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.

Build · I Do / We Do / You Do

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°.

Stuck? Revisit lesson § "Reason Phrases You Must Write" — the reason in brackets is part of the mark.

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 ____°.

Stuck? Revisit lesson § "Watch Me Solve It · Co-interior angles" — co-int. + co-int. = 180°.

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

Stuck on 3.8? If u ∥ v, the co-int. angles should sum to exactly 180°. Do 96° and 88° satisfy that?

How did this worksheet feel?

What I'll revisit before next class:

Answers — Do not peek before attempting

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°.)

HSCScience • Mathematics • Year 7 • Unit 3 • Lesson 12 • Worksheet 1 (Build)