Finding every angle

Tip for every question: first find the parallel lines (look for the arrow marks), then the transversal. Only then use the rules.

Parallel lines l and m, transversal t, angle 6 = 135°. Find the rest:

• angle 2 = 135° (corresponding to 6)
• angle 8 = 135° (vertically opposite 6)
• angle 4 = 135° (corresponding to 8)
• angle 5 = 180° − 135° = 45° (linear pair with 6), and so angles 1, 3, 7 are 45° too

Eight angles, just two sizes: that's what parallel lines do.

The rules also test whether lines are parallel. Angle a = 120° and angle f = 70°. a and b are a linear pair, so b = 60°. b corresponds to f. If the lines were parallel, b would equal f. But 60° ≠ 70°, so the lines are not parallel.

One more rule. Parallel lines, angle 3 = 50°. Angle 2 = 130° (linear pair), and angle 6 = 130° (corresponding to 2). Angles 3 and 6 are interior angles on the same side of the transversal, and 50° + 130° = 180°. Try other sizes: it always works. Interior angles on the same side of the transversal add up to 180°.

Use it: AB ∥ CD and AD ∥ BC, with ∠DAC = 65° and ∠ADC = 60°. With AB and CD as the parallels and AD as the transversal: ∠ADC + ∠DAB = 180°, so ∠DAB = 120°. Then 120° = 65° + ∠CAB, so ∠CAB = 55°. With AD and BC as the parallels and CD as the transversal: ∠BCD = 180° − 60° = 120°. In the same way, ∠ABC = 60°.