Quadrilaterals & Polygons

GE.F.1Parallelograms01/04

Properties of parallelograms.

Step01of 04

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. From that single definition, FIVE properties follow.

Opposite sides	→parallel AND congruent
Opposite angles	→congruent
Consecutive angles	→supplementary (sum to 180°)
Diagonals	→BISECT each other
Each diagonal	→divides the parallelogram into two congruent triangles

GE.F.1 — Properties of parallelograms.

Step 01 of 04

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. From that single definition, FIVE properties follow.

Opposite sides	→parallel AND congruent
Opposite angles	→congruent
Consecutive angles	→supplementary (sum to 180°)
Diagonals	→BISECT each other
Each diagonal	→divides the parallelogram into two congruent triangles

Step 02 of 04

Worked example. In parallelogram $A B C D$ : $∠ A = (3 x + 20) °$ , $∠ C = (5 x - 10) °$ . Find $x$ and the angle measures.

Opposite angles in a parallelogram are congruent:

3 x + 20 = 5 x - 10 \Rightarrow 30 = 2 x \Rightarrow x = 15

∠ A = ∠ C = 65°, ∠ B = ∠ D = 115°

Step 03 of 04

Five ways to PROVE a quadrilateral is a parallelogram. Any one of these is sufficient.

Both pairs of opposite sides parallel	→definition
Both pairs of opposite sides congruent	→converse property
Both pairs of opposite angles congruent	→converse property
Diagonals bisect each other	→converse property
ONE pair of opposite sides BOTH parallel and congruent	→sufficient

Step 04 of 04

Watch out for the trap. Just having one pair of parallel sides is NOT enough — that's a trapezoid, not a parallelogram. And just having one pair of CONGRUENT sides isn't enough either (could be a kite). You need BOTH conditions on the SAME pair, OR conditions on both pairs.

Key insight

Parallelogram = opposite sides parallel. Everything else (opposite sides equal, opposite angles equal, diagonals bisecting each other) follows automatically. To prove a quad is a parallelogram, hit ANY ONE of the five sufficient conditions.

GE.F.2 — Rectangles, rhombuses, squares.

Step 01 of 04

These three are all special parallelograms. Each adds an extra property on top of the parallelogram baseline.

Rectangle	→parallelogram + four right angles → diagonals are CONGRUENT
Rhombus	→parallelogram + four congruent sides → diagonals are PERPENDICULAR
Square	→rectangle AND rhombus → all four sides equal AND four right angles AND diagonals congruent + perpendicular

A square is the maximally constrained quadrilateral — it satisfies every property the others have, plus its own.

Step 02 of 04

The relationship hierarchy.

Every square	→is a rectangle, a rhombus, AND a parallelogram
Every rectangle	→is a parallelogram (but not always a rhombus or square)
Every rhombus	→is a parallelogram (but not always a rectangle or square)
NOT every parallelogram	→is a rectangle, rhombus, or square

When a problem says "given a rectangle," you automatically get every parallelogram property too — you don't have to re-prove them.

Step 03 of 04

Diagonals — the diagnostic that distinguishes the three.

Rectangle (only)	→diagonals congruent (equal length) but NOT perpendicular
Rhombus (only)	→diagonals perpendicular but NOT (in general) equal length
Square	→diagonals are BOTH equal length AND perpendicular

Diagonals always bisect each other (parallelogram property), so the diagonal length / perpendicularity is the only thing that changes.

Step 04 of 04

Worked example. The diagonals of rhombus $A B C D$ have lengths 6 and 8. Find the side length.

Diagonals of a rhombus are perpendicular and bisect each other, so they cut the rhombus into 4 right triangles with legs 3 and 4 (half each diagonal). Hypotenuse = side of rhombus:

side = 3^{2} + 4^{2} = 5

Key insight

Rectangle: equal diagonals. Rhombus: perpendicular diagonals. Square: both. The diagonals are the diagnostic — pair them with the parallelogram "diagonals bisect each other" baseline to handle every "find the side from the diagonal" problem.

GE.F.3 — Trapezoids and kites.

Step 01 of 04

These two are NOT parallelograms — they have different defining conditions and their own special properties.

Trapezoid	→EXACTLY one pair of parallel sides (called bases)
Isosceles trapezoid	→trapezoid where the non-parallel sides (legs) are congruent
Kite	→two pairs of CONSECUTIVE congruent sides (not opposite!)

Note on definitions: some textbooks allow a parallelogram to count as a trapezoid (using "AT LEAST one pair parallel"). Most US high-school texts use "EXACTLY one pair" — we follow that here.

Step 02 of 04

Trapezoid midsegment. The segment joining the midpoints of the two legs is parallel to both bases, and its length is the AVERAGE of the base lengths.

midsegment = \frac{b _{1} + b _{2}}{2}

Example: bases 8 and 14 → midsegment $(8 + 14) /2 = 11$ .

Step 03 of 04

Isosceles trapezoid bonus properties.

Base angles	→congruent (the two angles at each base are equal to each other)
Diagonals	→congruent

Kite properties.

Diagonals	→PERPENDICULAR
One diagonal	→BISECTS the other (the symmetry axis bisects the other diagonal)
One pair of opposite angles	→congruent (the angles between non-congruent sides)

Step 04 of 04

Areas — formulas worth memorizing.

Parallelogram	→ $A = bh$ (base × height)
Rectangle	→ $A = l w$
Square	→ $A = s^{2}$
Rhombus	→ $A = \frac{1}{2} d_{1} d_{2}$ (half product of diagonals)
Kite	→ $A = \frac{1}{2} d_{1} d_{2}$ (same as rhombus!)
Trapezoid	→ $A = \frac{1}{2} (b_{1} + b_{2}) h$ (average of bases × height)

Key insight

Trapezoid = exactly one pair of parallel sides. Kite = two pairs of consecutive congruent sides. Both have perpendicular-diagonal area formulas (kite, rhombus), and the trapezoid uses the average-of-bases shortcut for its midsegment AND its area.

GE.F.4 — Polygon angle sums.

Step 01 of 04

Sum of interior angles of an n-sided polygon:

S = (n - 2) \cdot 180°

The " $n - 2$ " comes from cutting the polygon into $n - 2$ triangles by drawing diagonals from one vertex. Each triangle contributes $180°$ .

Step 02 of 04

Quick reference.

Triangle (n=3)	→ $(3 - 2) (180°) = 180°$
Quadrilateral (n=4)	→ $(4 - 2) (180°) = 360°$
Pentagon (n=5)	→ $(5 - 2) (180°) = 540°$
Hexagon (n=6)	→ $(6 - 2) (180°) = 720°$
Octagon (n=8)	→ $(8 - 2) (180°) = 1080°$
Decagon (n=10)	→ $(10 - 2) (180°) = 1440°$

Step 03 of 04

Regular polygons have all sides AND all angles equal. Each interior angle of a regular n-gon:

\frac{( n - 2 ) \cdot 180°}{n}

For a regular hexagon: $720°/6 = 120°$ . For a regular octagon: $1080°/8 = 135°$ .

Step 04 of 04

Sum of EXTERIOR angles is always 360° for ANY convex polygon — regardless of how many sides. (Take one exterior angle at each vertex.)

For a regular n-gon, that means each exterior angle measures $360°/ n$ . Often the easier way to find the interior angle of a regular polygon: subtract the exterior from $180°$ .

Example: regular octagon. Each exterior = $360°/8 = 45°$ , so each interior = $180° - 45° = 135°$ . Same answer, less arithmetic.

Key insight

Interior sum:

(n - 2) \cdot 180°

. Each interior angle of a regular n-gon: divide by

n

. Sum of exterior angles is ALWAYS

360°

regardless of

n

— the easy back door to individual interior angles.

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