Angles on Lines, at Points and in Triangles
Three angle facts are tested repeatedly at 11+: angles on a straight line always sum to 180 degrees, angles meeting at a single point always sum to 360 degrees, and angles inside a triangle always sum to 180 degrees, with an isosceles triangle also having two equal base angles. Harder papers chain several of these facts together in one diagram, and sometimes write an unknown angle as an algebraic expression, such as 2x or (x + 20), so a student must form and solve an equation.
Method
- Learn the three core facts precisely: a straight line measures 180 degrees, angles around a point sum to 360 degrees, and angles in a triangle sum to 180 degrees.
- Learn the isosceles triangle rule: the two angles opposite the two equal sides are equal to each other.
- Mark every angle you are given directly onto the diagram, and label each unknown angle with a letter.
- Work outward from the angles you know, one fact at a time, noting which rule you used at each step, such as 'angles on a straight line'.
- When an unknown is written as an algebraic expression, form an equation using the correct angle fact, solve it for x, then substitute back to find the actual size of each angle.
- Check that your final angles really do obey the original rule, such as summing to 180 or 360, before giving your answer.
Worked example
Two angles lie on a straight line. One angle is 3x degrees and the other is (x + 40) degrees. Find the value of x and the size of each angle.
- Since the two angles lie on a straight line, they sum to 180 degrees: 3x + (x + 40) = 180.
- Simplify the left-hand side: 3x + x + 40 = 4x + 40, so 4x + 40 = 180.
- Subtract 40 from both sides: 4x = 140.
- Divide by 4: x = 35.
- Substitute back to find each angle: 3x = 3 x 35 = 105 degrees, and x + 40 = 35 + 40 = 75 degrees.
- Check: 105 + 75 = 180, which confirms the angles are correct.
Practice questions
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Q1Two angles on a straight line are 118 degrees and y degrees. Find y.Show answer
Answer: 62 (180 - 118)
Q2Four angles meet at a point. Three of them are 90, 85 and 70 degrees. Find the fourth angle.Show answer
Answer: 115 (360 - (90+85+70) = 360 - 245)
Q3A triangle has angles of 42 and 67 degrees. Find the third angle.Show answer
Answer: 71 (180 - (42+67) = 180 - 109)
Q4An isosceles triangle has an apex angle of 40 degrees. Find the size of each of the other two, equal, angles.Show answer
Answer: 70 degrees each ((180 - 40) / 2)
Q5An isosceles triangle has one of its two equal base angles equal to 50 degrees. Find the size of the third angle.Show answer
Answer: 80 degrees (180 - (50+50) = 180 - 100)
Q6Two angles lie on a straight line. One is 2x degrees and the other is 4x degrees. Find x.Show answer
Answer: 30 (2x + 4x = 180, so 6x = 180)
Q7Angles around a point are in the ratio 2 : 3 : 4. Find the size of the largest angle.Show answer
Answer: 160 degrees (9 equal parts make 360, so 1 part = 40, and the largest share is 4 x 40)
Q8A triangle has angles x, x and 2x degrees. Find x.Show answer
Answer: 45 (x + x + 2x = 180, so 4x = 180)
Exam-style questions
Written in the style of a 11+ exam paper, with a full mark scheme.
At point P on a straight line, three angles meet on the same side of the line: 2x degrees, 3x degrees and 70 degrees. Find the value of x and the size of the angle 3x.
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Triangle ABC is isosceles, with AB = AC. Angle BAC, the apex angle, is 4x degrees. The two base angles, angle ABC and angle ACB, are each (x + 15) degrees. Find x, and then find the size of angle BAC.
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Free printable worksheet
Want more practice on paper? Download the angles on lines, at points and in 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 32 of 11+ Maths Workbook 2, the whole course as one free printable PDF.
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