Angles in a Quadrilateral: Exploring Definitions, Properties, and Examples

Have you ever wondered about the angles in a quadrilateral – those four-sided shapes that appear in various forms all around us? In this article, we will dive deep into the world of angles in a quadrilateral, uncovering their definitions, properties, and fascinating facts. So, let's get started!

What Are Angles in a Quadrilateral?

Angles in a quadrilateral refer to the four angles formed at each vertex of a quadrilateral shape. These angles are known as interior angles. To understand the concept better, imagine drawing a diagonal line within a quadrilateral. This diagonal will divide the quadrilateral into two triangles. As we know, the sum of interior angles in a triangle is 180°. Therefore, it follows that the sum of angles in a quadrilateral is 360°.

Sum of angles in a quadrilateral is 360 degrees: visual proof

A quadrilateral, as the name suggests, is a two-dimensional shape with four sides, four vertices, and four interior angles. It is formed by connecting four non-collinear points.

4 non-collinear points forming a quadrilateral

In the figure above, let's consider the quadrilateral ABCD. The sides AB, BC, CD, and DA form the four sides of the quadrilateral. The points A, B, C, and D are the four vertices. The angles A, B, C, and D represent the four interior angles of this quadrilateral.

Interior Angles and Exterior Angles of a Quadrilateral

A quadrilateral consists of both interior angles and exterior angles. Let's explore the differences between these two types of angles.

Interior Angles of a Quadrilateral

The sum of the interior angles of a quadrilateral is always 360°. This property allows us to find the measure of a missing angle if we know the value of the other three angles.

Exterior Angles of a Quadrilateral

An exterior angle is formed by the intersection of any side of a polygon and the extension of the adjacent side. In a quadrilateral, the sum of all exterior angles is 360°.

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

Exterior angles of a quadrilateral is 180 degrees

According to the relationship between interior and exterior angles, we have:

• Angle A + Angle 1 = 180°
• Angle B + Angle 2 = 180°
• Angle C + Angle 3 = 180°
• Angle D + Angle 4 = 180°

Angles of a Quadrilateral Formulas

Let's explore some fundamental formulas related to the interior and exterior angles of a quadrilateral.

Interior Angles Sum Property

The sum of the interior angles of any polygon can be calculated using the formula: Sum = (n - 2) × 180°, where "n" represents the number of sides of the polygon.

For a quadrilateral (n = 4), the sum of the interior angles is (4 - 2) × 180°, resulting in 360°.

Note: If you know the measurements of three angles in a quadrilateral, you can calculate the fourth angle using the formula: 360° - (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°. You can use the following formulas:

• Exterior angle = 180° - Corresponding interior angle
• Interior angle = 180° - Corresponding exterior angle

Exterior Angles Sum of a Quadrilateral

The sum of the exterior angles of a quadrilateral is always 360°.

Let's consider angles W, X, Y, and Z as the exterior angles corresponding to angles A, B, C, and D (interior angles) of a quadrilateral. Therefore, the equation becomes: W + X + Y + Z = (180° - A) + (180° - B) + (180° - C) + (180° - D) = (180° + 180° + 180° + 180°) - (A + B + C + D) = 360° - (A + B + C + D).

Therefore, the sum of the exterior angles of a quadrilateral is 360°.

Interior and exterior angles of a quadrilateral

How to Find the Missing Angle in a Quadrilateral

To find the missing angle in a quadrilateral, follow these steps:

Step 1: Use angle properties to determine the known interior angles. Step 2: Add all the known interior angles together. Step 3: Subtract the sum from 360° to find the missing angle.

Let's solve an example to illustrate this process.

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

Angle measures in a quadrilateral

We know that the sum of the interior angles of a quadrilateral is 360°. Adding the given three angles, we obtain: $100° + 95° + 60° = 255°$.

By subtracting this sum from 360°, we find: $360° - 255° = x$.

Therefore, $x = 105°$.

Properties of Angles in a Quadrilateral

Let's explore the angle properties of some common types of quadrilaterals:

Rectangle

• All interior angles measure 90°.
• The diagonals bisect each other, forming four angles at the point of intersection: two acute angles and two obtuse angles.
• However, the diagonals do not bisect the angles at the vertices.
• Vertically opposite angles exist at the intersection of the diagonals.

Parallelogram

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

Square

• All interior angles are congruent and measure 90°.
• The diagonals bisect each other at right angles.

Rhombus

• Opposite angles are equal.
• Adjacent angles are supplementary.
• The diagonals of a rhombus bisect each other at right angles.
• The diagonals of a rhombus also bisect the vertex angles.

Take a look at the different types of quadrilaterals depicted below. You can create an anchor chart to explore the various properties of angles and sides in each type of quadrilateral. It's a fun way to dive deeper into the world of quadrilaterals!

Quadrilateral family

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. In such cases, all four vertices of the quadrilateral lie on the circumference of the circle. The sides of the quadrilateral form the chords of the circle.

Fascinating Fact: The sum of opposite angles in a cyclic quadrilateral is always 180°. In other words, opposite angles in a cyclic quadrilateral are supplementary.

Opposite angles in a cyclic quadrilateral are supplementary

Facts about Angles in a Quadrilateral

Here are some interesting facts to remember about angles in a quadrilateral:

• The sum of interior angles in a quadrilateral is always 360°.
• Opposite angles in a parallelogram are equal.
• Adjacent angles in a parallelogram are supplementary.
• Vertically opposite angles in a parallelogram are equal.
• All angles in a square are congruent and measure 90°.
• The diagonals of a rectangle bisect each other but do not bisect the vertex angles.

Conclusion

In this article, we explored the angles in a quadrilateral, their properties, interior and exterior angles, angles of a quadrilateral inscribed in a circle, and some important formulas. Armed with these insights, we can now apply these formulas to solve practical examples. So go ahead, delve into the world of quadrilaterals, and enjoy the intricacies of their angles!

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's assume the measures of the four angles as x, 2x, 3x, and 4x, respectively.

The sum of the interior angles of a quadrilateral is 360°.

$x + 2x + 3x + 4x = 360°$

$10x = 360°$

$x = 36°$

Thus, the measures of the four angles are:

$x = 36°$

$2x = 2(36°) = 72°$

$3x = 3(36°) = 108°$

$4x = 4(36°) = 144°$

Therefore, the angles of the quadrilateral are 36°, 72°, 108°, and 144°.

2. Find the exterior angle of a quadrilateral whose corresponding interior angle is 60°.

Solution: In a quadrilateral, the exterior angle and the corresponding interior angle form a linear pair.

Using the formula for the exterior angle of a quadrilateral:

Exterior angle = 180° - Interior angle

Exterior angle = 180° - 60°

= 120°

The exterior angle of the quadrilateral is 120°.

3. Find the corresponding interior angle of a quadrilateral if its exterior angle is 104°.

Solution: The relationship between the interior and exterior angles is that they form a linear pair.

Using the formula for the exterior angle of a quadrilateral:

Exterior angle = 180° - Interior angle

Interior angle = 180° - Exterior angle

Interior angle = 180° - 104°

= 76°

The corresponding interior angle of the quadrilateral is 76°.

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

Solution: ABCD is a cyclic quadrilateral, which means that opposite angles in it sum up to 180°.

Therefore, Angle D + x = 180°

$67° + x = 180°$

$x = 180° - 67°$

$x = 113°$

Practice Problems on Angles in a Quadrilateral

Now that we have covered the fundamental concepts and properties of angles in quadrilaterals, it's time for you to put your skills to the test. Here are a few practice problems to help solidify your understanding:

1. Find the missing angle in a quadrilateral whose three known angles measure 80°, 90°, and 120°.
2. In a parallelogram, one angle measures 60°. What are the measures of all the other angles in the parallelogram?
3. ABCD is a rhombus with one angle measuring 140°. Calculate the measures of all the other angles in the rhombus.

Frequently Asked Questions on Angles in a Quadrilateral

1. What are the properties of angles in a rectangle?

• All interior angles of a rectangle measure 90 degrees.
• The diagonals bisect each other, forming two acute angles and two obtuse angles at the point of intersection.
• The diagonals do not bisect the angles at the vertices.
• Vertically opposite angles exist at the intersection of the diagonals.

2. Do opposite angles in a parallelogram have equal measures?

• Yes, opposite angles in a parallelogram are equal in measure.

3. What are the interior angle measures of a square?

• All interior angles of a square are congruent and measure 90 degrees.

4. What is the sum of interior angles in a quadrilateral?

• The sum of interior angles in a quadrilateral is always 360 degrees.

Remember, exploring the world of angles in a quadrilateral can be fascinating and challenging at the same time. So, enjoy the journey, solve problems with confidence, and keep expanding your knowledge of this intriguing topic!