Polygons and Quadrilateral Properties

Polygon Interior and Exterior Angle Sums

Polygons are closed shapes made from straight line segments. Examples include triangles, quadrilaterals, pentagons, hexagons, and many more.

In this lesson, you will learn two very important facts about polygons:

• How to find the sum of the interior angles of any polygon.
• Why the sum of one exterior angle at each vertex is always \(360^\circ\).

These ideas help you solve many geometry problems, especially when some angles are missing.

1. Interior angles of a polygon

An interior angle is an angle inside a polygon, formed by two sides meeting at a vertex.

For example:

• A triangle has 3 interior angles.
• A quadrilateral has 4 interior angles.
• A pentagon has 5 interior angles.

2. Finding the sum of interior angles

There is a general formula for the sum of the interior angles of an \(n\)-sided polygon:

Here, \(n\) is the number of sides.

Why does this formula work?

You can divide a polygon into triangles by drawing diagonals from one vertex to all other non-adjacent vertices.

This creates exactly \(n-2\) triangles.

Since each triangle has an angle sum of \(180^\circ\), the total interior angle sum is:

Examples of the formula

• Triangle: \((3-2)\times 180 = 180^\circ\)
• Quadrilateral: \((4-2)\times 180 = 360^\circ\)
• Pentagon: \((5-2)\times 180 = 540^\circ\)
• Hexagon: \((6-2)\times 180 = 720^\circ\)

3. Exterior angles of a polygon

An exterior angle is formed when one side of a polygon is extended. It is outside the polygon.

At each vertex, the interior angle and its exterior angle form a straight line, so they add up to \(180^\circ\).

4. Sum of exterior angles

If you take one exterior angle at each vertex of any polygon, always going around the shape in the same direction, the sum is always:

Why is the exterior angle sum always \(360^\circ\)?

Imagine walking around the outside of a polygon. At each corner, you turn by the exterior angle. After going all the way around and returning to where you started, you have made one full turn.

One full turn is:

So the sum of the exterior angles of any polygon is always \(360^\circ\).

5. Connecting interior and exterior angle sums

Suppose a polygon has \(n\) sides.

There are \(n\) interior angles and \(n\) exterior angles, and each interior-exterior pair adds to \(180^\circ\). So:

Since the exterior sum is \(360^\circ\), we can write:

Now subtract \(360^\circ\):

This matches the formula from dividing the polygon into triangles.

6. Regular polygons

A regular polygon has all sides equal and all angles equal.

In a regular polygon:

• All interior angles are the same.
• All exterior angles are the same.

Since the exterior angles always add to \(360^\circ\), each exterior angle in a regular \(n\)-gon is:

Then each interior angle is:

You can also find each interior angle of a regular polygon by dividing the interior sum by the number of angles:

These two methods give the same answer.

Worked Example 1: Find the sum of the interior angles of an octagon

An octagon has \(8\) sides, so \(n=8\).

Answer: The sum of the interior angles is \(1080^\circ\).

Worked Example 2: Find one exterior angle of a regular nonagon

A nonagon has \(9\) sides.

For a regular polygon, each exterior angle is:

Answer: Each exterior angle is \(40^\circ\).

Worked Example 3: Find one interior angle of a regular hexagon

A regular hexagon has \(6\) equal exterior angles.

First find one exterior angle:

Now use the fact that an interior angle and exterior angle form a straight line:

Answer: Each interior angle is \(120^\circ\).

Worked Example 4: One interior angle of a regular polygon is \(150^\circ\). How many sides does the polygon have?

First find the exterior angle:

Now use the exterior angle formula for a regular polygon:

Solve for \(n\):

Answer: The polygon has \(12\) sides.

7. Common mistakes to avoid

• Mixing up interior and exterior angles: interior angles are inside the polygon; exterior angles are outside.
• Using \((n-2)\times 180\) for one angle: this formula gives the sum of all interior angles, not just one angle.
• Forgetting that exterior angles sum to \(360^\circ\): this is true for any polygon if you choose one exterior angle at each vertex consistently.
• Assuming all polygons are regular: you can only divide by \(n\) to get one angle when the polygon is regular.

8. Quick problem-solving steps

If you need the sum of interior angles:

1. Count the number of sides, \(n\).
2. Use $$ (n-2)\times 180^\circ $$

If you need one exterior angle of a regular polygon:

1. Count the sides, \(n\).
2. Use $$ \frac{360^\circ}{n} $$

If you need one interior angle of a regular polygon:

1. Find the exterior angle using $$ \frac{360^\circ}{n} $$
2. Subtract from \(180^\circ\)

9. Lesson summary

The sum of the interior angles of an \(n\)-sided polygon is:

This works because any polygon can be split into \(n-2\) triangles.

The sum of one exterior angle at each vertex of any polygon is always:

For a regular polygon, each exterior angle is \(\frac{360^\circ}{n}\), and each interior angle is \(180^\circ - \frac{360^\circ}{n}\).

Once you understand these rules, you can solve many angle problems involving polygons quickly and confidently.

Parallelogram Properties

A parallelogram is a quadrilateral, which means it is a shape with 4 sides. What makes a parallelogram special is that both pairs of opposite sides are parallel.

Parallelograms are important because many other shapes are connected to them. For example, rectangles, rhombuses, and squares are all special types of parallelograms.

In this lesson, you will learn the main properties of parallelograms and how to use them to find missing side lengths and angle measures.

1. Definition of a Parallelogram

If a quadrilateral has two pairs of opposite sides that are parallel, then it is a parallelogram.

If a parallelogram is named ABCD, then:

• \(AB \parallel CD\)

• \(BC \parallel AD\)

This parallel structure creates several useful properties.

2. Main Properties of Parallelograms

A parallelogram has four key properties you should know.

• Opposite sides are equal in length.

  If \(ABCD\) is a parallelogram, then:

  \(AB = CD\) and \(BC = AD\)

• Opposite angles are equal.

  That means:

  \(\angle A = \angle C\) and \(\angle B = \angle D\)

• Consecutive angles are supplementary.

  Consecutive angles are next to each other. Supplementary means they add to \(180^\circ\).

  So:

  $$\angle A + \angle B = 180^\circ$$

  $$\angle B + \angle C = 180^\circ$$

  $$\angle C + \angle D = 180^\circ$$

  $$\angle D + \angle A = 180^\circ$$

• Diagonals bisect each other.

  A diagonal is a segment connecting opposite corners. In a parallelogram, the diagonals cut each other into two equal parts.

  If diagonals \(AC\) and \(BD\) intersect at point \(M\), then:

  \(AM = MC\) and \(BM = MD\)

3. Visualizing the Properties

Imagine a parallelogram \(ABCD\):

• Sides \(AB\) and \(CD\) face each other and are equal.

• Sides \(BC\) and \(AD\) face each other and are equal.

• Angles across from each other match.

• Neighboring angles form a straight line together, so they add to \(180^\circ\).

• The diagonals cross in the middle, and each diagonal is split into two equal pieces.

4. Why These Properties Matter

These properties help you solve geometry problems. If you know one side, you may be able to find another side. If you know one angle, you can often find all the others. If you know part of a diagonal, you can find the rest.

Instead of memorizing separate facts, it helps to see the pattern:

• Across from each other: equal

• Next to each other: add to \(180^\circ\)

• Diagonals: split each other into equal halves

5. Worked Examples

Example 1: Finding a Missing Side

In parallelogram \(PQRS\), side \(PQ = 12\) cm. Find side \(RS\).

Step 1: Use the property that opposite sides of a parallelogram are equal.

Since \(PQ\) and \(RS\) are opposite sides,

$$RS = PQ$$

$$RS = 12$$

Answer: \(RS = 12\) cm

Example 2: Finding a Missing Angle

In parallelogram \(ABCD\), \(\angle A = 70^\circ\). Find \(\angle B\), \(\angle C\), and \(\angle D\).

Step 1: Opposite angles are equal.

So,

$$\angle C = \angle A = 70^\circ$$

Step 2: Consecutive angles are supplementary.

\(\angle A\) and \(\angle B\) are next to each other, so:

$$\angle A + \angle B = 180^\circ$$

$$70^\circ + \angle B = 180^\circ$$

$$\angle B = 110^\circ$$

Step 3: Opposite angles are equal again.

$$\angle D = \angle B = 110^\circ$$

Answer:

• \(\angle B = 110^\circ\)

• \(\angle C = 70^\circ\)

• \(\angle D = 110^\circ\)

Example 3: Using Algebra with Side Lengths

In parallelogram \(JKLM\), \(JK = 3x + 5\) and \(LM = 17\). Find \(x\).

Step 1: Opposite sides of a parallelogram are equal.

Since \(JK\) and \(LM\) are opposite sides,

$$3x + 5 = 17$$

Step 2: Solve the equation.

$$3x = 12$$

$$x = 4$$

Answer: \(x = 4\)

Example 4: Diagonals Bisect Each Other

In parallelogram \(WXYZ\), diagonals \(WY\) and \(XZ\) intersect at point \(M\). If \(WM = 9\), find \(MY\).

Step 1: Use the property that diagonals bisect each other.

This means point \(M\) is the midpoint of diagonal \(WY\).

So:

$$WM = MY$$

Step 2: Substitute the known value.

$$MY = 9$$

Answer: \(MY = 9\)

6. Common Mistakes to Avoid

• Do not assume all angles are equal. In a general parallelogram, only opposite angles are equal. Adjacent angles are usually different.

• Do not forget that consecutive angles add to \(180^\circ\), not \(360^\circ\).

• Do not confuse diagonals bisecting each other with diagonals being equal. In a general parallelogram, the diagonals are not always the same length.

• Match opposite parts correctly. Be careful when reading the order of the vertices.

7. Quick Check

Try these on your own:

1. In a parallelogram, one side is \(15\). What is the length of the opposite side?

2. If one angle is \(125^\circ\), what is the measure of an adjacent angle?

3. If one diagonal is split into segments of lengths \(x + 2\) and \(10\), what equation can you write?

Answers:

1. \(15\)

2. \(55^\circ\), because \(125^\circ + 55^\circ = 180^\circ\)

3. \(x + 2 = 10\)

8. Summary

A parallelogram is a quadrilateral with two pairs of parallel opposite sides. Its opposite sides are equal, opposite angles are equal, consecutive angles add to \(180^\circ\), and its diagonals bisect each other.

When solving problems, look for these patterns first. They allow you to find missing sides, missing angles, and unknown parts of diagonals quickly and accurately.

Rectangle, Rhombus, and Square Properties

In this lesson, you will learn the important properties of rectangles, rhombuses, and squares. These shapes are all special kinds of quadrilaterals, which means they each have 4 sides.

Understanding their properties helps you identify shapes, compare them, and solve geometry problems. A very important idea is that these shapes are also related to parallelograms, so they share some common features.

1. Review: What is a parallelogram?

A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Rectangles, rhombuses, and squares are all special parallelograms.

Because they are parallelograms, they all have these properties:

• Opposite sides are parallel.
• Opposite sides are equal in length.
• Opposite angles are equal.
• Consecutive angles add to \\(180^\circ\\).
• Diagonals bisect each other.

This means if the diagonals of a parallelogram cross at point \(M\), then each diagonal is cut into two equal parts at \(M\).

2. Rectangle properties

A rectangle is a parallelogram with 4 right angles. A right angle measures \\(90^\circ\\).

So a rectangle has all the properties of a parallelogram, plus these special properties:

• All 4 angles are \\(90^\circ\\).
• Opposite sides are equal.
• Its diagonals are equal in length.

If rectangle \(ABCD\) has diagonals \(AC\) and \(BD\), then:

However, the diagonals of a rectangle do not always cross at right angles, and they do not always cut the angles in half.

3. Rhombus properties

A rhombus is a parallelogram with all 4 sides equal.

So a rhombus has all the properties of a parallelogram, plus these special properties:

• All 4 sides are equal.
• Its diagonals bisect each other at right angles.
• Each diagonal bisects a pair of opposite angles.

If rhombus \(PQRS\) has diagonals \(PR\) and \(QS\), then:

• \(PR \perp QS\)
• Each diagonal cuts the other into 2 equal parts.

A rhombus does not always have right angles. That is an important difference between a rhombus and a rectangle.

4. Square properties

A square is a parallelogram that is both a rectangle and a rhombus.

This means a square has all the properties of both shapes.

• All 4 sides are equal.
• All 4 angles are \\(90^\circ\\).
• Opposite sides are parallel.
• Diagonals bisect each other.
• Diagonals are equal in length.
• Diagonals are perpendicular.
• Diagonals bisect the angles.

So if a quadrilateral has all equal sides and all right angles, it is a square.

5. Comparing rectangle, rhombus, and square

These three shapes overlap in important ways.

• A rectangle has right angles.
• A rhombus has all sides equal.
• A square has both right angles and all sides equal.

You can think of a square as the most specific of the three.

• Every square is a rectangle.
• Every square is a rhombus.
• Not every rectangle is a square.
• Not every rhombus is a square.

6. Diagonals: a key way to identify the shape

The diagonals give strong clues about the type of quadrilateral.

• In a rectangle, diagonals are equal and bisect each other.
• In a rhombus, diagonals are perpendicular, bisect each other, and bisect angles.
• In a square, diagonals do all of these things.

This is helpful when a problem gives information about diagonals instead of sides or angles.

7. Worked Examples

Example 1: Identifying a rectangle

A quadrilateral has opposite sides parallel, and all four angles are \\(90^\circ\\). What shape is it?

Step 1: Opposite sides parallel tells us it is a parallelogram.

Step 2: All four angles are right angles.

A parallelogram with four right angles is a rectangle.

Answer: The shape is a rectangle.

Example 2: Identifying a rhombus

A parallelogram has side lengths \\(6\\), \\(6\\), \\(6\\), and \\(6\\). What special quadrilateral is it?

Step 1: All four sides are equal.

Step 2: A parallelogram with all sides equal is a rhombus.

Answer: The shape is a rhombus.

Example 3: Is it a square?

A quadrilateral has all sides equal and one angle measures \\(90^\circ\\). Is it a square?

Step 1: All sides equal suggests a rhombus.

Step 2: One angle is \\(90^\circ\\).

In a parallelogram, if one angle is \\(90^\circ\\), then all angles are \\(90^\circ\\).

So the shape has all sides equal and all right angles.

Answer: Yes, it is a square.

Example 4: Using diagonals

A parallelogram has diagonals that are equal in length. What special shape could it be?

Step 1: In a general parallelogram, diagonals bisect each other, but they are not always equal.

Step 2: Equal diagonals are a special property of a rectangle.

Step 3: A square also has equal diagonals, since every square is a rectangle.

Answer: The shape could be a rectangle or a square.

8. Common mistakes to avoid

• Do not assume a rhombus has right angles. Only a square always has right angles.
• Do not assume a rectangle has all sides equal. Only a square always has all sides equal.
• Remember: equal diagonals suggest a rectangle, while perpendicular diagonals suggest a rhombus.
• A square fits into both categories: rectangle and rhombus.

9. Quick property chart

• Rectangle: opposite sides equal, opposite sides parallel, 4 right angles, diagonals equal, diagonals bisect each other
• Rhombus: all sides equal, opposite sides parallel, opposite angles equal, diagonals perpendicular, diagonals bisect each other, diagonals bisect angles
• Square: all sides equal, 4 right angles, opposite sides parallel, diagonals equal, diagonals perpendicular, diagonals bisect each other, diagonals bisect angles

10. Summary

Rectangles, rhombuses, and squares are all special parallelograms. A rectangle is defined by its right angles, a rhombus is defined by its equal sides, and a square has both of these features.

To identify the shape, look carefully at its sides, angles, and diagonals. The more properties a quadrilateral has, the more specifically it can be classified.

Trapezoids and kites are two important types of quadrilaterals. A quadrilateral is any polygon with 4 sides. In this lesson, you will learn how to recognize trapezoids and kites, understand their special properties, and use those properties to solve problems.

These shapes are not usually parallelograms, so their properties are different from rectangles, rhombuses, and squares. Knowing what makes each shape special helps you decide which rules to use.

Goal of this lesson: learn the side, angle, and diagonal properties of trapezoids and kites, especially isosceles trapezoids and the diagonal properties of kites.

1. Trapezoid basics

A trapezoid is a quadrilateral with one pair of parallel sides. The parallel sides are called the bases. The non-parallel sides are called the legs.

If a trapezoid is named so that the parallel sides are \\(\overline{AB}\\) and \\(\overline{CD}\\), then:

$$\overline{AB} \parallel \overline{CD}$$

The other sides, \\(\overline{AD}\\) and \\(\overline{BC}\\), are the legs.

• Bases: parallel sides
• Legs: non-parallel sides
• Base angles: angles next to the same base

Because the bases are parallel, some angle relationships come from parallel lines cut by a transversal. This means angles along the same leg are supplementary, or add to \\(180^\circ\\).

For example, if \\(\overline{AB} \parallel \overline{CD}\\), then:

$$m\angle A + m\angle D = 180^\circ$$

$$m\angle B + m\angle C = 180^\circ$$

2. Isosceles trapezoid

An isosceles trapezoid is a trapezoid whose legs are congruent.

So if \\(ABCD\\) is an isosceles trapezoid with bases \\(\overline{AB}\\) and \\(\overline{CD}\\), then:

$$\overline{AD} \cong \overline{BC}$$

Isosceles trapezoids have extra properties that ordinary trapezoids do not always have.

• The legs are congruent.
• The base angles on each base are congruent.
• The diagonals are congruent.

That means:

$$\angle A \cong \angle B$$

$$\angle D \cong \angle C$$

$$\overline{AC} \cong \overline{BD}$$

This diagonal property is very important. In an isosceles trapezoid, the diagonals have equal length. In a general trapezoid, this is not always true.

3. Angle properties in trapezoids

Even if a trapezoid is not isosceles, the consecutive angles on a leg are supplementary because the bases are parallel.

If \\(\overline{AB} \parallel \overline{CD}\\), then:

$$m\angle A + m\angle D = 180^\circ$$

$$m\angle B + m\angle C = 180^\circ$$

In an isosceles trapezoid, you can combine this with congruent base angles. This makes it easier to find missing angle measures.

For example, if \\(m\angle A = 70^\circ\\), then in an isosceles trapezoid:

• \\(m\angle B = 70^\circ\\)
• \\(m\angle D = 110^\circ\\)
• \\(m\angle C = 110^\circ\\)

4. Kite basics

A kite is a quadrilateral with two pairs of adjacent congruent sides. Adjacent sides are sides next to each other.

For example, if a kite is named \\(ABCD\\), one possible side relationship is:

$$\overline{AB} \cong \overline{AD}$$

$$\overline{BC} \cong \overline{CD}$$

This means one pair of equal sides meets at one vertex, and another pair of equal sides meets at another vertex.

A kite does not have both pairs of opposite sides parallel, so it is usually not a parallelogram.

5. Diagonal properties of a kite

The diagonals of a kite have a very special relationship. They are perpendicular, which means they intersect at a right angle.

If diagonals \\(\overline{AC}\\) and \\(\overline{BD}\\) intersect at point \\(E\\), then:

$$\overline{AC} \perp \overline{BD}$$

So the angle formed where they cross is \\(90^\circ\\).

Another important property is that one diagonal bisects the other diagonal. In a kite, the diagonal that connects the vertices where the congruent side pairs meet is the one that bisects the other diagonal.

If \\(\overline{AC}\\) is that diagonal, then:

$$BE = ED$$

But in general, you should not assume both diagonals bisect each other. That is a property of parallelograms, not kites.

Also, one diagonal of a kite bisects a pair of opposite angles. So a diagonal can split one vertex angle into two equal parts.

6. Comparing trapezoids and kites

• A trapezoid is identified by having one pair of parallel sides.
• An isosceles trapezoid has congruent legs, congruent base angles, and congruent diagonals.
• A kite is identified by having two pairs of adjacent congruent sides.
• A kite's diagonals are perpendicular.
• In a kite, one diagonal bisects the other.

This means that if a problem mentions congruent diagonals, you should think about an isosceles trapezoid. If it mentions perpendicular diagonals, you should think about a kite.

Worked Example 1: Find missing angles in an isosceles trapezoid

Suppose \\(ABCD\\) is an isosceles trapezoid with \\(\overline{AB} \parallel \overline{CD}\\), and \\(m\angle A = 68^\circ\\). Find the other three angles.

Step 1: Use the isosceles trapezoid base-angle property.

Angles on the same base are congruent, so:

$$m\angle B = 68^\circ$$

Step 2: Use supplementary angles along a leg.

$$m\angle A + m\angle D = 180^\circ$$

$$68^\circ + m\angle D = 180^\circ$$

$$m\angle D = 112^\circ$$

Step 3: Use the other pair of base angles.

Since it is isosceles, \\(\angle C \cong \angle D\\), so:

$$m\angle C = 112^\circ$$

Answer:

$$m\angle B = 68^\circ, \quad m\angle C = 112^\circ, \quad m\angle D = 112^\circ$$

Worked Example 2: Use diagonal property of an isosceles trapezoid

In isosceles trapezoid \\(PQRS\\), the diagonals are \\(\overline{PR}\\) and \\(\overline{QS}\\). If \\(PR = 15\\), find \\(QS\\).

Step 1: Recall the property.

In an isosceles trapezoid, the diagonals are congruent.

$$\overline{PR} \cong \overline{QS}$$

Step 2: Set the lengths equal.

$$QS = PR = 15$$

Answer: \\(QS = 15\\)

Worked Example 3: Use kite diagonal properties

Kite \\(ABCD\\) has diagonals \\(\overline{AC}\\) and \\(\overline{BD}\\) intersecting at \\(E\\). If \\(BE = 6\\) and \\(ED = 6\\), what can you conclude about diagonal \\(\overline{AC}\\)?

Step 1: Notice that \\(BE = ED\\).

This means point \\(E\\) is the midpoint of \\(\overline{BD}\\).

Step 2: Use the kite property.

In a kite, one diagonal bisects the other. So \\(\overline{AC}\\) bisects \\(\overline{BD}\\).

Step 3: State the conclusion.

Diagonal \\(\overline{AC}\\) cuts \\(\overline{BD}\\) into two equal parts, and the diagonals are perpendicular.

So:

$$\overline{AC} \perp \overline{BD}$$

and

$$BE = ED$$

Answer: \\(\overline{AC}\\) is the diagonal that bisects \\(\overline{BD}\\), and it is perpendicular to it.

Worked Example 4: Find a missing length in a kite

In kite \\(JKLM\\), diagonal \\(\overline{JL}\\) bisects diagonal \\(\overline{KM}\\) at point \\(N\\). If \\(KM = 18\\), find \\(KN\\) and \\(NM\\).

Step 1: Use the bisect property.

If \\(\overline{JL}\\) bisects \\(\overline{KM}\\), then it cuts it into two equal parts.

$$KN = NM$$

Step 2: Split the total length in half.

$$KN = NM = \frac{18}{2} = 9$$

Answer:

$$KN = 9 \quad \text{and} \quad NM = 9$$

Common mistakes to avoid

• Do not assume every trapezoid is isosceles. Only isosceles trapezoids have congruent legs, congruent base angles, and congruent diagonals.
• Do not assume both diagonals in a kite bisect each other. Only one diagonal bisects the other.
• Do not confuse congruent diagonals with perpendicular diagonals. Isosceles trapezoids have congruent diagonals, while kites have perpendicular diagonals.
• Remember the definition carefully. A trapezoid is about parallel sides; a kite is about adjacent congruent sides.

Quick check

1. A trapezoid has bases \\(\overline{WX}\\) and \\(\overline{YZ}\\). What do you know about those sides?
   Answer: \\(\overline{WX} \parallel \overline{YZ}\\)
2. In an isosceles trapezoid, if one diagonal is 12, what is the other diagonal?
   Answer: 12
3. In a kite, how do the diagonals intersect?
   Answer: Perpendicularly, at \\(90^\circ\\)
4. If one diagonal of a kite bisects the other diagonal of length 20, what are the two pieces?
   Answer: 10 and 10

Summary

A trapezoid has one pair of parallel sides called bases. In any trapezoid, angles along the same leg are supplementary. In an isosceles trapezoid, the legs are congruent, the base angles are congruent, and the diagonals are congruent.

A kite has two pairs of adjacent congruent sides. Its diagonals are perpendicular, and one diagonal bisects the other. When solving problems, look closely at whether the important clue is parallel sides, congruent diagonals, or perpendicular diagonals.

Coordinate Proofs for Quadrilaterals

In geometry, a coordinate proof uses points on the coordinate plane to prove that a figure has certain properties. For quadrilaterals, this means using coordinates, slopes, and distances to show whether a figure is a parallelogram, rectangle, rhombus, square, or another type of four-sided shape.

This is powerful because instead of just looking at a graph and guessing, you can prove what the shape is with math.

In this lesson, you will learn how to use:

• Slope to test if sides are parallel or perpendicular
• Distance to test if sides are congruent
• Midpoint to test if diagonals bisect each other

These tools help you classify quadrilaterals accurately.

1. Important quadrilateral properties

Before doing coordinate proofs, it helps to remember the key properties of common quadrilaterals.

• Parallelogram: both pairs of opposite sides are parallel
• Rectangle: a parallelogram with four right angles
• Rhombus: a parallelogram with four congruent sides
• Square: a rectangle and a rhombus, so it has four right angles and four congruent sides
• Kite: two pairs of adjacent congruent sides
• Trapezoid: one pair of parallel sides

In coordinate proofs, you do not just name the shape. You must show why it fits that category.

2. The formulas you need

Slope formula

For points \((x_1,y_1)\) and \((x_2,y_2)\), the slope is

$$m=\frac{y_2-y_1}{x_2-x_1}$$

You use slope to decide:

• Parallel lines have equal slopes.
• Perpendicular lines have slopes that are negative reciprocals.

For example, slopes of \(2\) and \(-\frac{1}{2}\) are perpendicular.

Distance formula

The distance between \((x_1,y_1)\) and \((x_2,y_2)\) is

$$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$$

You use distance to check whether sides or diagonals are congruent.

Midpoint formula

The midpoint of a segment with endpoints \((x_1,y_1)\) and \((x_2,y_2)\) is

$$\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)$$

You use midpoint to check whether diagonals bisect each other, which is an important property of parallelograms.

3. How to classify a quadrilateral on the coordinate plane

When given four points, follow a clear process.

1. Plot the points or list the sides in order.
2. Find slopes of the sides to test for parallel or perpendicular lines.
3. Find distances of sides if you need to test for congruent sides.
4. Find midpoints of diagonals if you need to test whether they bisect each other.
5. Use the results to classify the quadrilateral.

4. What each test can prove

• If both pairs of opposite sides are parallel, the figure is a parallelogram.
• If it is a parallelogram and one angle is a right angle, the figure is a rectangle.
• If it is a parallelogram and all sides are congruent, the figure is a rhombus.
• If it has both right angles and all sides congruent, the figure is a square.
• If exactly one pair of sides is parallel, it is a trapezoid.

Be careful: one fact alone is sometimes not enough. For example, having congruent sides does not automatically make a figure a square. You also need right angles.

Worked Example 1: Proving a parallelogram

Classify the quadrilateral with vertices \(A(1,1)\), \(B(5,1)\), \(C(7,4)\), and \(D(3,4)\).

Step 1: Find the slopes of the sides

Slope of \(\overline{AB}\):

$$m_{AB}=\frac{1-1}{5-1}=\frac{0}{4}=0$$

Slope of \(\overline{CD}\):

$$m_{CD}=\frac{4-4}{3-7}=\frac{0}{-4}=0$$

So \(\overline{AB}\parallel\overline{CD}\).

Slope of \(\overline{BC}\):

$$m_{BC}=\frac{4-1}{7-5}=\frac{3}{2}$$

Slope of \(\overline{DA}\):

$$m_{DA}=\frac{1-4}{1-3}=\frac{-3}{-2}=\frac{3}{2}$$

So \(\overline{BC}\parallel\overline{DA}\).

Step 2: Use the slopes

Both pairs of opposite sides are parallel, so the quadrilateral is a parallelogram.

Conclusion: \(ABCD\) is a parallelogram.

Worked Example 2: Proving a rectangle

Classify the quadrilateral with vertices \(A(0,0)\), \(B(4,0)\), \(C(4,3)\), and \(D(0,3)\).

Step 1: Check opposite sides for parallel lines

$$m_{AB}=\frac{0-0}{4-0}=0$$

$$m_{CD}=\frac{3-3}{0-4}=0$$

So \(\overline{AB}\parallel\overline{CD}\).

$$m_{BC}=\frac{3-0}{4-4}=\text{undefined}$$

$$m_{DA}=\frac{0-3}{0-0}=\text{undefined}$$

So \(\overline{BC}\parallel\overline{DA}\).

This shows the figure is a parallelogram.

Step 2: Check for a right angle

\(\overline{AB}\) has slope \(0\), so it is horizontal.

\(\overline{BC}\) has undefined slope, so it is vertical.

A horizontal line and a vertical line are perpendicular, so angle \(B\) is a right angle.

A parallelogram with one right angle is a rectangle.

Conclusion: \(ABCD\) is a rectangle.

Worked Example 3: Proving a rhombus

Classify the quadrilateral with vertices \(A(0,2)\), \(B(2,4)\), \(C(4,2)\), and \(D(2,0)\).

Step 1: Check whether opposite sides are parallel

$$m_{AB}=\frac{4-2}{2-0}=1$$

$$m_{CD}=\frac{0-2}{2-4}=\frac{-2}{-2}=1$$

So \(\overline{AB}\parallel\overline{CD}\).

$$m_{BC}=\frac{2-4}{4-2}=\frac{-2}{2}=-1$$

$$m_{DA}=\frac{2-0}{0-2}=\frac{2}{-2}=-1$$

So \(\overline{BC}\parallel\overline{DA}\).

The figure is a parallelogram.

Step 2: Check whether all sides are congruent

$$AB=\sqrt{(2-0)^2+(4-2)^2}=\sqrt{4+4}=\sqrt{8}$$

$$BC=\sqrt{(4-2)^2+(2-4)^2}=\sqrt{4+4}=\sqrt{8}$$

$$CD=\sqrt{(2-4)^2+(0-2)^2}=\sqrt{4+4}=\sqrt{8}$$

$$DA=\sqrt{(0-2)^2+(2-0)^2}=\sqrt{4+4}=\sqrt{8}$$

All four sides are congruent.

A parallelogram with four congruent sides is a rhombus.

Conclusion: \(ABCD\) is a rhombus.

Worked Example 4: Proving a square

Classify the quadrilateral with vertices \(A(1,1)\), \(B(4,1)\), \(C(4,4)\), and \(D(1,4)\).

Step 1: Check for parallel sides

$$m_{AB}=\frac{1-1}{4-1}=0$$

$$m_{CD}=\frac{4-4}{1-4}=0$$

So \(\overline{AB}\parallel\overline{CD}\).

$$m_{BC}=\frac{4-1}{4-4}=\text{undefined}$$

$$m_{DA}=\frac{1-4}{1-1}=\text{undefined}$$

So \(\overline{BC}\parallel\overline{DA}\).

The figure is a parallelogram.

Step 2: Check for a right angle

\(\overline{AB}\) is horizontal and \(\overline{BC}\) is vertical, so they are perpendicular.

So the figure is a rectangle.

Step 3: Check side lengths

$$AB=\sqrt{(4-1)^2+(1-1)^2}=\sqrt{9}=3$$

$$BC=\sqrt{(4-4)^2+(4-1)^2}=\sqrt{9}=3$$

$$CD=\sqrt{(1-4)^2+(4-4)^2}=\sqrt{9}=3$$

$$DA=\sqrt{(1-1)^2+(1-4)^2}=\sqrt{9}=3$$

All four sides are congruent.

Since the figure is both a rectangle and a rhombus, it is a square.

Conclusion: \(ABCD\) is a square.

5. Using diagonals in coordinate proofs

Sometimes a problem asks you to prove a quadrilateral is a parallelogram by using its diagonals.

A quadrilateral is a parallelogram if its diagonals bisect each other. That means the diagonals have the same midpoint.

Suppose the vertices are \(A(0,0)\), \(B(6,0)\), \(C(8,4)\), and \(D(2,4)\).

The diagonals are \(\overline{AC}\) and \(\overline{BD}\).

Midpoint of \(\overline{AC}\):

$$\left(\frac{0+8}{2},\frac{0+4}{2}\right)=(4,2)$$

Midpoint of \(\overline{BD}\):

$$\left(\frac{6+2}{2},\frac{0+4}{2}\right)=(4,2)$$

Since the diagonals have the same midpoint, they bisect each other.

Conclusion: The quadrilateral is a parallelogram.

6. Tips for writing a good coordinate proof

• Show your calculations clearly. Do not skip slopes, distances, or midpoints.
• Name the property you proved. For example, say “opposite sides are parallel.”
• Connect the property to the shape. For example, “Since both pairs of opposite sides are parallel, the quadrilateral is a parallelogram.”
• Work in order. First identify side relationships, then check angles or side lengths if needed.
• Be careful with undefined slope. Undefined slope means the line is vertical.

7. Common mistakes to avoid

• Mixing up the order of subtraction in the slope formula
• Forgetting that equal slopes mean parallel lines
• Forgetting that negative reciprocal slopes mean perpendicular lines
• Assuming a figure is a square just because it looks like one
• Not checking enough properties to fully classify the quadrilateral

8. Quick strategy guide

If you want to prove a figure is a:

• Parallelogram: show both pairs of opposite sides are parallel, or show the diagonals bisect each other
• Rectangle: prove it is a parallelogram and has a right angle
• Rhombus: prove it is a parallelogram and all sides are congruent
• Square: prove it has four congruent sides and a right angle, or prove it is both a rectangle and a rhombus
• Trapezoid: show exactly one pair of opposite sides is parallel

Summary

Coordinate proofs for quadrilaterals use the coordinate plane to prove what kind of four-sided figure you have. The main tools are slope, distance, and midpoint. Slope helps you test for parallel and perpendicular sides, distance helps you test for congruent sides, and midpoint helps you test whether diagonals bisect each other.

By combining these facts, you can classify a quadrilateral as a parallelogram, rectangle, rhombus, square, trapezoid, or another type of figure. The key is to show each step clearly and connect your calculations to the properties of the shape.

Area of Regular Polygons

In this lesson, you will learn how to find the area of a regular polygon. A polygon is a closed shape made of straight sides, such as a triangle, square, pentagon, or hexagon.

A regular polygon is a polygon in which all sides are equal and all angles are equal. Examples include a square, a regular pentagon, and a regular hexagon.

The area of a regular polygon can be found using a special segment called the apothem and the polygon's perimeter.

Key Formula

The formula for the area of a regular polygon is:

where:

• A is the area,
• a is the apothem,
• p is the perimeter.

What is the apothem?

The apothem is the distance from the center of the regular polygon to the midpoint of one of its sides. It is drawn straight out from the center and is perpendicular to the side.

You can think of the apothem as a kind of "height" for the small triangles inside the polygon.

Why does this formula work?

A regular polygon can be divided into several congruent triangles by drawing segments from the center to each vertex.

Each triangle has:

• a base equal to one side of the polygon,
• a height equal to the apothem.

The area of one triangle is:

If you add the areas of all the triangles together, the total of all the side lengths is the perimeter. That gives the full formula:

Steps for finding the area of a regular polygon

1. Find the perimeter by multiplying the number of sides by the side length.
2. Find or use the given apothem.
3. Substitute into the formula \(A = \frac{1}{2}ap\).
4. Simplify and include square units.

Example 1: Regular Pentagon

A regular pentagon has side length \(8\text{ cm}\) and apothem \(5.5\text{ cm}\). Find its area.

Step 1: Find the perimeter

A pentagon has 5 sides, so:

Step 2: Use the area formula

Step 3: Simplify

So, the area is:

Example 2: Regular Hexagon

A regular hexagon has side length \(12\text{ m}\) and apothem \(10.4\text{ m}\). Find its area.

Step 1: Find the perimeter

A hexagon has 6 sides, so:

Step 2: Use the formula

Step 3: Simplify

So, the area is:

Example 3: Finding Area When You First Need the Perimeter

A regular octagon has side length \(7\text{ in}\) and apothem \(8.45\text{ in}\). Find its area.

Step 1: Find the perimeter

An octagon has 8 sides, so:

Step 2: Substitute into the formula

Step 3: Simplify

So, the area is:

Example 4: Finding a Missing Apothem

The area of a regular polygon is \(270\text{ cm}^2\). Its perimeter is \(36\text{ cm}\). Find the apothem.

Start with the formula:

Substitute the known values:

Simplify \(\frac{1}{2} \times 36\):

Now divide both sides by 18:

So, the apothem is:

Important things to remember

• The formula \(A = \frac{1}{2}ap\) works only for regular polygons.
• The perimeter is the total distance around the polygon.
• The apothem must go from the center to the midpoint of a side.
• Area is always written in square units, such as \(\text{cm}^2\), \(\text{m}^2\), or \(\text{in}^2\).

Common mistakes to avoid

• Do not confuse the side length with the perimeter.
• Do not forget to multiply the side length by the number of sides to get the perimeter.
• Do not leave off the \(\frac{1}{2}\) in the formula.
• Do not use this formula for irregular polygons.

Quick Check

• A regular nonagon has side length \(4\text{ cm}\) and apothem \(5.5\text{ cm}\). What is its perimeter?
• Then use \(A = \frac{1}{2}ap\) to find its area.

Solution to Quick Check

A nonagon has 9 sides, so:

Now find the area:

So, the area is:

Summary

To find the area of a regular polygon, use the formula \(A = \frac{1}{2}ap\), where \(a\) is the apothem and \(p\) is the perimeter. This formula works because a regular polygon can be split into equal triangles, and the apothem acts like the height of each triangle.

Always make sure the polygon is regular, find the perimeter correctly, and write your final answer in square units.