Mathematics • Year 10 • Unit 3 • Lesson 9
Angles in Parallel Lines — Mixed Challenge
Pull together every idea from Lesson 9: corresponding (F), alternate (Z), co-interior (C), and the converse "equal corresponding angles ⇒ parallel". Always supply a reason. Spot a classmate's plausible mistake, then design your own angle-chase diagram.
1. Mixed problems — name the rule, then solve
Each question uses a different idea from Lesson 9. State the rule in brackets after every angle you find. 3 marks each
1.1 AB ∥ CD, transversal cuts both. An angle on AB is 84°. Find the corresponding angle on CD, the alternate angle on CD, and the co-interior partner on CD. State the rule for each.
1.2 Two lines are cut by a transversal. The two co-interior angles are 95° and 85°. Are the lines parallel? Justify.
1.3 AB ∥ CD with a transversal. The angle just above AB on the right of the transversal is 3x + 20. The alternate angle just below CD on the left of the transversal is 5x − 10. Find x and the value of both angles.
1.4 AB ∥ CD. A co-interior pair is given by 2y + 30 and 3y + 10. Form an equation, solve for y, and state both angles.
1.5 Three parallel lines are cut by one transversal. The angle at the top intersection is 72°. Find the alternate angle at the middle intersection and the corresponding angle at the bottom intersection.
1.6 AB ∥ CD with a transversal. The angle just below CD and to the right of the transversal is given as 4z. Its co-interior partner on AB is 2z + 60. Solve for z, then state every angle at both intersections.
2. Find the mistake
Another Year 10 student is asked: "Two parallel lines are cut by a transversal. One co-interior angle is 70°. Find the other." Their working is below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working:
Line 1: Co-interior angles are angles between two parallel lines, on the same side of the transversal.
Line 2: Co-interior angles on parallel lines are equal.
Line 3: So the other co-interior angle = 70°.
(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 corrected final answer.
Stuck? Co-interior angles are SUPPLEMENTARY (sum to 180°), not equal. That's what makes the C pattern different from the F and Z patterns.3. Open-ended challenge — design your own angle chase
This question has many valid answers. Be creative but show every reason. 4 marks
3.1 Design your own angle-chase problem involving two parallel lines and one transversal. Your problem must:
(i) Provide exactly ONE angle as a starting fact (somewhere between 20° and 160°, not 90°).
(ii) Ask the solver to find at least three other angles, each using a different rule (corresponding, alternate, co-interior — or vertically opposite for one of them).
(iii) Include a labelled sketch (rough is fine) so the reader knows which angle is which.
Then solve your own problem, listing every angle with its reason in brackets. End with a sanity check: every angle on each intersection must be between 0° and 180°, and the four angles at each intersection should sum to 360°.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — 84° plus three partners
Corresponding angle on CD = 84° (corresponding angles on parallel lines).
Alternate angle on CD = 84° (alternate angles on parallel lines).
Co-interior partner on CD = 180° − 84° = 96° (co-interior angles supplementary on parallel lines).
1.2 — Co-interior 95° and 85°
95° + 85° = 180°. Co-interior angles on parallel lines are supplementary, and these do sum to 180°. By the converse of the co-interior theorem, the two lines are parallel.
1.3 — Algebraic alternate pair
Alternate angles are equal: 3x + 20 = 5x − 10 → 30 = 2x → x = 15.
Both angles = 3(15) + 20 = 65° (check: 5(15) − 10 = 65° ✓).
1.4 — Algebraic co-interior pair
Co-interior sums to 180°: (2y + 30) + (3y + 10) = 180 → 5y + 40 = 180 → 5y = 140 → y = 28.
Angles: 2(28) + 30 = 86° and 3(28) + 10 = 94°. Check: 86 + 94 = 180 ✓.
1.5 — Three parallel lines
Alternate angle at middle intersection = 72° (alternate angles on parallel lines).
Corresponding angle at bottom intersection = 72° (corresponding angles on parallel lines).
The angle "slides down" each parallel line keeping the same value.
1.6 — Co-interior with algebra
4z + (2z + 60) = 180 → 6z + 60 = 180 → 6z = 120 → z = 20.
So 4z = 80° (below CD, right of transversal) and 2z + 60 = 100° (above AB, right of transversal — co-interior partner).
At each intersection: 80°, 100°, 80°, 100° (and 100°, 80°, 100°, 80°). Sums to 360° ✓.
2 — Find the mistake
(a) The mistake is on Line 2.
(b) Co-interior angles on parallel lines are supplementary (sum to 180°), not equal. Equality is the rule for corresponding angles (F) and alternate angles (Z); co-interior angles (C) are the odd one out — they add to 180°.
(c) Corrected working:
Co-interior angles are angles between two parallel lines, on the same side of the transversal.
Co-interior angles on parallel lines are supplementary — they sum to 180°.
So the other co-interior angle = 180° − 70° = 110°.
3 — Open-ended challenge (sample solution)
Problem. Lines AB and CD are parallel. A transversal cuts both. The angle just above AB and to the right of the transversal is 130°. Find (a) the angle just above CD on the right of the transversal, (b) the angle just below CD on the left of the transversal, (c) the angle just above CD on the left of the transversal, (d) the angle just below AB on the right of the transversal.
Solution.
(a) 130° (corresponding angles on parallel lines — F pattern).
(b) 130° (alternate angles on parallel lines — Z pattern, or vertically opposite to (a)).
(c) 180° − 130° = 50° (co-interior angles on parallel lines — C pattern, paired with the 130° above AB on the left of the transversal which is supplementary to 130°).
(d) 180° − 130° = 50° (angles on a straight line AB are supplementary).
Sanity check. Upper intersection: 130°, 50°, 130°, 50° → sum 360° ✓. Lower intersection: same pattern 130°, 50°, 130°, 50° → 360° ✓.
Marking: 1 for a clear sketch and one starting fact, 1 for using at least three different rules, 1 for stating a reason in brackets for every angle found, 1 for the 360° sanity check. Full marks for any valid problem and solution meeting all four.