Mathematics • Year 10 • Unit 3 • Lesson 9
Angles in Parallel Lines — Skill Drill
Build fluency with the three Lesson 9 rules for a transversal cutting two parallel lines: corresponding angles are equal (F pattern), alternate angles are equal (Z pattern), co-interior angles are supplementary (C pattern, sum to 180°). One step at a time, from a fully worked example to independent practice.
1. I do — fully worked example
Read every step. Each one has a short reason on the right so you can see why, not just what.
Problem. Two parallel lines AB and CD are cut by a transversal. One angle at the upper intersection (above AB and to the right of the transversal) is marked as 125°. Find every other angle at both intersections, and state the reason for each.
Step 1 — Vertically opposite angle on AB.
Angle below AB and to the left of the transversal = 125° (vertically opposite angles)
Reason: vertically opposite angles are always equal — no parallel lines needed.
Step 2 — Two remaining angles on AB (on a straight line with 125°).
180° − 125° = 55° (angles on a straight line are supplementary)
Reason: AB is a straight line, so adjacent angles at the intersection add to 180°.
Step 3 — Corresponding angle on CD.
Angle above CD and to the right of the transversal = 125° (corresponding angles, AB ∥ CD)
Reason: corresponding angles in matching positions are equal when lines are parallel (F pattern).
Step 4 — Fill in the rest of CD using the same rules.
Vertically opposite to 125° = 125°. Other two angles = 180° − 125° = 55°.
Answer: Each intersection has the same pattern: 125°, 55°, 125°, 55°.
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. A transversal cuts two parallel lines AB and CD. One co-interior angle at the upper intersection is marked as 110°. Find its co-interior partner on CD, the alternate angle to 110° on CD, and the corresponding angle to 110° on CD. State a reason for each.
Step 1 — Co-interior partner. Co-interior angles are on the ____________________ side of the transversal and ____________________ the two parallel lines.
Co-interior partner = 180° − ______ = ______° (co-interior angles supplementary, AB ∥ CD)
Step 2 — Alternate angle to 110° on CD.
Alternate angles are equal when lines are parallel → alternate angle = ______° (Z pattern)
Step 3 — Corresponding angle to 110° on CD.
Corresponding angles are equal when lines are parallel → corresponding angle = ______° (F pattern)
Step 4 — Quick check that the three answers are consistent.
110 + co-interior partner = ______ ✓ (must total 180)
3. You do — independent practice
Show your working in the space under each problem. For every angle you find, state the reason in brackets (e.g. "corresponding angles on parallel lines"). The first four are foundation (one rule). The middle two are standard (two rules). The last two are extension (multi-step angle chase).
Foundation — apply one rule
3.1 Two parallel lines are cut by a transversal. One angle is 65°. Find the corresponding angle on the other line. State the reason. 1 mark
3.2 Two parallel lines are cut by a transversal. One angle is 105°. Find its alternate angle on the other line. State the reason. 1 mark
3.3 Two parallel lines are cut by a transversal. One co-interior angle is 130°. Find its co-interior partner. State the reason. 1 mark
3.4 Two angles are vertically opposite. One is 47°. State the other and give the reason. 1 mark
Standard — combine two rules
3.5 Two parallel lines are cut by a transversal. One angle on the upper line is 72°. Find (a) its co-interior partner on the lower line, and (b) the alternate angle to that partner. Justify each with a reason. 2 marks
3.6 A transversal makes two angles, x and y, on the same line, with x + y = 180° (they sit on a straight line). If x = 38°, find y, and then find the corresponding angle to x on the parallel line below. State reasons. 2 marks
Extension — multi-step angle chase
3.7 Three parallel lines AB ∥ CD ∥ EF are cut by a single transversal. One angle at the AB intersection is 48°. Find the alternate angle at the EF intersection. State each step with its reason. 3 marks
3.8 Two lines are cut by a transversal. One pair of alternate angles is measured as 68° and 70°. Are the lines parallel? Justify using the converse theorem. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (110° co-interior, transversal)
Step 1: co-interior angles are on the same side of the transversal and between the parallel lines. Co-interior partner = 180° − 110 = 70°.
Step 2: alternate angle to 110° on CD = 110° (Z pattern).
Step 3: corresponding angle to 110° on CD = 110° (F pattern).
Step 4: 110 + 70 = 180 ✓.
3.1 — Corresponding to 65°
65° (corresponding angles on parallel lines are equal).
3.2 — Alternate to 105°
105° (alternate angles on parallel lines are equal — Z pattern).
3.3 — Co-interior partner of 130°
180° − 130° = 50° (co-interior angles on parallel lines are supplementary — C pattern).
3.4 — Vertically opposite to 47°
47° (vertically opposite angles are equal). Note: this rule does not require parallel lines.
3.5 — 72°: co-interior + alternate combo
(a) Co-interior partner = 180° − 72° = 108° (co-interior angles supplementary, parallel lines).
(b) Alternate angle to that 108° = 108° (alternate angles equal, parallel lines).
3.6 — Straight line, then corresponding
y = 180° − 38° = 142° (angles on a straight line are supplementary).
Corresponding to x = 38° on the parallel line: 38° (corresponding angles on parallel lines).
3.7 — Three parallel lines, alternate chase
Step 1: angle at AB = 48°.
Step 2: corresponding angle on CD = 48° (corresponding angles, AB ∥ CD).
Step 3: corresponding angle on EF = 48° (corresponding angles, CD ∥ EF).
Step 4: alternate angle to that 48° on EF = 48° (alternate angles, parallel lines).
The whole chase is essentially "the angle keeps the same value as you slide it from line to line".
3.8 — Are the lines parallel?
For parallel lines, alternate angles must be equal. Here the alternate angles are 68° and 70°, which are not equal. By the converse of the alternate-angle theorem, the two lines are not parallel.