Special Types of Quadrilaterals

166. The Parallelogram's Balance · Opposite sides & angles always match

Opposite sides are equal, opposite angles match, and diagonals bisect each other.

The Parallelogram Theorem has four parts: (1) opposite sides are equal, (2) opposite angles are equal, (3) adjacent angles are supplementary (sum to 180°), and (4) diagonals bisect each other. All four hold simultaneously for any parallelogram.

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Selina ICSE: Special Types of Quadrilaterals

What this lesson covers

Try to break it

Drag A, B, or D. The shape stretches and shears, but AB ∥ DC, AD ∥ BC, opposite sides stay equal, and opposite angles stay equal. Try to drag a vertex so one pair of opposite sides stops being parallel — the figure refuses; the fourth vertex follows along to keep the parallelism.

How you build it

Construct a parallelogram.

Place point A — the first corner of the base.
Place point B — the second corner of the base.
Draw segment AB — the base of the parallelogram.
Place point D above AB — the third corner.
Draw segment AD — one slant side of the parallelogram.
Parallel tool: click point B, then click side AD — draws the line through B parallel to AD. Side BC will lie on it.
Parallel tool: click point D, then click base AB — draws the top side DC parallel to AB.
Mark point C where the two parallel lines cross — click snaps to the exact crossing. AB ∥ DC and AD ∥ BC by construction.

The proof, step by step

Prove that opposite sides and opposite angles of a parallelogram are equal and its diagonals bisect each other.

Draw diagonal AC to split the parallelogram into ΔABC and ΔADC.
∠BAC = ∠DCA (Alternate angles, AB || DC)
∠BCA = ∠DAC (Alternate angles, AD || BC)
AC = AC (Common side)
ΔABC ≅ ΔADC by ASA congruence rule.
Therefore, AB = DC, BC = AD, and ∠B = ∠D (CPCT).

Worked example

In a parallelogram ABCD, if ∠A = 70°, find the measure of ∠C and ∠B.

Opposite angles of a parallelogram are equal, so ∠C = ∠A = 70°. Adjacent angles are supplementary, so ∠B = 180° - 70° = 110°.

∠C = 70°, ∠B = 110° — correct
∠C = 110°, ∠B = 70°
∠C = 70°, ∠B = 70°
∠C = 110°, ∠B = 110°