Mathematics • Year 7 • Unit 3 • Lesson 12
Using Parallel Line Properties — Mixed Challenge
Mix the rules and their converses: find unknown angles, watch for the classic "co-int. should sum, not equal" trap, and tackle an open-ended proof. Every step needs a Year-7-quality reason phrase.
1. Mixed problems
Show working. Cite the rule AND the parallel lines. 2 marks each
1.1 AB ∥ CD. Corresponding angles measure 78° and (2x − 4)°. Find x.
1.2 ℓ ∥ m. Alternate angles measure (5y − 15)° and (3y + 25)°. Find y and the angle.
1.3 AB ∥ CD. Co-interior angles measure (4z)° and (2z + 30)°. Find z.
1.4 A transversal cuts lines p and q. Corresponding angles are 88° and 92°. Are p and q parallel? Justify with a sentence.
1.5 A transversal cuts lines u and v. Alternate angles are 73° and 73°. Are u and v parallel? Justify with the converse rule.
1.6 AB ∥ CD. At the top crossing the angle above-left of the transversal is 105°. Find the angle at the bottom crossing below-right of the transversal. Show two steps with reasons.
2. Find the mistake
Exactly one step contains a mistake. Identify it, explain why it's wrong, then redo. 3 marks
Student's question: ℓ ∥ m. Alternate angles measure (3x + 10)° and (x + 50)°. Find x.
Step 1: Alternate angles, between the lines, opposite sides → Z. ✓
Step 2: Alternate angles are SUPPLEMENTARY:
Step 3: (3x + 10) + (x + 50) = 180
Step 4: 4x + 60 = 180 → 4x = 120 → x = 30.
Step 5: ∴ x = 30 (alt. ∠s, ℓ ∥ m).
(a) Which step contains the mistake?
(b) Explain the mistake.
(c) Write the corrected working in full, including the corrected x.
Stuck? Z = alternate = EQUAL. Only the C (co-int.) is supplementary.3. Open-ended challenge — prove or disprove parallel
This problem has a single correct conclusion but multiple valid routes to it. 4 marks
3.1 A transversal cuts two lines, p and q. The angle the transversal makes with p above-right is (4x + 5)°. The angle the transversal makes with q below-left is (2x + 35)°.
(a) If p ∥ q, what relationship must hold between these two angles? (Hint: above-right at one and below-left at the other — same position? opposite?)
(b) Use that relationship to write an equation and solve for x.
(c) State the angle and the conclusion (p ∥ q or not).
(d) Write the full reason phrase with the appropriate converse rule.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Corresponding 78° and (2x − 4)°
2x − 4 = 78 (corr. ∠s, AB ∥ CD)
2x = 82 → x = 41. Check: 2(41) − 4 = 78 ✓.
1.2 — Alternate (5y − 15)° and (3y + 25)°
5y − 15 = 3y + 25 (alt. ∠s, ℓ ∥ m)
2y = 40 → y = 20.
Angle = 5(20) − 15 = 85°. Check: 3(20) + 25 = 85 ✓.
1.3 — Co-interior (4z)° and (2z + 30)°
4z + 2z + 30 = 180 (co-int. ∠s, AB ∥ CD)
6z = 150 → z = 25.
Angles: 100° and 80°. Check: 100 + 80 = 180 ✓.
1.4 — Corresponding 88° and 92°: parallel?
For p ∥ q, corresponding angles must be EQUAL. 88 ≠ 92, so p and q are NOT parallel.
1.5 — Alternate 73° and 73°: parallel?
The two alternate angles are equal (73° = 73°). By the converse of the alternate-angles rule, equal alternate angles force the lines to be parallel. ∴ u ∥ v.
1.6 — Two-step chase (105°)
Step 1: corresponding angle at the bottom crossing above-left = 105° (corr. ∠s, AB ∥ CD).
Step 2: angle below-right at the bottom crossing is vertically opposite to that 105° = 105° (vert. opp. ∠s).
Final: 105°.
2 — Find the mistake
(a) The mistake is on Step 2.
(b) Alternate angles are EQUAL, not supplementary. The "sum to 180°" rule belongs to CO-INTERIOR angles. The student has mixed up the Z rule with the C rule.
(c) Corrected working:
Step 1: Alternate angles → Z. ✓
Step 2 (fixed): Alternate angles are EQUAL: 3x + 10 = x + 50.
Step 3 (fixed): 2x = 40 → x = 20.
Step 4 (fixed): Angle = 3(20) + 10 = 70°. Check: 20 + 50 = 70° ✓.
∴ x = 20 (alt. ∠s, ℓ ∥ m).
3 — Open-ended prove parallel
(a) Above-right at the top crossing and below-left at the bottom crossing are alternate angles (they sit on opposite sides of the transversal and between the two lines IF you trace through). If p ∥ q, they must be EQUAL.
Equivalent reasoning: they are also a "Z" via rotation symmetry.
(b) 4x + 5 = 2x + 35 (alt. ∠s, p ∥ q, assumed)
2x = 30 → x = 15.
(c) Angle = 4(15) + 5 = 65°. Check: 2(15) + 35 = 65° ✓. The two angles ARE equal, so by the converse of the alternate-angles rule, the lines are parallel.
(d) Conclusion: p ∥ q, justified by "equal alternate angles ⇒ lines parallel (converse of alt. ∠s rule)".
Marking: 1 for identifying the relationship; 1 for equation and x = 15; 1 for angle = 65°; 1 for citing the converse rule and concluding p ∥ q.