Mathematics • Year 7 • Unit 3 • Lesson 12
Using Parallel Line Properties
Move from spotting angle families to USING them: find unknown angles AND use the converse rules to prove lines are parallel. Every step ends with a reason in brackets, that's the Year 7 standard.
1. I do, fully worked example
This is the full Year-7 layout: write the equation, cite the rule, name the parallel lines.
Problem. Lines AB ∥ CD. A transversal cuts both. A pair of corresponding angles measures 68° and (x + 12)°. Find x.
Step 1, Identify the shape.
Same position at each crossing → F-shape → corresponding angles.
Reason: corresponding angles match position at each parallel line.
Step 2, Apply the rule.
Corresponding angles are EQUAL when the lines are parallel:
x + 12 = 68 (corr. ∠s, AB ∥ CD)
Step 3, Solve.
x = 68 − 12 = 56.
Step 4, State.
∴ x = 56°.
Answer: x = 56°.
2. We do, fill in the missing steps
The structure is the same as Section 1. Fill in the rule, the equation and the answer. 4 marks
Problem. Lines EF ∥ GH. Co-interior angles measure (2y − 10)° and 100°. Find y.
Step 1, Shape: same side of transversal, between the lines → ____ shape → ___________________ angles.
Step 2, Rule: co-interior angles are __________________ (equal / supplementary).
Step 3, Equation:
(2y − 10) + ____ = 180 (_____. ∠s, EF ∥ GH)
Step 4, Solve:
2y − 10 + 100 = 180 → 2y + 90 = 180 → 2y = ____ → y = ____.
Step 5, Check: the angle (2y − 10)° = ____° ; together with 100° it sums to ____°.
3. You do, independent practice
Show working under each problem. Cite the rule in brackets.
Foundation, find the angle
3.1 AB ∥ CD. Alternate angles measure 64° and x. Find x. 1 mark
3.2 ℓ ∥ m. Co-interior angles are 109° and y. Find y. 1 mark
3.3 PQ ∥ RS. Corresponding angles are 87° and z. Find z. 1 mark
3.4 AB ∥ CD. An angle on the top line is 142°. The vertically opposite angle (at the same crossing) and the alternate angle (at the bottom crossing), both share the same value. State it. 1 mark
Standard, short algebra
3.5 ℓ ∥ m. Corresponding angles are (x + 20)° and 95°. Find x. 2 marks
3.6 AB ∥ CD. Co-interior angles are (3x)° and (2x + 30)°. Find x and the two angles. 2 marks
Extension, using the converse
3.7 A transversal cuts lines p and q. At line p, the angle above-right of the transversal is 78°. At line q, the angle above-right of the transversal is also 78°. Are p and q parallel? Explain using the converse of the corresponding-angles rule. 2 marks
3.8 A transversal cuts lines u and v. Co-interior angles measure 96° and 88°. Are u and v parallel? Show the calculation that justifies your answer. 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2, We do (co-interior, (2y − 10)° and 100°)
Step 1: C shape → co-interior angles.
Step 2: supplementary.
Step 3: (2y − 10) + 100 = 180 (co-int. ∠s, EF ∥ GH).
Step 4: 2y + 90 = 180 → 2y = 90 → y = 45.
Step 5: (2y − 10)° = 80°; check: 80° + 100° = 180° ✓.
3.1, Alternate 64°
Alt. ∠s equal: x = 64° (alt. ∠s, AB ∥ CD).
3.2, Co-interior 109°
Co-int. supplementary: y = 180 − 109 = 71° (co-int. ∠s, ℓ ∥ m).
3.3, Corresponding 87°
Corr. ∠s equal: z = 87° (corr. ∠s, PQ ∥ RS).
3.4, Vert. opp. and alternate to 142°
Vert. opp. at top crossing = 142° (vert. opp. ∠s). Alternate at bottom crossing = 142° (alt. ∠s, AB ∥ CD).
3.5, Corresponding (x + 20)° and 95°
x + 20 = 95 (corr. ∠s, ℓ ∥ m)
x = 75. Angle = 75 + 20 = 95° ✓.
3.6, Co-interior (3x)° and (2x + 30)°
3x + 2x + 30 = 180 (co-int. ∠s, AB ∥ CD)
5x = 150 → x = 30.
Angles: 3(30) = 90° and 2(30) + 30 = 90°. Check: 90 + 90 = 180 ✓.
3.7, Are p and q parallel?
Yes. The two corresponding angles (above-right of the transversal at each crossing) are both 78°, they are equal. By the converse of the corresponding-angles rule, equal corresponding angles force the two lines to be parallel. ∴ p ∥ q.
3.8, Are u and v parallel? (co-int. 96° and 88°)
If u ∥ v, then co-int. angles must sum to 180°. Check: 96 + 88 = 184 ≠ 180.
Because the sum is NOT 180°, u and v are not parallel. (If they were parallel, the second angle would have to be 180 − 96 = 84°.)