Mathematics • Year 7 • Unit 3 • Lesson 11
Parallel Lines and Transversals
Build fluency with the three angle families formed when a transversal cuts two parallel lines: alternate (Z, equal), co-interior (C, sum 180°) and corresponding (F, equal). Spot the shape, name the rule, find the unknown.
1. I do — fully worked example
Read each step. The reason on the right tells you which of the three rules is being used.
Problem. Lines AB and CD are parallel. A transversal cuts both. One co-interior angle is 118°. Find the other co-interior angle x.
Step 1 — Spot the shape.
Both angles are on the SAME side of the transversal, BETWEEN the parallel lines → that's a C → co-interior.
Reason: trace a "C" with your finger linking the two angles.
Step 2 — Pick the right rule.
Co-interior angles are SUPPLEMENTARY: they sum to 180°.
x + 118 = 180 (co-int. ∠s, AB ∥ CD)
Reason: co-int. ≠ equal; they add up. Don't confuse with alternate.
Step 3 — Solve.
x = 180 − 118 = 62°.
Reason: subtract from 180.
Step 4 — State with reason.
∴ x = 62° (co-int. ∠s, AB ∥ CD)
Reason: always cite the rule AND name the parallel lines.
Answer: x = 62°.
2. We do — fill in the missing steps
Same structure, but with blanks. Write in the rule, the equation and the answer. 4 marks
Problem. Lines ℓ and m are parallel (ℓ ∥ m). A transversal cuts both. An angle on the top line above-right of the transversal is 74°. Angle y is on the bottom line, also above-right of the transversal. Find y.
Step 1 — Spot the shape. Both angles are in the SAME position (above-right) at their crossings. The shape traced is an _______ (letter).
Step 2 — Name the family. Same-position angles are called _______________ angles. They are _______________ (equal / supplementary).
Step 3 — Write the equation with the reason.
y = _______° (_______. ∠s, ℓ ∥ m)
Step 4 — State the answer.
∴ y = _______°.
3. You do — independent practice
Show working under each problem. State the reason in brackets (alt. / co-int. / corr.) and name the parallel lines.
Foundation — one rule, one step
3.1 AB ∥ CD. A transversal cuts both. Alternate angles are 56° and x. Find x. 1 mark
3.2 AB ∥ CD. Corresponding angles are 113° and x. Find x. 1 mark
3.3 ℓ ∥ m. Co-interior angles are 95° and x. Find x. 1 mark
3.4 ℓ ∥ m. Alternate angles are x and 132°. Find x. 1 mark
Standard — name the family AND find the angle
3.5 AB ∥ CD. Two angles measure 47° and x. They are on OPPOSITE sides of the transversal, BETWEEN the parallel lines. (a) Which family? (b) Find x. 2 marks
3.6 ℓ ∥ m. Two angles measure 124° and x. They are on the SAME side of the transversal, BETWEEN the parallel lines. (a) Which family? (b) Find x. 2 marks
Extension — combine rules
3.7 AB ∥ CD. A transversal makes an angle of 63° above the top line and to the right of the transversal. Find the angle BELOW the bottom line, directly under it on the right of the transversal. (Hint: corresponding first, then vertically opposite.) 3 marks
3.8 ℓ ∥ m. Two co-interior angles measure (2x + 10)° and (x + 50)°. Form an equation and find x. Then state both angles. 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (corresponding angles, 74°)
Step 1: shape is an F.
Step 2: corresponding angles; they are equal.
Step 3: y = 74° (corr. ∠s, ℓ ∥ m).
Step 4: ∴ y = 74°.
3.1 — Alternate angles, 56°
Alternate angles are equal: x = 56° (alt. ∠s, AB ∥ CD).
3.2 — Corresponding angles, 113°
Corresponding angles are equal: x = 113° (corr. ∠s, AB ∥ CD).
3.3 — Co-interior angles, 95°
Co-int. angles sum to 180°: x + 95 = 180, so x = 85° (co-int. ∠s, ℓ ∥ m).
3.4 — Alternate angles, 132°
Alternate angles are equal: x = 132° (alt. ∠s, ℓ ∥ m).
3.5 — Opposite sides, between the lines
(a) Alternate (Z-shape).
(b) Equal: x = 47° (alt. ∠s, AB ∥ CD).
3.6 — Same side, between the lines
(a) Co-interior (C-shape).
(b) Supplementary: x + 124 = 180, so x = 56° (co-int. ∠s, ℓ ∥ m).
3.7 — Two-step chase (63°)
Step 1: corresponding angle below the bottom line on the right is 63° (corr. ∠s, AB ∥ CD).
Step 2: the angle BELOW the bottom line directly under it on the right is vertically opposite the corresponding angle, so it is also 63° (vert. opp. ∠s).
Final: 63°.
3.8 — Algebra with co-int. angles
(2x + 10) + (x + 50) = 180 (co-int. ∠s, ℓ ∥ m)
3x + 60 = 180
3x = 120, so x = 40.
Angles: 2(40) + 10 = 90° and 40 + 50 = 90°. Check: 90 + 90 = 180 ✓.