# Angles in a Quadrilateral – Definition, Properties, Examples, Facts

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## What Are Angles in a Quadrilateral?

Angles in a quadrilateral are the four angles formed at each vertex. These angles are called interior angles of a quadrilateral.

If you draw a diagonal, you can see that a quadrilateral gets divided into two triangles. We know the sum of the interior angles of a triangle is 180°. It follows that the sum of angles in a quadrilateral is 360°.

A quadrilateral is defined as a two-dimensional shape with four sides, four vertices, and four interior angles. A quadrilateral is a four-sided polygon shape formed by four non-collinear points.

In the figure given above, ABCD is a quadrilateral.

AB, BC, CD, and DA are four sides of the quadrilateral.

A, B, C, and D are four vertices.

$\angle A, \angle B, \angle C$, and $\angle D$ are the four interior angles of this quadrilateral.

## Interior Angles and Exterior Angles of a Quadrilateral

There are 4 interior angles and 4 exterior angles in a quadrilateral.

Let’s understand the difference between the interior and exterior angles of a quadrilateral.

The sum of the interior angles of a quadrilateral is $360^{\circ}$. If there is one missing angle, we can use this property to find the measure of the missing angle.

An exterior angle is formed by the intersection of any of the sides of a polygon and extension of the adjacent side. The sum of all the exterior angles of a quadrilateral is $360^{\circ}$.

Take a look at the quadrilateral ABCD. Angles 1, 2, 3, and 4 are the exterior angles. Angles A, B, C, and D are interior angles.

$\text{Interior Angle} + \text{Corresponding Exterior Angle} = 180^{\circ}$

$\angle A + \angle 1 = 180^{\circ}$

$\angle B + \angle 2 = 180^{\circ}$

$\angle C + \angle 3 = 180^{\circ}$

$\angle D + \angle 4 = 180^{\circ}$

## Angles of a Quadrilateral Formulas

Let’s take a look at some basic formulas related to the interior and exterior angles of a quadrilateral.

Interior Angles Sum Property

The sum of interior angles of a polygon $= Sum = (n \;-\; 2) \times 180^{\circ}$

where “n” is the number of sides of the given polygon.

In a quadrilateral, $n = 4$.

Sum of interior angles of a quadrilateral $= (4 \;−\; 2) 180^{\circ}$

Sum of interior angles of a quadrilateral $= 360^{\circ}$

Note: If 3 angles of a quadrilateral are known, then the 4th angle can be calculated using the formula: $360^{\circ} \;-\;$ (Sum of the other 3 interior angles).

Interior angle and corresponding exterior angle

The sum of an interior angle and its corresponding exterior angle is always 180.

• Exterior angle $= 180^{\circ} \;-\;$ Corresponding Interior angle
• Interior angle $= 180^{\circ} \;-\;$ Corresponding Exterior angle

Exterior Angles Sum of a Quadrilateral

The sum of exterior angles of a quadrilateral is $360^{\circ}$.

Let angles W, X, Y, Z be the exterior angles corresponding to the interior angles A, B, C, D of a quadrilateral respectively.

Thus, $W + X + Y + Z = (180^{\circ} \;-\; A) + (180^{\circ} \;-\; B) + (180^{\circ} \;-\; C) + (180^{\circ} \;-\; D)$

$W + X + Y + Z = (180^{\circ} + 180^{\circ} + 180^{\circ} + 180^{\circ}) \;-\; (A + B + C + D)$

$W + X + Y + Z = (180^{\circ} + 180^{\circ} + 180^{\circ} + 180^{\circ}) \;-\; 360^{\circ}$

(…interior angles of a quadrilateral add up to $360^{\circ}$.)

$W + X + Y + Z = 360^{\circ}$

Thus, sum of exterior angles of a quadrilateral $= 360^{\circ}$

## How to Find the Missing Angle in a Quadrilateral

Step 1: Use angle properties to determine interior angles.

Step 2: Add all known interior angles.

Step 3: Subtract the angle sum from $360^{\circ}$.

Example: Find the missing angle in the quadrilateral given below.

We know that the sum of the interior angles of a quadrilateral is $360^{\circ}$.

Adding the given 3 angles, we get

$100^{\circ} + 95^{\circ} + 60^{\circ} = 255^{\circ}$

Subtracting the sum from $360^{\circ}$, we get

$360^{\circ} \;-\; 255^{\circ} = x$

Thus,  $x = 105$

## Properties of Angles in a Quadrilateral

Let us look at the angle properties of a few common quadrilaterals.

Rectangle

• All interior angles measure $90^{\circ}$.
• Diagonals bisect each other and form four angles at the point of intersection, two acute angles and two obtuse angles.
• The diagonals do not bisect the angles at vertices.
• Vertically opposite angles at the intersection of the diagonals.

Parallelogram

• Opposite angles are equal.
• Adjacent angles or consecutive angles are supplementary.
• Vertically opposite angles are formed at the intersection of the diagonals.

Square

• All interior angles are congruent and measure $90^{\circ}$.
• Diagonals bisect each other at right angles.

Rhombus

• Opposite angles are the equal.
• Diagonals of a rhombus bisect each other at right angles.
• Diagonals of a rhombus bisect vertex angles.

Take a look at the different types of quadrilaterals shown below. You can make an anchor chart to explore different properties of angles and sides in each of the given quadrilaterals & their types. Try it out!

## Angles of a Quadrilateral Inscribed in a Circle

When a quadrilateral is inscribed in a circle, it is known as a cyclic quadrilateral or a chordal quadrilateral. It is a quadrilateral that has all its four vertices lying on the circumference of a circle. In a cyclic quadrilateral, the four sides of the quadrilateral form the chords of the circle.

The sum of opposite angles in a cyclic quadrilateral is $180^{\circ}$. In other words, opposite angles in a cyclic quadrilateral are supplementary.

• Angles in a quadrilateral add up to $360^{\circ}$.
• Any quadrilateral with four right angles is a rectangle.
• A quadrilateral with four right angles and four equal sides is a square.

## Conclusion

In this article, we learned about the angles of a quadrilateral, its properties, interior and exterior angles, angles of a quadrilateral inscribed in a circle and some important formulas. Let us apply those formulas to solve some examples!

## Solved Examples on Angles in a Quadrilateral

1. The angles of a quadrilateral are in the ratio of 1 : 2 : 3 : 4. Find the measure of each angle.

Solution:

The given angle ratio is 1 : 2 : 3 : 4.

Let the measures of the four angles be $x,\; 2x,\; 3x$, and  $4x$.

The sum of interior angles of a quadrilateral is $360^{\circ}$.

$x + 2x + 3x + 4x = 360^{\circ}$

$10 x = 360^{\circ}$

$x = 36^{\circ}$

Thus, the measure of four angles are:

$x = 36^{\circ}$

$2x = 2(36^{\circ}) = 72^{\circ}$

$3x = 3(36^{\circ}) = 108^{\circ}$

$4x = 4(36^{\circ}) = 144^{\circ}$

Therefore, the angles of a quadrilateral are $36^{\circ},\;72^{\circ},\; 108^{\circ}$ and $144^{\circ}$.

2. Find the exterior angle of a quadrilateral whose corresponding interior angle is $60^{\circ}$.

Solution:

We know that the interior and exterior angles of a quadrilateral form a linear pair.

Thus, using the formula for the exterior angle of a quadrilateral,

Exterior angle $= 180^{\circ}\;-\;$ Interior angle

Exterior angle $= 180^{\circ} \;-\; 60^{\circ}$

$= 120^{\circ}$

The exterior angle of the quadrilateral is $120^{\circ}$.

3. Find the corresponding interior angle of a quadrilateral if its exterior angle is $104^{\circ}$.

Solution:

We know that the interior angle and the corresponding exterior angle of a quadrilateral form a linear pair.

Thus, using the formula for the exterior angle of a quadrilateral.

Exterior angle $= 180^{\circ} \;-\;$ Interior angle

Interior angle $= 180^{\circ} \;-\;$  Exterior angle

Interior angle $= 180^{\circ} \;-\; 104$

Interior angle $= 76^{\circ}$

The corresponding interior angle of the quadrilateral is $76^{\circ}$.

4. ABCD is a cyclic quadrilateral with center O. Find x.

Solution:

Opposite angles of a cyclic quadrilateral sum up to 180.

Thus, $\angle D + x = 180^{\circ}$

$67^{\circ} + x = 180^{\circ}$

$x = 180^{\circ} \;-\; 67^{\circ}$

$x = 113^{\circ}$

## Practice Problems on Angles in a Quadrilateral

1

### Which of the following is not a quadrilateral?

Parallelogram
Square
Triangle
Rhombus
CorrectIncorrect
A triangle is not a quadrilateral as it has only three sides.
2

### What is the sum of interior angles of a quadrilateral?

$180^{\circ}$
$160^{\circ}$
$90^{\circ}$
$270^{\circ}$
CorrectIncorrect
Correct answer is: $160^{\circ}$
The interior angles of a quadrilateral always sum up to $360^{\circ}$.
3

### Exterior angle and the corresponding interior angle of a quadrilateral add up to _______ angle.

$180^{\circ}$
$360^{\circ}$
$90^{\circ}$
$270^{\circ}$
CorrectIncorrect
Correct answer is: $180^{\circ}$
Exterior angle and the corresponding interior angle of a quadrilateral form a $180^{\circ}$ angle.
4

### If an interior angle of a quadrilateral is 70, then its corresponding exterior angle will be _______.

$100^{\circ}$
$20^{\circ}$
$110^{\circ}$
$290^{\circ}$
CorrectIncorrect
Correct answer is: $110^{\circ}$
If an interior angle of a quadrilateral is 70, then its corresponding exterior angle will be, $180^{\circ} \;-\; 70^{\circ} = 110^{\circ}$.
5

### The sum of the opposite angles of a cyclic quadrilateral is _______.

$90^{\circ}$
$180^{\circ}$
$270^{\circ}$
$360^{\circ}$
CorrectIncorrect
Correct answer is: $180^{\circ}$
The opposite angles in a cyclic quadrilateral are supplementary. The sum of the opposite angles is equal to $180^{\circ}$.

No, all angles of a quadrilateral cannot be acute angles. Acute angles are the angles less than $90^{\circ}$. If all the angles in a quadrilateral be less than $90^{\circ}$, the sum of interior angles will never be $360^{\circ}$. All angles of quadrilateral can be $90^{\circ}$, which will form a rectangle (or a square) but not less than $90^{\circ}$.