Angles of a parallelogram
Neighbours add up to 180°; opposites are equal.
How many sizes?
A leaning parallelogram has sharp corners and wide corners. Its four angles are linked to each other.
How many different sizes do you expect among the four angles of a parallelogram?
What this lesson covers
The idea
In a parallelogram, adjacent angles add up to 180°, being interior angles on the same side of a transversal, and opposite angles are equal.
How many sizes?
A leaning parallelogram has sharp corners and wide corners. Its four angles are linked to each other.
How many different sizes do you expect among the four angles of a parallelogram?
- Four different sizes
- Two different sizes
- Just one size
Measure the angles
The angles are hidden. Tap ∠A, ∠B, ∠C, ∠D to measure them. Which angles are equal? Which pairs add up to the same total? Then drag B or D to make a new parallelogram.
Pairs of angles
In a parallelogram, adjacent angles add up to 180°, being interior angles on the same side of a transversal, and opposite angles are equal.
Why 180°? AD cuts the parallel lines AB and DC. ∠A and ∠D are interior angles on the same side, so ∠A + ∠D = 180°. In the same way ∠A + ∠B = 180°.
Why equal? Let ∠A = x. Then ∠D = 180° − x, and ∠C = 180° − ∠D = 180° − (180° − x) = x. So ∠C = ∠A.
Notes
In a parallelogram, adjacent angles add up to 180°, being interior angles on the same side of a transversal, and opposite angles are equal.
Check yourself
In parallelogram ABCD, ∠A = 70°. How large is the adjacent angle ∠B?
Answer: 110 °
Adjacent angles of a parallelogram add up to 180°, so ∠B = 180° − 70° = 110°.
In parallelogram ABCD, ∠A = 3x + 10° and ∠C = 70°. What is x?
Answer: 20
Opposite angles of a parallelogram are equal, so 3x + 10 = 70. Then 3x = 60 and x = 20.
In parallelogram ABCD, why is ∠A + ∠D = 180°?
In a parallelogram ABCD, ∠B is 2 times ∠A. How large is ∠A?
Answer: 60 °
Let ∠A = x. Then ∠B = 2x, and x + 2x = 180°, so 3x = 180° and x = 60°. ∠A = 60° and ∠B = 120°.
- AD is a transversal of the parallel sides AB and DC, and ∠A, ∠D are interior angles on the same side — correct. Yes! Interior angles on the same side of a transversal to parallel lines add up to 180°.
- Because all the angles of a quadrilateral add up to 360°, so every pair adds up to 180°. The four angles add up to 360°, but that alone does not make each pair 180°. The parallel sides do.
- Because ∠A and ∠D are opposite angles. ∠A and ∠D are neighbours. The opposite angle of ∠A is ∠C, and it is equal to ∠A.