Mathematics • Year 7 • Unit 3 • Lesson 11
Parallel Lines — Mixed Challenge
Pull all three families together: alternate (Z), co-interior (C), corresponding (F). Spot the family, name the rule, watch out for the common Z-vs-C confusion, and finish with an open-ended puzzle.
1. Mixed problems — name the family, find the angle
For each, state the family in brackets and cite the parallel lines. 2 marks each
1.1 AB ∥ CD. A transversal makes alternate angles of 73° and x. Find x.
1.2 ℓ ∥ m. A transversal makes co-interior angles of 137° and y. Find y.
1.3 PQ ∥ RS. Corresponding angles measure 104° and z. Find z.
1.4 AB ∥ CD. At the top crossing, an angle of 58° sits above the top line on the right of the transversal. Find the angle directly below 58° at the same crossing.
1.5 ℓ ∥ m. Two angles, one on each parallel line, are on the SAME side of the transversal and OUTSIDE the parallel lines (one above the top line, one below the bottom line). One is 81°. Find the other. (Hint: these are co-exterior — they have the same equal property as corresponding.)
1.6 AB ∥ CD. Alternate angles measure (3x − 5)° and (x + 45)°. Form an equation, find x, then state the angle. Show every step.
2. Find the mistake
Another Year 7 student has tried this question. Exactly one step contains a mistake. Spot it, explain why it's wrong, then re-do the working. 3 marks
Student's question: AB ∥ CD. A transversal cuts both. Co-interior angles measure 118° and x. Find x.
Step 1: Co-interior angles → C-shape. ✓
Step 2: Co-interior angles are EQUAL (Z-rule).
Step 3: ∴ x = 118°.
Step 4: Answer: x = 118° (co-int. ∠s, AB ∥ CD).
(a) Which step contains the mistake?
(b) Explain the mistake in one or two sentences.
(c) Write out the corrected working in full.
Stuck? Z = alternate = equal. C = co-interior = supplementary (sum 180°). They are DIFFERENT rules.3. Open-ended challenge — chase the angles
This question has only one correct value for the final angle, but more than one valid reasoning path. Show your reasoning clearly. 4 marks
3.1 Two parallel lines AB and CD are cut by a transversal. At the top crossing on line AB, an angle of 124° sits above the line on the LEFT of the transversal. Without measuring, work out the angle at the bottom crossing on line CD that sits BELOW the line on the RIGHT of the transversal.
(i) Sketch the diagram with both crossings, the transversal, the 124° angle clearly marked, and the unknown angle labelled x.
(ii) Find x using AT LEAST TWO different reasoning chains (e.g. one using alternate, one using corresponding + vertically opposite).
(iii) State the final value of x with a reason.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Alternate, 73°
Alt. ∠s are equal: x = 73° (alt. ∠s, AB ∥ CD).
1.2 — Co-interior, 137°
Co-int. ∠s sum to 180°: y = 180 − 137 = 43° (co-int. ∠s, ℓ ∥ m).
1.3 — Corresponding, 104°
Corr. ∠s are equal: z = 104° (corr. ∠s, PQ ∥ RS).
1.4 — Straight-line pair
The angle directly below 58° at the same crossing is on a straight line with 58°: 180 − 58 = 122° (∠s on a straight line).
1.5 — Outside the lines, same side (81°)
Top-outside angle = corresponding to the inside-bottom angle, both via the F-shape extended above and below. The two outside-same-side angles are corresponding (just mirrored) and are EQUAL. Other angle = 81° (corr. ∠s extended, ℓ ∥ m).
1.6 — Alternate, algebra
Alt. ∠s are equal: 3x − 5 = x + 45 (alt. ∠s, AB ∥ CD)
2x = 50, so x = 25.
Angle = 3(25) − 5 = 70° (and check: 25 + 45 = 70° ✓).
2 — Find the mistake
(a) The mistake is on Step 2.
(b) Co-interior angles (C-shape) are supplementary (sum to 180°), NOT equal. The "equal" rule is for ALTERNATE angles (Z-shape) and CORRESPONDING angles (F-shape). The student has mixed up the rules.
(c) Corrected working:
Step 1: Co-interior angles → C-shape. ✓
Step 2 (fixed): Co-interior angles are SUPPLEMENTARY: x + 118 = 180.
Step 3 (fixed): x = 180 − 118 = 62°.
Step 4 (fixed): ∴ x = 62° (co-int. ∠s, AB ∥ CD).
3 — Open-ended chase
Diagram: two parallel lines, transversal cutting both. 124° marked above the top line on the left of the transversal. x marked below the bottom line on the right of the transversal.
Chain 1 (corresponding + vertically opposite):
Corresponding to 124° at the bottom crossing: above the bottom line on the LEFT = 124° (corr. ∠s, AB ∥ CD).
Vertically opposite that = below the bottom line on the RIGHT = 124° (vert. opp. ∠s).
So x = 124°.
Chain 2 (straight-line + alternate):
At the top crossing, the angle below the top line on the RIGHT is 180 − 124 = 56° (∠s on a straight line).
Alternate to that 56° at the bottom crossing = above the bottom line on the LEFT = 56° (alt. ∠s, AB ∥ CD).
Straight-line pair with x: x + 56 = 180, so x = 124° (∠s on a straight line).
Both chains agree: x = 124°.
Marking: 1 for clear diagram; 2 for any valid 2-step chain reaching 124°; 1 for a second different chain agreeing.