Contents

# Interior Angle|Definition & Meaning

## Definition

Angles that are inside a shape are referred to as interior angles, and interior angles can also refer to angles that are located in the region enclosed by two parallel lines and a transversal.

From the basic polygon, a triangle, through the polygon with an unlimited number of sides,** n**, the sides of polygons **converge** in space. Each intersection of sides produces a vertex with both an interior and an exterior angle. Polygon’s interior angles are contained **within** the polygon.

## What Is Interior Angle?Â

In geometry, there are **two** ways to create internal angles. One occurs **within** a polygon, while the other occurs when a **transversal** cuts **parallel lines**. On the basis of their dimensions, distinct types of angles are distinguished. Other sorts of angles are characterized as** pair angles** because they exist in pairs to demonstrate a particular attribute. Interior angles are an example.

Interior angles can be defined in two ways:

- Interior Angles of a Polygon
- Interior Angles of Parallel Lines

## Interior Angles of a Polygon

The figure created by combining the **two** rays at their same terminal is referred to as an angle in mathematics. An interior angle is a shape’s internal angle. The closed shape with sides and vertices is a **polygon**. All of the inner angles of a regular polygon are **equal** to one another. For instance, a square has interior angles that are all exactly right angles, or** 90 degrees**.

A polygon’s internal angles are **proportional** to the number of sides. Angles are typically expressed in terms of degrees or radians. Therefore, if a polygon has **four** sides, it also has **four** angles. Additionally, the total internal angles of various polygons vary.

We refer to the angles contained **within** a shape, most commonly a polygon, as its inner angles.

The figure above shows the interior angle of a square each interior angle is **90Âº**. The point where the lines join with one another is the vertex. In a square, each vertex has a **90Âº** angle.

## Interior Angles of Parallel Lines

Interior angles are another name for angles that are in the space bounded by** parallel lines** which are crossed by a line that is called a **transversal line**. All the angles within these parallel lines along the transversal line are **interior angles** whereas angles out sIde the parallel line are **exterior angles**.

The lines** L1** and** L2** are parallel to one another in the below diagram, and the lines **T1** and **T2** are the transversal lines that are cutting both parallel lines. The angles **âˆ a**, **âˆ b**, **âˆ c**, and** âˆ d** all fall into the category of internal angles.

## Interior Angle’s Types

When **two** parallelÂ lines are intersected by a transversal, two different kinds of interior angles are created. These two kinds of interior angles are known as c**o-interior angles** and**Â alternate interior angles**.

### Alternate Interior Angles

These angles are created when a **transversal** intersects two **parallel** lines. The opposing sides of a transversal produce these non-adjacent angles pair. The alternate interior angles paired in the initial image are **1** and **3**, and** 2** and **4**. If a transversal cuts two parallel lines, their lengths are **equal**.

### Co-Interior Angles

On the identical side of the transversal, such angles seem to be the pair of interior non-adjacent angles. Co-interior angle pairs are** (1 and 4)**, and **(2 and 3)**.Â Additionally known as consecutive interior angles or same-side interior angles, these angles are interior. Co-interior angles also make up a pair of supplementary angles because the sum of two of them is **180 degrees**.

Here **âˆ 1**,** âˆ 2**, **âˆ 3**, and **âˆ 4** are interior angles between two parallel lines **L1** and **L2**, and one transversal line** T1**.

## Interior AnglesÂ of a Triangle

Every vertex of a triangle contains **three** inner angles. These inner angles always add up to **180 degrees**. The incenter is the place where the bisectors of all these angles intersect. Due to the fact that the total of the internal angles in a triangle equals **180Â°**, there is simply one right angle and obtuse angle available in any triangle.

A triangle with three acute interior angles is known as an **acute triangle**, a triangle that has one interior angle that is obtuse is called an o**btuse triangle**, and a triangle including one interior angle that is right-angled is known as a **right-angled triangle**.

## Interior Angle Formula

All the sides of a polygon generate a **vertex**, and this vertex has an exterior and interior angle, from the basic polygon, let’s assume a triangle, to the indefinitely complicated polygon with **n** sides, such as an octagon. The total of a triangle’s three internal angles is **180 degrees**, according to the angle sum theorem. The summation of the interior angles of any polygon is obtained by multiplying two fewer than the number of sides by **180 degrees**.

**Sum of interior angle = (n-2)x 180Âº**

Here, n is the sides of any shape. Applying this formula to the regular pentagon, the pentagon has 5 sides so putting values in the formula, we get:

**Sum of interior angles of a pentagon = (5-2) x180Âº**

**Sum of interior angle = 3×180Âº**

**Sum of interior angle= 540Âº**

### Example 1

### Solution

**n= 8,**because the octagon has 8 sides, we have:

**Sum of interior angles = (n-2)x 180Âº**

**Sum of interior angle= (8-2) x 180Âº**

**=1080Âº**

### Example 2

Find the unknown interior angle xÂº of two parallel lines.

### Solution

The unknown angle **xÂº** is with a transversal line.

**85Âº + xÂº = 180Âº**

**xÂº = 180Âº – 85Âº**

**xÂº = 95Âº**

*All the figures above are created on Geogebra.*