Mathematics Year 7: Alternate interior angles and vertically opposite angles
Mathematics, Year 7 (Portugal): Which angles are equal when two lines cross and when a transversal cuts two parallel lines, and why.
Two pairs that make equal angles
When two lines cross they form four angles. The ones facing each other at the vertex are vertically opposite, and they have the same measure. Now take two parallel lines cut by a transversal: eight angles appear. The alternate interior angles lie between the parallels, on opposite sides of the transversal, one on each line, and they have the same measure. If the lines are not parallel, that stops being true for alternate interior angles. Vertically opposite angles, though, are equal whenever two lines cross.
Why they are equal
Vertically opposite: if angle a and its neighbour b make a straight angle, a + b = 180°; the other neighbour c also satisfies b + c = 180°. So a = 180° − b = c. Alternate interior: you can see it by folding paper or dragging lines in dynamic geometry, and it is what justifies that the angles of a triangle add up to 180°. Through one vertex, draw the parallel to the opposite side: two of the angles there are alternate interior with two angles of the triangle, and all three together make a straight angle.
A case: two parallels and a transversal
Two parallel lines r and s are cut by a transversal t at P (on r) and Q (on s). Between the parallels, the angle at P to the right of t measures 64°. Its alternate interior angle is the one at Q, between the parallels and to the left of t: it also measures 64°. The neighbour of each of them, on the same line, measures 180° − 64° = 116°, and the vertically opposite angles repeat those values. So at each point there are two angles of 64° and two of 116°. Check: 4 × 64° + 4 × 116° = 256° + 464° = 720°, two full turns.
The trap: equal only between parallels, and only for the right pair
Two reasonable confusions. First: taking two angles that look alternate interior as equal without knowing the lines are parallel. If you measure 64° on one and 70° on the other, the lines are not parallel: the equality only holds for parallels. Second: picking the wrong pair. The two interior angles on the same side of the transversal are not equal: in the example above, 64° and 116° add up to 180°. Before saying «they are equal», ask: between the parallels? On opposite sides of the transversal?
Where you see this
When you open a pair of scissors, the angle between the handles is vertically opposite to the angle between the blades, so the blades open exactly as much as the handles. Where a path crosses two parallel edges of a street, it makes equal alternate interior angles with them. In dynamic geometry, such as GeoGebra, you can drag the transversal and watch the angles change together while staying equal.
Test what you learned
Two lines cross and form four angles. One of them measures 50°. What do the other three measure?
- a) 50°, 50° and 50°
- b) 50°, 130° and 130°
- c) 50°, 40° and 40°
- d) 130°, 130° and 130°
Correct answer: b) 50°, 130° and 130° — Right: the opposite angle is 50°, and each neighbour is 180° − 50° = 130°. Check: 50 + 50 + 130 + 130 = 360°.
Parallel lines are cut by a transversal. Between the parallels, an angle at P measures 72°. At Q there are two angles between the parallels, one on each side of the transversal; one of them is the alternate interior angle of 72°. How much does the other measure?
- a) 72°
- b) 18°
- c) 108°
- d) 288°
Correct answer: c) 108° — Right: the alternate interior angle is 72° and the two angles at Q make a straight angle, so the other is 180° − 72° = 108°.
Parallel lines are cut by a transversal at P and Q. At P, the angle between the parallels, to the right of the transversal, measures 64°. Rui says the angle at Q between the parallels, also to the right, measures 64° too. Is he right?
- a) Yes: with parallel lines, all the angles between them are equal
- b) No: it measures 116°; only the alternate interior angle, on the other side of the transversal, measures 64°
- c) No: it measures 26°, the complement of 64°
- d) Yes: they are vertically opposite angles
Correct answer: b) No: it measures 116°; only the alternate interior angle, on the other side of the transversal, measures 64° — Right: the alternate interior angle (left of the transversal) is 64°, so the one on the right, its neighbour on line s, is 180° − 64° = 116°.
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