Mathematics • Year 8 • Unit 3 • Lesson 14
Angles in Parallel Lines — Mixed Challenge
Six mixed problems mixing F, Z, C, and vertically opposite rules. One "find the mistake" and one open-ended diagram-building challenge.
1. Mixed problems — choose the right rule
Decide which rule applies (corresponding F, alternate Z, co-interior C, vertically opposite, or angles on a straight line) BEFORE you set up the calculation. Every angle answer needs a written reason. 3 marks each
1.1 Two parallel lines are cut by a transversal. One angle is 137°. Find (a) the corresponding angle, (b) the co-interior angle on the same side, (c) the vertically opposite angle. State the rule for each.
1.2 Alternate angles are (5x − 15)° and (2x + 30)°. Find x and each angle.
1.3 Co-interior angles are (3x + 10)° and (x + 30)°. Find x and each angle.
1.4 Two parallel lines L1 and L2 are cut by a transversal. At L1, the angle on the upper-left of the intersection is 64°. Find all eight angles (state each rule used).
1.5 A transversal cuts two parallel lines and makes a 90° angle with one of them. What angle does it make with the other? Justify with a parallel-line rule.
1.6 Three parallel lines are cut by a single transversal. The angle between the transversal and the TOP parallel line is 50° (measured on the right). Find the corresponding angle on the right at each of the three parallel lines, and explain why all three are equal.
2. Find the mistake
A Year 8 student tried to solve: "Co-interior angles in a pair of parallel lines are (2x + 30)° and (x + 60)°. Find x." Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, and re-do the working. 3 marks
Student's working:
Line 1: Co-interior angles are equal (parallel lines)
Line 2: 2x + 30 = x + 60
Line 3: x = 30
Line 4: So both angles = 2(30) + 30 = 90°.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the correct value of x and both angles.
Stuck? Revisit lesson § Card 6 — co-interior angles are NOT equal. They are SUPPLEMENTARY (add to 180°). Only alternate and corresponding angles are equal.3. Open-ended challenge — design your own parallel-line diagram
This question has more than one valid answer. 4 marks
3.1 Sketch a diagram with TWO parallel lines and ONE transversal. Label EIGHT angles a, b, c, d, e, f, g, h (four at each intersection).
Then choose ONE angle to be exactly 43° and find ALL the other seven angles. For EACH of the other seven, state the rule used (e.g. "vertically opposite", "corresponding", "alternate", "co-interior", or "angles on a straight line").
Bonus: after solving, write down how many DISTINCT angle values appear in the diagram, and explain why (in 1 sentence).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — One angle = 137°
(a) Corresponding = 137° (corresponding angles, parallel lines).
(b) Co-interior = 180° − 137° = 43° (co-interior angles add to 180°).
(c) Vertically opposite = 137° (vertically opposite angles).
1.2 — Alternate (5x−15)° and (2x+30)°
Alternate angles are equal: 5x − 15 = 2x + 30 → 3x = 45 → x = 15. Each angle = 5(15) − 15 = 60°.
1.3 — Co-interior (3x+10)° and (x+30)°
Co-interior add to 180°: (3x + 10) + (x + 30) = 180 → 4x + 40 = 180 → 4x = 140 → x = 35. Angles: 3(35)+10 = 115° and 35+30 = 65°.
1.4 — One angle at L1 = 64°
At L1: upper-left = 64°; upper-right = 116° (straight line); lower-right = 64° (vertically opposite); lower-left = 116° (vertically opposite).
At L2: corresponding to L1 → upper-left = 64°, upper-right = 116°, lower-right = 64°, lower-left = 116°. All eight angles are either 64° or 116°.
1.5 — Transversal perpendicular to one parallel line
It also makes 90° with the other parallel line (corresponding angles equal). If one is perpendicular, both intersections form right angles.
1.6 — Three parallel lines + one transversal
The corresponding angle on the right at each of the three parallel lines is 50°. They are all equal because corresponding angles (parallel lines) are equal — and this rule extends to any number of parallel lines cut by the same transversal.
2 — Find the mistake
(a) The mistake is on Line 1 (carried into Line 2).
(b) Co-interior angles are NOT equal — they are SUPPLEMENTARY (they add to 180°). The student set the two expressions equal when they should have set their sum equal to 180.
(c) Corrected: Co-interior angles add to 180°. (2x + 30) + (x + 60) = 180 → 3x + 90 = 180 → 3x = 90 → x = 30. Angles: 2(30)+30 = 90° and 30+60 = 90° ✓ (the transversal is perpendicular to both parallel lines).
3 — Diagram with one angle = 43° (sample solution)
At the first intersection (say where the transversal hits L1):
a = 43°; b = 180° − 43° = 137° (straight line); c = 43° (vertically opposite to a); d = 137° (vertically opposite to b).
At the second intersection (where the transversal hits L2):
e = 43° (corresponding to a); f = 137° (corresponding to b); g = 43° (vertically opposite to e); h = 137° (vertically opposite to f).
Distinct values: only TWO distinct angles appear (43° and 137°). This is because vertically opposite angles match, and corresponding angles match — so the entire diagram collapses into just two possibilities: the original and its supplement.
Marking: 2 marks for correctly labelling all 8 angles with values; 1 mark for naming the rules used; 1 mark for correctly identifying that only 2 distinct values appear, with reason.