Contents

**Included Side|Definition & Meaning**

**Definition**

Included sides are the **sides linking two angles** in **triangles** and other **polygons**. The angle between two lines is regarded as ‘included’ between two lines. The included side is **also** **referred** to as the **leg** that **connects** **two** **angles**. **Triangles** or **polygons** are **made** up **of** **these** **sides** that connect two angles and are shared by both angles.

**Finding Properties of Triangle From Included Sides **

Figure 1 – Included Side of a Triangle ABC

Because they can be used to **establish** various **aspects** of **triangles**, **included** **sides** are **significant** **in** **geometry**. These characteristics include the **triangle’s** **area**, **perimeter**, and **angles**.

**Area of Triangle**

Figure 2 – Area of Triangle from included side

**Utilizing** **ratios** is one way that **included** **sides** can be used to ascertain a **triangle’s** **characteristics**. For instance, the **area** of a **triangle** can be **calculated** using the **ratio** of the **lengths** of its **included** **sides**. A triangle’s area is determined by multiplying its base by its height and dividing that result by two. This is known as the **area** **formula** for **triangles**.

**Perimeter of Triangle**

The **perimeter** of a **triangle** can **also** be **calculated** **using** the triangle’s **included** **sides**. The perimeter of a triangle equals the sum of the lengths of its three sides. The lengths of all **three** **sides** are simply **added** to **determine** a **triangle’s** **perimeter**.

**Angles of Triangle**

Figure 3 – Finding Angle of triangle from included side

A triangle’s **included** **sides** can be **utilized** to **calculate** its **angles** in **addition** to these other attributes. The **intersection** of **two** **sides** **creates** each **angle** in a triangle, and the length of the included side determines the angle’s measurement. The **angle** will be **greater** the **longer** the **included** **side** is.

**Law of Cosines**

The **Law** **of** **Cosines** is another **theorem** that **deals** with **included** **sides** and is used to **calculate** a **side’s** **length** in a triangle **when** the other **two** **sides** **lengths** and the **size** of the **included** **angle** are **known**. According to the Law of Cosines, **one** of the **sides** of the triangle **square** is **equal** to the **addition** **of** **squares** of the **rest** of the **two** **sides**, **less** by **twice** the **product** of the **sides **other two lengths and the **cosine** of the **included** **angle**.

**Law of Sines**

There are other **geometric** **formulas** that incorporate sides and angles. One of these formulas, known as the **Law** **of** **Sines**, is used to **determine** a **triangle’s** **angle’s** measure when **both** its **included** **angle** and **side** **lengths** are **known**. The law of Sines states that the ratio of a triangle’s length to the sine of the included angle is equal to the ratio of its length to the sine of the other side.

**Included Side Vs. Included Angles**

**Included Side**

A **side** that is **encompassed** **between** **two** other triangle **sides** is said to be an **included** **side**. It is a **side** of the triangle that is **not** the **base**. A triangle has **three** included sides, and they are significant in geometry because they may be used to calculate the **triangle’s** area, **perimeter**, and **angles**, among other things.

**Included Angle**

On the other hand, an **included** **angle** is an angle that is **formed** by the **intersection** of **two** **sides** of a **triangle**. It is an **angle** that is **enclosed** **between** the **two** **sides** of the triangle. In a triangle, there are **three** **included** **angles**, and these angles are **significant** in **geometry** because they can be used to calculate the triangle’s area, perimeter, and angles, among other things.

**Differences**

One of the main **differences** between included sides and included angles is that **included** **sides** are **lengths**, while **included** **angles** are **measures** of **rotation**. **Included** **sides** are measured in **units** of **length**, such as **inches** or **centimeters**, while **included** **angles** are measured in **units** of **angle**, such as **degrees** or **radians**. The **included** **side** is used to **solve** **problems** involving **triangles** and other **geometric** **shapes**, while the **included** **angle** is used to **describe** the **angles** and **relationships** **between** **sides** of a shape.

**“Angle Side Angle” Theorem**

The angle-side-angle (ASA) theorem states that **two** **triangles** are **congruent** if **two** **angles** and **one** **of** their **included** **sides** are **congruent** (equal) to **two** **angles** and the other triangle’s included side. This **suggests** that the **two** **triangles** are **identical** **in** **terms** of **size** and **shape**.

The theorem states that two **triangles** are **congruent** in relation to the **included** **side** if the **lengths** of their **included** **sides** are **equal**. For instance, if the **included** **side** of one **triangle** is the **same** **length** as the included side of **another** **triangle** and vice versa.

For example, consider **two** **triangles** **ABC** and **DEF**. If the **measure** of **angle** **A** is **equal** to the **measure** of **angle** **D**, the **measure** of **angle** **C** is **equal** to the **measure** of **angle** **F**, and the **length** of side **BC** is **equal** to the **length** of side **EF**, then **triangles** **ABC** and **DEF** are **congruent**.

**Summary**

In summary, a **side** that **lies** **between** the **two** **angles** created by the other two sides of a triangle is **referred** to as an **included** **side** in geometry. The **triangle’s** **median** is **another** **name** for the **included** **side**, the **length** of the **included** **side** of a triangle can be used to define **specific** **trigonometric** **functions** these functions are employed to address issues involving **triangles** and **circles** and are defined in terms of the ratios of a triangle’s sides. As well as **provide** important **information** about the **triangle** and its **properties**. The included side of a triangle also plays a **significant** **role** in the study of **geometry** and **trigonometry**.

**Solved Examples of Included Sides**

**Example 1**

**Consider** a **triangle** with sides of **lengths** **5, 12, and 13**. This triangle is a **right** **triangle** because the side lengths satisfy the **Pythagorean** **theorem** (the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides). Let’s call this **triangle** **ABC**. Side **AB** has a **length** of **5** and is opposite **angle** **BAC**. Side **AC** has a **length** of **12** and is **opposite** **angle** **CAB**. Side **BC** has a **length** of **13** and is the **hypotenuse**, which is **opposite** the **right** **angle**. **Find** the **included** **side**.

**Solution**

In this example, **side** **BC** is the **included** **side**. It is **bounded** by **angles** **BAC** and **CAB** and is **opposite** the **right** **angle**. The **length** of the **included** **side** (side BC) can be **used** to **calculate** the **measure** of the **included** **angles** using **trigonometric** **functions**.

**Example 2**

Figure 4 – Example of finding Area of triangle from included side

**Consider** a **triangle** **ABC** having **three** included **sides** **a,b, and c**. **Find** the **area** of a **triangle** using the concept of finding the area of a triangle using included sides.

**Solution**

Area of triangle = $\dfrac{1}{2}$ x Base x Height

Area of triangle = $\dfrac{1}{2}$ x 5 x 4

Area of triangle = $\dfrac{1}{2}$ x 20

Area of triangle = 10

*All mathematical drawings and images were created with GeoGebra.*