
Exam-style questions
Written in the style of a KS3 Maths exam paper, with a full mark scheme.
Angles ABC and CBD lie on a straight line AD. Angle ABC = (3x + 10) degrees and angle CBD = (x + 30) degrees. Work out the value of x, and hence find the size of angle ABC.
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Triangle PQR is isosceles, with PQ = PR. Angle P = (x + 20) degrees and angle Q = 2x degrees. (a) Explain why angle R = angle Q. (b) Find the value of x, and state the size of each angle in the triangle.
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Free printable worksheet
Want more practice on paper? Download the angles: lines, points and triangles worksheet pack - 6 pages of exam-style questions with a full mark scheme. One email opens every download in this browser for 14 days - no account, no card. Print it for personal and classroom use.
This topic is chapter 4 of KS3 Maths Workbook 2, the whole course as one free printable PDF.
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