This set has 12 original questions on the five lessons of angles and geometric reasoning. They run from easy to harder. Every question uses facts stated in the question, so no diagram needs measuring.
How should you use this set?
Work on paper. For every angle you find, write the value and the reason in words, as you would in an exam.
Then open the answer and compare both parts. A right number with a missing reason is only half done.
If you want to keep track of repeated errors, the mistake log and retest queue lets you record them on your own device. The non-calculator working trainer can check the arithmetic.
Questions
Q1 (parallel lines). AB is parallel to CD. Two co-interior angles are x° and 118°. Find x.
Show answer
Co-interior angles add to 180°, so x = 180 − 118 = 62°.
Q2 (parallel lines). Two alternate angles between parallel lines are (2x + 15)° and (3x − 20)°. Find x and the angle.
Show answer
Alternate angles are equal: 2x + 15 = 3x − 20, so x = 35. The angle is 2(35) + 15 = 85°, and 3(35) − 20 = 85° confirms it. x = 35, angle = 85°.
Q3 (triangle). Two angles of a triangle are 48° and 67°. Find the third.
Show answer
48 + 67 = 115, so the third angle is 180 − 115 = 65°, because the angles in a triangle add to 180°.
Q4 (isosceles triangle). In triangle PQR, PQ = PR and ∠QPR = 34°. Find ∠PQR.
Show answer
The base angles are equal, so each is (180 − 34) ÷ 2 = 146 ÷ 2 = 73°. Check: 73 + 73 + 34 = 180.
Q5 (exterior angle). In triangle ABC, ∠ABC = 53° and ∠BAC = 56°. Side BC is extended beyond C to D. Find ∠ACD.
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The exterior angle equals the sum of the two opposite interior angles, so ∠ACD = 53 + 56 = 109°. Check: ∠ACB = 180 − 53 − 56 = 71°, and 180 − 71 = 109°.
Q6 (regular polygon). Find each interior angle of a regular 12-sided polygon.
Show answer
Exterior angle = 360 ÷ 12 = 30°, so interior = 180 − 30 = 150°. Check: (12 − 2) × 180 = 1800, and 1800 ÷ 12 = 150.
Q7 (polygon, number of sides). Each interior angle of a regular polygon is 162°. How many sides does it have?
Show answer
Exterior angle = 180 − 162 = 18°, so n = 360 ÷ 18 = 20. Check: (20 − 2) × 180 = 3240, and 3240 ÷ 20 = 162.
Q8 (polygon with algebra). The angles of a pentagon are x°, (x + 10)°, (x + 20)°, (x + 30)° and 2x°. Find x and the largest angle.
Show answer
The sum is (5 − 2) × 180 = 540. So x + x + 10 + x + 20 + x + 30 + 2x = 540, which gives 6x + 60 = 540 and x = 80. The angles are 80, 90, 100, 110 and 160, and they add to 540. x = 80, largest angle = 160°.
Q9 (circle). O is the centre of a circle, and A, B, C lie on the circle with C on the major arc. ∠AOB = 124°. Find ∠ACB, with a reason.
Show answer
The angle at the centre is twice the angle at the circumference, both standing on arc AB. So ∠ACB = 124 ÷ 2 = 62°.
Q10 (tangent). TA is a tangent to a circle with centre O, touching at A. ∠AOT = 64°. Find ∠ATO.
Show answer
A tangent is perpendicular to the radius at A, so ∠OAT = 90°. Then ∠ATO = 180 − 90 − 64 = 26°, because the angles in a triangle add to 180°.
Q11 (diagram assumption). Two lines are crossed by a third line. One angle is marked 65° and another is marked x. The lines look parallel but have no arrows, and the question says nothing about them. (a) Can you find x? (b) What changes if the question states that the lines are parallel, with x and 65° alternate angles?
Show answer
(a) No. Looking parallel is not a stated fact, so no rule links the two angles. (b) If the lines are stated as parallel, alternate angles are equal, so x = 65°.
Q12 (multi-step). In triangle ABC, AB = AC. A straight line DAE is parallel to BC, with D on the side of B. ∠DAB = 55°. Find ∠ABC, ∠ACB and ∠BAC, with reasons.
Show answer
∠ABC = 55°, because DE is parallel to BC and these are alternate angles. ∠ACB = 55°, because AB = AC, so the base angles are equal. ∠BAC = 180 − 55 − 55 = 70°, because the angles in a triangle add to 180°. Check: ∠DAB + ∠BAC + ∠CAE = 55 + 70 + 55 = 180, a straight line.
If you got these wrong
| Where it went wrong | Go back to |
|---|---|
| Q1, Q2, Q12: wrong angle pair or wrong rule for parallel lines | Use parallel-line angle relationships |
| Q3, Q4, Q5: angle sum, base angles or exterior angle | Explain an angle using triangle properties |
| Q6, Q7, Q8: interior sum or exterior angle mix-up | Work with interior and exterior polygon angles |
| Q9, Q10: wrong circle fact, or doubling instead of halving | Apply a circle angle relationship with a reason |
| Q11: using something because it looks true | Separate a diagram assumption from a stated fact |
Retry a fresh question on the same skill a few days later, and log any repeated slip in the mistake log. The next module, similarity, congruence and scale, builds on this reasoning.
If the same errors keep returning, our teachers can look at your written working in online one-to-one Mathematics tuition.