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 at the bottom crossing below the bottom parallel line and to the RIGHT of the transversal. (Hint: first use the corresponding angle above-right at the bottom crossing, then use the straight-line pair.) 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: the corresponding angle above-right at the bottom crossing is 63° (corr. ∠s, AB ∥ CD).
Step 2: the requested below-right angle forms a straight-line pair with it, so 180 − 63 = 117° (angles on a straight line).
Final: 117°.
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 ✓.