Geometry
Quadrilaterals
Use the properties of a parallelogram and the mid-point theorem.
A parallelogram is defined by one thing only: both pairs of opposite sides are parallel. Everything else people say about it — opposite sides equal, opposite angles equal, diagonals cutting each other in half — is not part of the definition. Each one is a theorem, proved from that single fact, and the proofs all run through congruent triangles.
One diagonal, two congruent triangles
Draw a diagonal across a parallelogram and it makes two triangles. Alternate angles at each pair of parallel sides make two pairs of angles equal, and the diagonal is a side of both triangles. That is ASA, so the triangles are congruent — and their matching sides are the parallelogram's opposite sides.
Have a play
A parallelogram with one diagonal drawn across it. Find the two pieces the diagonal makes, and the diagonal itself.
Tap any part of the picture.
Worked example
Why do two neighbouring angles of a parallelogram add up to 180°?
Take sides AB and DC. They are parallel, and AD cuts across both of them.
A line crossing two parallel lines is a transversal, so the parallel-line angle facts apply.
Try it together
In parallelogram ABCD, angle A is 70°. Find angles B and D.
The corners are named in order round the shape, so A and B are next to each other and B and D are opposite.
1.Angles A and B are neighbours. How many degrees do they add up to?
The mid-point theorem
Join the mid-points of two sides of a triangle and the segment you get is parallel to the third side and exactly half its length. It works backwards too: a line from the mid-point of one side, drawn parallel to a second side, always lands on the mid-point of the third.
Have a go
Have a go on your own: in triangle XYZ the mid-points of XY and XZ are joined, and YZ is 20 cm long. How long is the joining segment, in centimetres?
Hint: Half of the third side.
Ready to practise?
Eight questions on what you have just read. Nothing is timed, and you can play as many times as you like.