Math resources Geometry

Angles in polygons

Quadrilateral angles

# Quadrilateral angles

Here you will learn about quadrilateral angles, including the sum of quadrilateral angles, how to find missing angles, and how to use these angle facts to generate equations and solve problems.

Students will first learn about quadrilateral angles as part of geometry in high school.

## What are quadrilateral angles?

Quadrilateral angles are the four angles that occur at each vertex within a four-sided shape; these angles are called interior angles of a quadrilateral.

The sum of the interior angles of a quadrilateral is 360^{\circ}. You can prove this using the angle sum of a triangle.

The rectangle above is split into two triangles by joining two vertices together across the diagonal. As the sum of angles in a triangle is 180^{\circ}, you can add two lots of 180^{\circ} together, making the angle sum of a quadrilateral equal to 360^{\circ}.

This is the same for all types of quadrilaterals:

Step-by-step guide: Angles of a triangle

### Angle properties of quadrilaterals

The four angles in any quadrilateral always add to 360^{\circ}, but there are a few key properties of quadrilaterals that can help us calculate other angles.

## Common Core State Standards

How does this relate to high school math?

• High School – Geometry – Congruence (H.G.CO.C.11)
Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

## How to find missing quadrilateral angles

In order to find missing quadrilateral angles:

1. Use angle properties to determine any interior angles.
2. Add all known interior angles.
3. a: Subtract the angle sum from \bf{360}^{\circ}.
b: Form and solve the equation.

## Quadrilateral angles examples

### Example 1: trapezoid

ABCD is a trapezoid. Calculate the size of angle BCD, labeled x.

1. Use angle properties to determine any interior angles.

Angle fact:

• The line AD is perpendicular to lines AB and CD , so angle BAD=90^{\circ}.

2Add all known interior angles.

90+90+110=290^{\circ}

3a: Subtract the angle sum from \bf{360}^{\circ}.

Here, 360-290=70^{\circ}

So x=70^{\circ}.

### Example 2: irregular quadrilateral

Find the value of the missing angle x.

Use angle properties to determine any interior angles.

Add all known interior angles.

a: Subtract the angle sum from \bf{360}^{\circ}.

### Example 3a: parallelogram with one interior angle (form and solve)

Calculate the missing angle for the following parallelogram.

Use angle properties to determine any interior angles.

Add all known interior angles.

b: Form and solve the equation.

### Example 3b: parallelogram with one interior angle (form and solve)

Calculate the missing angle for the following parallelogram.

Use angle properties to determine any interior angles.

Add all known interior angles.

b: Form and solve the equation.

### Example 4: exterior angle given

Calculate the measure of the missing angle for the following quadrilateral.

Use angle properties to determine any interior angles.

Add all known interior angles.

a: Subtract the angle sum from \bf{360}^{\circ}.

### Example 5: forming and solving equations

By finding the value for x, calculate the value of each angle in the kite drawn below.

Use angle properties to determine any interior angles.

Add all known interior angles.

b: Form and solve the equation.

### Example 6: forming and solving equations

By finding the value for x, calculate the value of each angle in the quadrilateral.

Use angle properties to determine any interior angles.

Add all known interior angles.

b: Form and solve the equation.

### Teaching tips for quadrilateral angles

• Encourage students to classify different types of quadrilaterals (square, rectangle, parallelogram, trapezoid, rhombus) and compare their angle properties.

• Provide worksheets that involve finding unknown angles in quadrilaterals and applying properties of specific quadrilateral types.

• Students should be familiar with different types of quadrilaterals from elementary school ( 3 rd grade, 4 th grade, 5 th grade) but if needed, provide a quick intro to each type including their properties.

### Easy mistakes to make

• Mistaking the sum of quadrilateral angles with the angles in a triangle
The angle sum is remembered incorrectly as 180^{\circ}, rather than 360^{\circ}. The sum of angles in a triangle is equal to 180^{\circ}.

• Joining all the diagonals
When recalling the angle sum in a quadrilateral, students join all the diagonals together, creating 4 triangles. This makes their angle sum 720^{\circ} which is also incorrect.

• Using an incorrect angle fact
A common mistake is to use the incorrect angle fact or make an incorrect assumption to overcome a problem.
For example, here the trapezoid is assumed to be symmetrical (an isosceles trapezoid) so the interior angles are easy to deduce. This is not always true and so you should use co-interior angles instead.

Here, the angle x should be equal to 105^{\circ} and y should be equal to 60^{\circ} due to co-interior angles in parallel lines.

### Practice quadrilateral angles questions

1. ABCD is a rhombus. Given that ADC=84^{\circ}, calculate the value of a.

84^{\circ}

96^{\circ}

6^{\circ}

168^{\circ}

Diagonally opposite angles in a rhombus are equal

Co-interior angles add to equal 180^{\circ}.

180-84=96^{\circ}

2. ABCD is a trapezoid. Use the information below to calculate the value of b.

89^{\circ}

72^{\circ}

199^{\circ}

91^{\circ}

Co-interior angles add to equal 180^{\circ}.

180-89=91^{\circ}

3. ABCD is a parallelogram. Calculate the size of the angle BCD.

100^{\circ}

50^{\circ}

40^{\circ}

200^{\circ}

The sum of co-interior angles is 180^{\circ} so

\begin{aligned}5x+4x&=180 \\\\ 9x&=180 \\\\ x&=20 \end{aligned}

BCD=5x=100^{\circ}

4. ABCD is a quadrilateral. Given that CE is a straight line, calculate the interior angle at D marked x.

155^{\circ}

279^{\circ}

223^{\circ}

137^{\circ}

Angles on a straight line add to equal 180^{\circ}.

180-124=56^{\circ}

Angles of a quadrilateral add up to 360^{\circ}.

\begin{aligned}39+56+42+x&=360 \\\\ x+130&=360 \\\\ x&=223^{\circ} \end{aligned}

5. ABCD is an isosceles trapezoid. Calculate the value of y.

125^{\circ}

55^{\circ}

155^{\circ}

175^{\circ}

ADC=BCD so

\begin{aligned}2x+25&=3x-25 \\\\ 2x+50&=3x \\\\ x&=50^{\circ} \end{aligned}

\begin{aligned}y&=180-(3\times{50-25}) \\\\ &=180-125 \\\\ &=55^{\circ} \end{aligned}

6. ABCD is an irregular quadrilateral where BE is a straight line through C. Calculate the exact size of the angle y.

84^{\circ}

6^{\circ}

168^{\circ}

100^{\circ}

Angles on a straight line add to equal 180^{\circ}.

y=180-(140-2x)=2x+40

x+30+x+5x+20+2x+40=9x+90

Quadrilateral angles add to equal 360^{\circ}.

9x+90=360^{\circ}

\begin{aligned}9x+90-90&=360-90 \\\\ 9x&=270 \\\\ x&=30^{\circ} \end{aligned}

As x=30^{\circ},~y=2x+40=2\times{30+40}=100^{\circ}.

## Quadrilateral angles FAQs

What is a quadrilateral?

A quadrilateral is a closed figure/polygon with four sides and four vertices (corners). The sum of its interior angles is always 360 degrees.

Quadrilaterals can be classified into various types based on their side lengths, angles, and symmetry, including squares, rectangles, parallelograms, rhombuses, trapezoids, and kites. Each type of quadrilateral has its own set of properties and characteristics.

How are quadrilateral angles different from the interior angles of a polygon?

Quadrilateral angles are different from angles of other polygons primarily in their sum.

The sum of the interior angles of a quadrilateral is always 360 degrees, while for other polygons, the sum is calculated using the formula (𝑛−2) \times 180 degrees, where 𝑛 is the number of sides.

For example, a triangle’s angles add up to 180 degrees, and a pentagon’s angles add up to 540 degrees.

Where are the adjacent angles in a quadrilateral?

In a quadrilateral, adjacent angles are pairs of angles that share a common side (adjacent side). Since a quadrilateral has four sides and four angles, each angle has two adjacent angles.

Is a protractor needed for measuring quadrilateral angles?

While some angles in special types of quadrilaterals (like rectangles or squares) can be inferred without measurement due to their known properties (e.g., all angles in a rectangle are 90 degrees), a protractor is necessary for precise measurement, especially in irregular quadrilaterals where the angles are not equal or known in advance.

## The next lessons are

• Congruence and similarity
• Transformations
• Mathematical proof

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