# How to Find Angles of Quadrilateral Shapes?

Quadrilaterals are four-sided polygons with four vertices and four interior angles. The unknown angles of a quadrilateral can be easily calculated. In this step-by-step guide, you learn more about finding angles of quadrilateral shapes.

A quadrilateral has \(4\) angles. If we know the \(3\) angles of a quadrilateral, we can find the \(4th\) angle of a quadrilateral

**Step by step guide to finding angles of quadrilateral shapes**

There are four interior angles in a quadrilateral that add up to \(360\) degrees. This value is obtained using the sum of the angles of a quadrilateral. According to the angle sum property of a polygon, the sum of the interior angles of a polygon can be calculated by the number of triangles formed in it. These triangles are formed by drawing diagonals from a single vertex. To make it easier, it can be calculated with a formula that says that if a polygon has \(n\) sides, there will be a triangle \((n – 2)\) inside it.

The sum of the interior angles of a polygon can be calculated by the following formula:

\(S = (n-2) × 180°\),

where \(n\) represents the number of sides of a given polygon.

**Interior and exterior angles of the quadrilateral**

There are \(4\) interior and \(4\) exterior angles in a quadrilateral. To understand the difference between the interior and exterior angles of a quadrilateral, consider the following figure.

**Interior angles of a quadrilateral**

The angles inside a quadrilateral are called **interior angles.** The sum of the interior angles of a quadrilateral is \(360°\). This helps to calculate the unknown angles of a quadrilateral. If a quadrilateral is square or rectangular, we know that all its interior angles are \(90°\) each.

**Exterior angles of a quadrilateral**

The angles that form between one side of a quadrilateral and another line that extends from an adjacent side are called the **exterior angles**. If we look at the figure above, we see that the exterior angle and the interior angle form a straight line, and hence, they make a linear pair. Therefore, if the interior angle of a quadrilateral is known, we can find the value of the corresponding exterior angle. If the quadrilateral is square or rectangular, all its exterior angles will be \(90°\) each.

**Angles of quadrilateral formula**

There are some basic formulas for the interior and exterior angles of a quadrilateral:

- \(\color{blue}{Exterior\:angle\:=\:180°\:-\:Interior\:angle}\). This formula is used when the interior angle of a quadrilateral is known and the corresponding exterior angle value is required. Since both of them form a linear pair, they are supplementary, meaning that their sum is always equal to \(180°\). This formula can also be used to find the interior angle if the corresponding exterior angle is given. In that case, the formula would be \(\color{blue}{Interior\:angle\:=\:180°\:-\:Exterior\:angle}\).
- If \(3\) angles of a quadrilateral are known, the fourth angle can be calculated using the formula: \(\color{blue}{360\:-\:\left(sum\:of \:3\:other\:interior\:angles\right)}\).
- The sum of interior angles of a quadrilateral \(\color{blue}{=\:Sum=\left(n\:−\:2\right)\:×\:180°}\), where \(n\) represents the number of sides of the given polygon. In a quadrilateral, \(n = 4\), so after substituting the value of \(n\) as \(4\), we get, \(Sum\:=\:\left(4\:−\:2\right)\:×\:180°\:=\:360°\)

**Angles of a quadrilateral inscribed in a circle**

When a quadrilateral is inscribed in a circle, it is known as a cyclic quadrilateral. A cyclic quadrilateral is a quadrilateral that lies inside a circle and touches all the vertices of that circle. There are many theorems about the angles of a quadrilateral inscribed in a circle.

“The opposite angles in a circular quadrilateral are supplementary, that is, the sum of the opposite angles is \(180\) degrees,” says a theorem of opposite angles of a cyclic quadrilateral. Notice the figure below, which shows the opposite angles in a cyclic quadrilateral sum up to \(180°\).

**Finding Angles of Quadrilateral Shapes** **– Example 1:**

If \(3\) interior angles of a quadrilateral are given as \(76°, 99°\), and \(112°\), find the \(4th\) angle.

**Solution:**

The \(4th\) angle of the quadrilateral can be calculated using the formula: \(360\:-\:\left(Sum\:of\:the\:other\:3\:interior\:angles\right)\)

Unknown \(4th\) angle \(= 360\:-\:\left(76°\:+\:99°+\:112°\right)\)

\(4th\) angle \(= 360\:- 287=\:73°\)

**Exercises for** **Finding Angles of Quadrilateral Shapes**

**Find the measure of each angle indicated.**

- \(\color{blue}{80^{\circ }}\)
- \(\color{blue}{69^{\circ }}\)
- \(\color{blue}{70^{\circ }}\)

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