Do the sides ever meet?

The idea

The opposite sides of a rectangle are parallel, because the interior angles on the same side of a transversal add up to 90° + 90° = 180°.

Lines that never meet

Two lines start from the two ends of a segment AB, on the same side. They lean at angles ∠A and ∠B. Parallel lines never meet, however far we extend them.

If the two lines never meet, what must be true about ∠A + ∠B?

• It is 90°
• It is 180°
• It is 360°

Turn the lines

Drag the round handles, or use the − and + buttons. Watch the angle bar: when do the lines stop meeting?

180° means parallel

The opposite sides of a rectangle are parallel, because the interior angles on the same side of a transversal add up to 90° + 90° = 180°.

The lines never meet exactly when the two angles on the same side of AB add up to 180°.

In a rectangle, AB cuts AD and BC. ∠A + ∠B = 90° + 90° = 180°, so AD ∥ BC. In the same way AB ∥ DC.

Notes

The opposite sides of a rectangle are parallel: the interior angles on the same side of a transversal add up to 90° + 90° = 180°.

Check yourself

Two lines are cut by a transversal. The interior angles on the same side are 100° and 80°. Are the lines parallel?

AB is a transversal of two lines. ∠A = 65° on one side. For the lines to be parallel, how large must ∠B be on the same side?

Answer: 115 °

∠A + ∠B must be 180°, so ∠B = 180° − 65° = 115°.

Why are the sides DC and AB of a rectangle ABCD parallel?

In quadrilateral ABCD, ∠A = 80° and ∠B = 100°. Which pair of sides must be parallel?

• Yes, because 100° + 80° = 180° — correct. Yes! Interior angles on the same side that add up to 180° make the lines parallel.
• No, because the angles are not equal. The angles on the same side need not be equal. They must add up to 180°.
• No, because they add up to more than 90°. 90° is not the test. The two angles must add up to 180°.
• Because AB = DC. Equal length alone does not make lines parallel. Two equal lines can lean towards each other.
• Because ∠A + ∠D = 90° + 90° = 180°, on the same side of the transversal AD — correct. Yes! AD cuts AB and DC, and the interior angles on its same side add up to 180°.
• Because the diagonals are equal. Equal diagonals do not tell us about parallel sides. The angles do.
• AB and DC. AB is the transversal here. The two angles tell us about the lines AD and BC that it cuts.
• AD and BC — correct. Yes! AB cuts AD and BC, and 80° + 100° = 180° on the same side.
• AD and DC. AD and DC meet at D, so they cannot be parallel.