In a trapezoid, the two angles that are on the same leg (one on the top base, one on the bottom base) are called ‘adjacent angles’. These adjacent angles are supplementary, which means their measures sum up to 180°, as we will now show.
In a trapezoid ABCD, prove that the adjacent angles are supplementary.
We need to show that m∠BAD + m∠CDA = 180°, and that m∠ABC + m∠DCB = 180°.
We have a trapezoid without any special features (that is, it is not an isosceles trapezoid and not a right trapezoid). So, all we know about this trapezoid is that the two bases are parallel. This is what we will need to use in our proof.
If this looks familiar, it is because we have already proven it for the general case of two parallel lines intersected by transversal line – here in the consecutive interior angles theorem.
So today, we just need to see that a trapezoid is no more than two parallel lines (the bases) intersected by two transversal lines (the legs) – and then apply the theorem, twice.
Proof: in trapezoids, adjacent angles are supplementary
(1) ABCD is a trapezoid //given
(2) AB||CD //definition of trapezoid
(3) m∠BAD + m∠CDA = 180° //consecutive interior angles theorem
(4) m∠ABC + m∠DCB = 180° //consecutive interior angles theorem
Alternatively, we could have applied the theorem to just the first set of angles. And since the sum of the angles in a simple convex quadrangle – and that includes trapezoids – is 360° – the other set of angles must be 360°-180°= 180°.