Properties of Quadrilaterals

⬛ Properties of Quadrilaterals

Part 1 of 5 — The Quadrilateral Family

Topics in This Part

🔑 Key Concept: Every quadrilateral — square, kite, trapezoid, or a lopsided four-sided blob — has interior angles that add up to exactly 360°. That single fact powers most of Part 1.

What Is a Quadrilateral?

A quadrilateral is a closed figure with four straight sides and four vertices. The prefix quad- means four; -lateral means sides.

Key vocabulary you'll use throughout this lesson:

A quadrilateral is convex if every interior angle is less than $180°$ (no "dents"). Unless told otherwise, assume the quadrilaterals in this lesson are convex.

💡 A quadrilateral has exactly two diagonals. Many of the special properties ahead are really statements about how those two diagonals behave.

The Angle-Sum Property

Draw one diagonal in any quadrilateral and you split it into two triangles. Each triangle's angles sum to $180°$, so:

$\text{Sum of interior angles} = 2 \times 180° = 360°$

This holds for every quadrilateral, no matter how irregular.

Worked Example

Three angles of a quadrilateral are $80°$, $100°$, and $95°$. Find the fourth.

$x = 360° - (80° + 100° + 95°) = 360° - 275° = 85°$

✅ Check: $80° + 100° + 95° + 85° = 360°$ ✓

Concept Check 🎯

A Useful Consequence

The angle-sum fact alone already locks in some shapes. If a convex quadrilateral has three right angles, the fourth is forced to be a right angle too:

$360° - 3(90°) = 360° - 270° = 90°$

So you can never build a four-sided figure with exactly three right angles — it always becomes a rectangle.

💡 Keep this in your back pocket: angle-chasing problems almost always start by writing "the four angles add to $360°$."

Find the Missing Angle 🧮

Each quadrilateral's angles must total $360°$. Find the unknown angle (just the number, no degree symbol).

1) Angles $120°, 75°, 65°,$ and $x$. Find $x$. 2) Angles $100°, 100°, 100°,$ and $x$. Find $x$. 3) A quadrilateral has three right angles and one angle $x$. Find $x$.

The Quadrilateral Family Tree

Quadrilaterals form a hierarchy — special shapes are special because they add properties on top of more general ones.

🔑 Read it top-down: a square is a rhombus, a rhombus is a parallelogram, and a parallelogram is a quadrilateral. The reverse is not true — most parallelograms are not squares.

💡 Because the tree flows downward, a square inherits every property of rectangles, rhombuses, and parallelograms. We'll cash in on that inheritance in Parts 3–5.

Place It in the Family 🔽

Choose the answer that correctly describes each relationship.

⬛ Properties of Quadrilaterals

Part 2 of 5 — Parallelograms

🔑 The Idea: A parallelogram has both pairs of opposite sides parallel. From that one definition, four powerful properties about its sides, angles, and diagonals follow automatically.

The Four Properties of a Parallelogram

In parallelogram $ABCD$ (with $\overline{AB} \parallel \overline{DC}$ and $\overline{AD} \parallel \overline{BC}$):

⚠️ Watch property 4 carefully: the diagonals bisect each other, but in a general parallelogram they are not equal and not perpendicular. Those extra features only appear in special parallelograms (Part 3).

Why consecutive angles are supplementary

$\overline{AD} \parallel \overline{BC}$ makes $\angle A$ and $\angle B$ co-interior (same-side) angles on the transversal $\overline{AB}$, so they sum to $180°$.

Concept Check 🎯

Worked Example: Solving for Sides

In parallelogram $PQRS$, side $PQ = 3x + 2$ and the opposite side $SR = 5x - 8$. Find $x$ and the length of $PQ$.

Opposite sides of a parallelogram are congruent, so set them equal:

$3x + 2 = 5x - 8$ $2 + 8 = 5x - 3x$ $10 = 2x \;\Rightarrow\; x = 5$

Now substitute back:

$PQ = 3(5) + 2 = 17$

✅ Check: $SR = 5(5) - 8 = 17$. The opposite sides match ✓

Solve the Parallelogram 🧮

Use the parallelogram properties to solve.

1) Opposite sides are $2x + 1$ and $x + 9$. Find $x$. 2) In a parallelogram, one angle is $115°$. Find a consecutive angle. 3) A diagonal is bisected at point $M$. If one half is $12$, how long is the whole diagonal?

Proving a Quadrilateral Is a Parallelogram

The properties also work in reverse — any one of these conditions is enough to guarantee a parallelogram:

1. Both pairs of opposite sides are congruent.
2. Both pairs of opposite angles are congruent.
3. The diagonals bisect each other.
4. One pair of opposite sides is both parallel and congruent.

💡 Condition 4 is a favorite on tests: you only need to check one pair of sides, as long as that pair is parallel and equal in length.

Is It a Parallelogram? 🔽

Decide whether each condition guarantees a parallelogram.

⬛ Properties of Quadrilaterals

Part 3 of 5 — Rectangles, Rhombuses & Squares

🔑 Inheritance: Each special parallelogram keeps all four parallelogram properties and then adds its own. A square, sitting at the bottom of the tree, gets every property of all of them.

The Rectangle

A rectangle is a parallelogram with four right angles.

It keeps everything a parallelogram has, and adds one signature property:

🔑 Rectangle diagonals are congruent: in rectangle $ABCD$, $AC = BD$.

Worked Example: Diagonal of a Rectangle

A rectangle is $6$ by $8$. Its diagonal is the hypotenuse of a right triangle:

$d = \sqrt{6^2 + 8^2} = \sqrt{36 + 64} = \sqrt{100} = 10$

Rectangle Drills 🧮

1) A rectangle has sides $9$ and $12$. Find the length of a diagonal. 2) In a rectangle, one diagonal is $20$. How long is the other diagonal? 3) The diagonals of a rectangle meet at center $M$. If a full diagonal is $26$, how far is each vertex from $M$?

The Rhombus

A rhombus is a parallelogram with four congruent sides.

It also keeps every parallelogram property and adds two new ones:

🔑 Rhombus diagonals are perpendicular ($\overline{AC} \perp \overline{BD}$) and they bisect the angles of the rhombus.

⚠️ Don't mix them up: a rectangle has congruent diagonals; a rhombus has perpendicular diagonals. Only the square has both.

Area from Diagonals

Because the diagonals are perpendicular, a rhombus's area is:

$A = \frac{1}{2} \, d_1 d_2$

If $d_1 = 6$ and $d_2 = 8$: $\;A = \frac{1}{2}(6)(8) = 24$.

Concept Check 🎯

The Square — The Best of Both

A square is a parallelogram that is both a rectangle and a rhombus. So it inherits every special property:

🔑 A square is the only quadrilateral whose diagonals are congruent, perpendicular, and bisect each other — and each diagonal cuts the corner angles into two $45°$ halves.

Diagonal of a Square

For a square with side $s$, the diagonal is:

$d = s\sqrt{2}$

A square with side $5$ has diagonal $5\sqrt{2} \approx 7.07$.

Match the Diagonal Behavior 🔽

For each shape, choose how its diagonals behave (assume a general example of each shape).

⬛ Properties of Quadrilaterals

Part 4 of 5 — Trapezoids & Kites

🔑 Off the parallelogram branch: Trapezoids and kites are not parallelograms. Each has its own special structure — one pair of parallel sides (trapezoid) or two pairs of adjacent equal sides (kite).

The Trapezoid

A trapezoid has at least one pair of parallel sides, called the bases. The non-parallel sides are the legs.

The Midsegment Rule

The midsegment is parallel to the bases and its length is the average of the bases:

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

Example: bases $10$ and $16$ give midsegment $\dfrac{10 + 16}{2} = 13$.

Area of a Trapezoid

$A = \frac{1}{2}(b_1 + b_2)\,h$

where $h$ is the perpendicular distance between the bases.

Trapezoid Drills 🧮

1) Bases $8$ and $14$. Find the midsegment length. 2) A trapezoid has bases $5$ and $11$ and height $4$. Find its area. 3) The midsegment is $9$ and one base is $6$. Find the other base.

The Isosceles Trapezoid

An isosceles trapezoid has congruent legs. This symmetry creates extra properties:

🔑 In an isosceles trapezoid: the legs are congruent, each pair of base angles is congruent, and the diagonals are congruent.

⚠️ A trapezoid's two angles along the same leg are co-interior on the parallel bases, so they are supplementary — they add to $180°$.

Concept Check 🎯

The Kite

A kite has two pairs of consecutive (adjacent) congruent sides — but its opposite sides are not equal.

🔑 A kite's diagonals are perpendicular, just like a rhombus — but only one diagonal is bisected (the axis of symmetry bisects the other).

Area of a Kite

Like the rhombus, a kite's perpendicular diagonals give:

$A = \frac{1}{2}\, d_1 d_2$

💡 A rhombus is actually a special kite (all four sides equal) — that's why both use the same diagonal area formula.

Trapezoid vs. Kite 🔽

Match each clue to the shape it best describes.

⬛ Properties of Quadrilaterals

Part 5 of 5 — Mixed Practice & Mastery Check

You can now (1) use the $360°$ angle sum, (2) apply parallelogram properties, (3) tell rectangles, rhombuses, and squares apart, and (4) handle trapezoids and kites. Let's pull it all together.

Quick Reference

⚠️ The classic trap: rectangle ⇒ congruent diagonals, rhombus ⇒ perpendicular diagonals, square ⇒ both. Memorize this triangle of facts and most diagonal questions become instant.

Mixed Computation 🧮

1) A parallelogram has one angle of $130°$. Find a consecutive angle. 2) A square has side $10$. Find its diagonal as a decimal, rounded to the nearest hundredth. (Use $d = s\sqrt{2}$.) 3) A rhombus has diagonals $10$ and $24$. Find its area.

Name That Quadrilateral 🔽

Use the most specific name that the clues force.

One Last Strategy

When a problem hands you only diagonal clues, reach for the diagonal triangle:

• Bisect each other → it's a parallelogram (at least).
• + Congruent → rectangle.
• + Perpendicular → rhombus.
• + Congruent AND perpendicular → square.

🔑 Always claim the most specific shape the clues allow — that's what the final questions are testing.

Mixed Practice 🎯

Exit Quiz ✅

Answer all three to finish the lesson.