Chapter 9: Quadrilaterals

High School Geometry · Unit 5: Quadrilaterals · 3 interactive diagrams · 8 practice problems

Learning Objectives

Apply the interior angle sum formula $(n-2) \cdot 180°$ to polygons and quadrilaterals
Identify and classify quadrilaterals using the hierarchy of types
State and apply all five properties of parallelograms
Use six methods to prove a quadrilateral is a parallelogram
Distinguish properties of rectangles, rhombuses, and squares
Apply the Trapezoid Midsegment Theorem and properties of isosceles trapezoids and kites

9.1 Polygons and Quadrilateral Basics

Definition: Polygon and Quadrilateral

A polygon is a closed figure formed by three or more straight sides. A quadrilateral is a polygon with exactly four sides, four vertices, and four angles.

A diagonal of a quadrilateral is a segment connecting two non-consecutive (non-adjacent) vertices.

Theorem: Quadrilateral Angle Sum = 360°

The sum of the interior angles of any polygon with $n$ sides is $(n-2) \cdot 180°$.

For a quadrilateral ($n = 4$): $(4-2) \cdot 180° = 2 \cdot 180° = \mathbf{360°}$.

Every quadrilateral's four interior angles always add up to 360°, regardless of shape. The main types of quadrilaterals form a hierarchy:

Parallelogram — two pairs of parallel sides
Rectangle — parallelogram with four right angles
Rhombus — parallelogram with four congruent sides
Square — parallelogram with four right angles AND four congruent sides (both rectangle and rhombus)
Trapezoid — exactly one pair of parallel sides
Isosceles Trapezoid — trapezoid with congruent legs
Kite — two pairs of consecutive congruent sides (not opposite)

Example 9.1 — Finding a Missing Angle

A quadrilateral has angles $95°$, $85°$, $110°$, and $x°$. Find $x$.

Using the Angle Sum Theorem: $95 + 85 + 110 + x = 360$

$290 + x = 360 \Rightarrow x = \mathbf{70°}$

Example 9.2 — Interior Angles of a Regular Hexagon

A regular hexagon has 6 sides. Find each interior angle.

Sum of interior angles: $(6-2) \cdot 180° = 720°$

Each angle of a regular hexagon: $\dfrac{720°}{6} = \mathbf{120°}$

TRY IT

A quadrilateral has angles in the ratio $2:3:4:6$. Find each angle. (Total = 360°)

Show Answer

Sum of ratio parts: $2+3+4+6=15$. Each part = $360°/15 = 24°$.
Angles: $2 \times 24 = \mathbf{48°}$, $3 \times 24 = \mathbf{72°}$, $4 \times 24 = \mathbf{96°}$, $6 \times 24 = \mathbf{144°}$.

Quadrilateral ABCD with labeled vertices and both diagonals AC and BD shown.

Figure 9.1 — Quadrilateral with Vertices and Diagonals

9.2 Parallelograms

Definition: Parallelogram

A parallelogram is a quadrilateral with two pairs of parallel sides. In parallelogram $ABCD$: $AB \parallel CD$ and $BC \parallel AD$.

5 Properties of Parallelograms

Opposite sides are congruent: $AB \cong CD$ and $BC \cong AD$
Opposite angles are congruent: $\angle A \cong \angle C$ and $\angle B \cong \angle D$
Consecutive angles are supplementary: $\angle A + \angle B = 180°$, $\angle B + \angle C = 180°$, etc.
Diagonals bisect each other: The diagonals intersect at their midpoints
A diagonal divides it into two congruent triangles: $\triangle ABC \cong \triangle CDA$

Proving a Quadrilateral Is a Parallelogram

A quadrilateral is a parallelogram if any ONE of the following conditions holds:

Both pairs of opposite sides are parallel
Both pairs of opposite sides are congruent
Both pairs of opposite angles are congruent
One pair of opposite sides is both parallel and congruent
The diagonals bisect each other
Consecutive angles are supplementary

Example 9.3 — Using Opposite Sides Equal

$ABCD$ is a parallelogram. $AB = 2x + 3$ and $CD = 5x - 6$. Find $x$ and $AB$.

Opposite sides of a parallelogram are congruent: $AB = CD$

$2x + 3 = 5x - 6 \Rightarrow 9 = 3x \Rightarrow x = 3$

$AB = 2(3) + 3 = \mathbf{9}$

Example 9.4 — Using Angle Properties

Parallelogram $ABCD$ has $\angle A = 65°$. Find $\angle B$, $\angle C$, and $\angle D$.

$\angle B = 180° - 65° = \mathbf{115°}$ (consecutive angles are supplementary)
$\angle C = 65°$ (opposite angles are congruent, $\angle C \cong \angle A$)
$\angle D = 115°$ (opposite angles are congruent, $\angle D \cong \angle B$)

TRY IT

In parallelogram $PQRS$, diagonals meet at $T$. If $PT = 3x - 1$ and $TR = x + 7$, find $x$ and $PR$.

Show Answer

Diagonals bisect each other, so $PT = TR$:
$3x - 1 = x + 7 \Rightarrow 2x = 8 \Rightarrow x = 4$
$PT = 3(4) - 1 = 11$, so $PR = 2 \times 11 = \mathbf{22}$

★

AP Tip: Know all 5 properties of parallelograms AND all 6 ways to prove a quadrilateral is a parallelogram. Both property-application questions and proof-based questions appear on standardized tests. The most commonly tested proving method is: one pair of opposite sides both parallel AND congruent.

9.3 Special Parallelograms: Rectangles, Rhombuses, and Squares

Definitions: Special Parallelograms

A rectangle is a parallelogram with four right angles. Special property: its diagonals are congruent.
A rhombus is a parallelogram with four congruent sides. Special properties: its diagonals are perpendicular bisectors of each other; each diagonal bisects a pair of opposite angles.
A square is a parallelogram with four right angles AND four congruent sides. A square has ALL properties of both a rectangle and a rhombus.

Special Parallelogram Properties Summary

Property	Parallelogram	Rectangle	Rhombus	Square
Diagonals bisect each other	✓	✓	✓	✓
Diagonals congruent	✗	✓	✗	✓
Diagonals perpendicular	✗	✗	✓	✓
Diagonals bisect angles	✗	✗	✓	✓

Example 9.5 — Rectangle Diagonals

Rectangle $ABCD$ has $AC = 3x - 7$ and $BD = x + 9$. Find $x$ and the length of each diagonal.

Diagonals of a rectangle are congruent: $AC = BD$

$3x - 7 = x + 9 \Rightarrow 2x = 16 \Rightarrow x = 8$

Each diagonal $= 3(8) - 7 = \mathbf{17}$

Example 9.6 — Rhombus Side Length and Perimeter

Rhombus $KLMN$ has $KN = 2x + 3$ and $KL = 5x - 9$. Find $x$ and the perimeter.

All sides of a rhombus are congruent: $KN = KL$

$2x + 3 = 5x - 9 \Rightarrow 12 = 3x \Rightarrow x = 4$

$KN = 2(4) + 3 = 11$. Perimeter $= 4 \times 11 = \mathbf{44}$

TRY IT

A square has diagonal length 10. Find the side length and perimeter.

Show Answer

In a square with side $s$, the diagonal $= s\sqrt{2}$.
$s\sqrt{2} = 10 \Rightarrow s = \dfrac{10}{\sqrt{2}} = 5\sqrt{2}$
Perimeter $= 4s = 20\sqrt{2} \approx 28.28$

Rectangle (purple), Rhombus (blue), and Square (orange) — each showing both diagonals.

Figure 9.2 — Rectangle, Rhombus, and Square with Diagonals

9.4 Trapezoids and Kites

Definitions: Trapezoid and Kite

A trapezoid is a quadrilateral with exactly one pair of parallel sides, called the bases. The non-parallel sides are called legs.
An isosceles trapezoid has congruent legs. Special properties: base angles are congruent; diagonals are congruent.
A kite is a quadrilateral with two pairs of consecutive congruent sides (not opposite sides). Properties: the diagonals are perpendicular; one diagonal bisects the other; one pair of opposite angles is congruent.

Trapezoid Midsegment Theorem

The midsegment of a trapezoid connects the midpoints of the two legs. It is parallel to both bases, and its length equals the average of the two bases:

$$m = \frac{b_1 + b_2}{2}$$

Example 9.7 — Midsegment of a Trapezoid

Trapezoid $ABCD$ has $AB \parallel CD$, $AB = 18$, and $CD = 10$. Find the midsegment length.

$m = \dfrac{AB + CD}{2} = \dfrac{18 + 10}{2} = \mathbf{14}$

Example 9.8 — Angles of an Isosceles Trapezoid

An isosceles trapezoid has lower base angles of $65°$. Find the other two angles.

In an isosceles trapezoid, base angles are congruent, so both lower base angles are $65°$.

Consecutive angles between a pair of parallel sides are supplementary:

Upper base angles $= 180° - 65° = \mathbf{115°}$ each.

Example 9.9 — Perimeter of a Kite

Kite $ABCD$ has $AB = BC = 5$ and $CD = DA = 8$. Find the perimeter.

$P = 2(5) + 2(8) = 10 + 16 = \mathbf{26}$

TRY IT

A trapezoid has a midsegment of length 12 and one base of length 8. Find the other base.

Show Answer

$m = \dfrac{b_1 + b_2}{2}$, so $12 = \dfrac{8 + b_2}{2}$
$24 = 8 + b_2 \Rightarrow b_2 = \mathbf{16}$

Isosceles trapezoid (left, purple) with midsegment, and kite (right, blue) with perpendicular diagonals.

Figure 9.3 — Isosceles Trapezoid with Midsegment and Kite

Practice Problems

Parallelogram $ABCD$ has $\angle A = 3x + 10$ and $\angle C = 5x - 14$. Find $x$ and all four angles.

Show Solution

Opposite angles are congruent: $3x + 10 = 5x - 14 \Rightarrow 24 = 2x \Rightarrow x = 12$.
$\angle A = \angle C = 3(12) + 10 = \mathbf{46°}$.
$\angle B = \angle D = 180° - 46° = \mathbf{134°}$.

The diagonals of parallelogram $PQRS$ meet at $T$. $PT = 4x - 3$ and $TR = 2x + 7$. Find $x$ and $PR$.

Show Solution

Diagonals bisect each other: $PT = TR$
$4x - 3 = 2x + 7 \Rightarrow 2x = 10 \Rightarrow x = 5$
$PT = 4(5) - 3 = 17$, so $PR = 2 \times 17 = \mathbf{34}$.

Prove that $ABCD$ is a parallelogram given $AB = CD = 8$ and $AB \parallel CD$.

Show Solution

One pair of opposite sides ($AB$ and $CD$) is both congruent ($AB = CD = 8$) and parallel ($AB \parallel CD$). By the parallelogram condition — one pair of opposite sides both parallel and congruent — $ABCD$ is a parallelogram. $\square$

Rectangle $EFGH$ has $EG = 4x + 6$ and $FH = 6x - 2$. Find the length of each diagonal.

Show Solution

Diagonals of a rectangle are congruent: $EG = FH$
$4x + 6 = 6x - 2 \Rightarrow 8 = 2x \Rightarrow x = 4$
Each diagonal $= 4(4) + 6 = \mathbf{22}$.

A rhombus has diagonals of length 16 and 12. Find the side length of the rhombus.

Show Solution

Diagonals of a rhombus are perpendicular bisectors of each other.
Half-diagonals: $8$ and $6$. Each side is the hypotenuse of a right triangle:
$s = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = \mathbf{10}$.

Trapezoid $ABCD$ has $AB \parallel CD$, $AB = 20$, $CD = 12$. Find the midsegment length. If the height is 8, find the area using $A = \frac{1}{2}(b_1 + b_2)h$.

Show Solution

Midsegment: $m = \dfrac{20 + 12}{2} = \mathbf{16}$.
Area: $A = \dfrac{1}{2}(20 + 12)(8) = \dfrac{1}{2}(32)(8) = \mathbf{128}$ square units.

The midsegment of a trapezoid is 15 and one base is 9. Find the other base and the ratio of the midsegment to each base.

Show Solution

$15 = \dfrac{9 + b_2}{2} \Rightarrow 30 = 9 + b_2 \Rightarrow b_2 = \mathbf{21}$.
Ratio of midsegment to bases: $15:9 = 5:3$ and $15:21 = 5:7$.

AP Problem: Quadrilateral $WXYZ$ has vertices $W=(0,0)$, $X=(5,0)$, $Y=(6,4)$, $Z=(1,4)$. (a) Use slope to classify $WXYZ$. (b) Find the lengths of diagonals $WY$ and $XZ$. (c) Is it a rectangle?

Show Solution

(a) Slope of $WX$: $\frac{0-0}{5-0}=0$. Slope of $ZY$: $\frac{4-4}{6-1}=0$. So $WX \parallel ZY$.
Slope of $WZ$: $\frac{4-0}{1-0}=4$. Slope of $XY$: $\frac{4-0}{6-5}=4$. So $WZ \parallel XY$.
Both pairs of opposite sides parallel → parallelogram.
$WX=5$, $ZY=5$; $WZ=\sqrt{1+16}=\sqrt{17}$, $XY=\sqrt{1+16}=\sqrt{17}$ → sides unequal, not a rhombus or square.
(b) $WY = \sqrt{(6-0)^2+(4-0)^2}=\sqrt{52}=2\sqrt{13}$. $XZ=\sqrt{(1-5)^2+(4-0)^2}=\sqrt{32}=4\sqrt{2}$.
(c) Diagonals not equal ($2\sqrt{13} \neq 4\sqrt{2}$), so not a rectangle. $WXYZ$ is a parallelogram only.

📋 Chapter Summary

Parallelogram Properties

Parallelogram

Opposite sides parallel and congruent. Opposite angles congruent. Consecutive angles supplementary. Diagonals bisect each other.

Rectangle

A parallelogram with four right angles. Diagonals are congruent (and bisect each other). All rectangle properties hold.

Rhombus

A parallelogram with four congruent sides. Diagonals are perpendicular and bisect the angles. All parallelogram properties hold.

Square

A rectangle AND a rhombus. Four right angles AND four congruent sides. Diagonals are congruent, perpendicular, and bisect the angles.

Other Quadrilaterals

Trapezoid

Exactly one pair of parallel sides (the bases). Midsegment $= \frac{1}{2}(\text{base}_1 + \text{base}_2)$. Isosceles trapezoid: legs congruent, diagonals congruent.

Kite

Two pairs of consecutive congruent sides. One pair of congruent opposite angles. Diagonals are perpendicular; one diagonal bisects the other.

📘 Key Terms

ParallelogramA quadrilateral with both pairs of opposite sides parallel.

RectangleA parallelogram with four right angles. Diagonals are congruent.

RhombusA parallelogram with four congruent sides. Diagonals are perpendicular.

SquareA quadrilateral with four right angles AND four congruent sides. Both a rectangle and a rhombus.

TrapezoidA quadrilateral with exactly one pair of parallel sides.

KiteA quadrilateral with two pairs of consecutive congruent sides. Diagonals are perpendicular.

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