171. The Isosceles Trap · Equal sides, equal diagonals, parallel bases

Try to break it

Drag B to widen or narrow the top side AB. The slanted sides AD and BC stay equal, the base angles stay equal, and the diagonals AC and BD always match in length — that's what makes it isosceles. Try to tilt that symmetry — you can't.

How you build it

Construct an isosceles trapezium.

• Draw the base segment DC of length 320.
• Draw a line parallel to DC, 220 units above it — this is the level where the top base AB will sit.
• Mark point O between DC and the parallel line, roughly in the centre — the two diagonals will cross here.
• Draw a line through D and O. Extended upward, it will hit the parallel at point B.
• Mark point B where the line through D and O meets the parallel above.
• Draw a line through C and O. Because O lies between D and C, this mirrors the first diagonal — giving an isosceles trapezium.
• Mark point A where the line through C and O meets the parallel above.
• Draw segment AB — the top base. Since A and B both lie on the parallel, AB is parallel to DC.
• Draw segment AD — the left leg.
• Draw segment BC — the right leg. Check that AD = BC: the trapezium is isosceles!

The proof, step by step

Prove that the non-parallel sides and the diagonals of an isosceles trapezium are equal.

• In isosceles trapezium ABCD, AB || DC and AD = BC.
• Base angles are equal: ∠ADC = ∠BCD.
• Consider triangles ADC and BCD. Side DC is common to both.
• By SAS congruence criterion, ΔADC ≅ ΔBCD.
• Therefore, corresponding parts are equal: AC = BD.

Worked example

In an isosceles trapezium ABCD with AB parallel to DC, if angle A measures 105°, what is the measure of angle C?

In an isosceles trapezium, consecutive angles between parallel sides sum to 180°. So, ∠A + ∠D = 180°. Given ∠A = 105°, we get ∠D = 75°. Since base angles are equal in an isosceles trapezium, ∠C = ∠D = 75°.

• 65°
• 105°
• 115°
• 75° — correct