02.06 Quadrilateral Proofs

Four sides, then name the pairs

A polygon is closed, three or more straight sides, no crossing. A quadrilateral has four. Opposite sides do not share a vertex. Adjacent sides share a vertex. Diagonals join opposite vertices and cut the figure into triangles. Those triangles are how the proofs work.

A parallelogram is the one whose opposite sides are parallel. From that one fact the rest follow: opposite sides congruent, opposite angles congruent, consecutive angles supplementary, diagonals bisect each other.

The parallelogram toolkit

Opposite sides: same slope (parallel) and same length. Opposite angles match. Neighbors add to 180° because they are same-side interior on a pair of parallels.

Diagonals cut each other in half. They do not have to be equal, and they do not have to be perpendicular. Those extras belong to special parallelograms.

Number example: exterior angle 40° next to ∠ABC in parallelogram ABCD. Linear pair makes m∠ABC = 140°. Opposite angles match, so m∠ADC = 140°.

Another: AE = ½x − 14 and CE = 37 on a diagonal that bisects. Set them equal: ½x − 14 = 37, x = 102. The reason is "diagonals of a parallelogram bisect each other," not a random Property of Equality with no geometry named.

Lookalike diagonals: bisect each other (any parallelogram). Congruent (rectangle or square). Perpendicular (rhombus, square, kite). A quiz will offer all three as reasons for the same picture.

Prove it is a parallelogram

You only need one of these:

Both pairs of opposite sides parallel
Both pairs of opposite sides congruent
One pair of opposite sides both congruent and parallel
Both pairs of opposite angles congruent
Diagonals bisect each other

Opposite-angles route: the four interior angles add to 360°. Two pairs of matches means each consecutive pair is 180°, so same-side interior are supplementary, so both pairs of sides are parallel.

Rectangle, rhombus, square

Rectangle: parallelogram with four right angles. Extra: diagonals are congruent.

Rhombus: parallelogram with four congruent sides. Extra: diagonals are perpendicular, and they bisect the vertex angles.

Square: both. Four right angles and four congruent sides, so both extra diagonal facts.

If a parallelogram already has one pair of adjacent sides congruent, Transitive plus opposite sides pulls all four sides equal, so it is a rhombus.

In square ABCD, AC = 3(x + 8) and BD = 7 + 4x. Diagonals of a square (or rectangle) are congruent, so 3x + 24 = 7 + 4x, x = 17.

In rhombus ADCB, m∠DCB = 46°. A diagonal bisects that angle, so each half is 23°. The other triangle at that vertex is a right triangle (perpendicular diagonals), so the remaining piece is 90 − 23 = 67°. That 67° is a usual lookalike next to 46° and 134° (the consecutive supplementary).

Kite and trapezoid

Kite: two pairs of consecutive congruent sides. Vertex angles sit between a matching pair. The vertex diagonal bisects those angles and is the perpendicular bisector of the other diagonal. Kites are not parallelograms.

Trapezoid: exactly one pair of parallel sides (the bases). Legs are the nonparallel sides. Base angles share a base. Isosceles trapezoid: legs congruent, base angles congruent, diagonals congruent.

Kite vs parallelogram: kite matches consecutive sides. Parallelogram matches opposite sides. A kite's diagonals are perpendicular; only one of them is bisected.

Hinge Theorem, quickly

Two triangles with two pairs of matching sides. The one with the larger included angle has the larger third side. Think of a door: wider opening, wider gap across the room.

This is not congruence. It is a comparison. The converse goes the other way: larger third side means larger included angle.

Your turn

Same length or same angle in two choices. The discriminator is which parallelogram fact you named.