An **Isosceles Triangle** is a specific type of triangle distinguished by the property that **it has two sides of equal length.** In this geometric configuration, the **angles opposite these equal sides are also equal in measure**. This distinctive feature sets isosceles triangles apart from other triangle types, such as **scalene or equilateral triangles**, and leads to various interesting geometric properties.

In this article, we will walk you through the complete about **Isosceles Triangle** with some practice problems so that you can grab the concepts completely good.

## What Is Isosceles Triangle?

An isosceles triangle is a geometric figure that derives its name from its defining characteristic — the presence of two sides of equal length.

The term **“isosceles”** originates from the Greek words **“isos,” meaning “equal,” and “skelos,” meaning “leg.”** Thus, **an isosceles triangle** literally translates to **a triangle with equal legs**.

Suppose in this triangle △ABC, if sides AB and AC are equal, then △ABC is an isosceles triangle where ∠B = ∠C.

The theorem that describes the isosceles triangle is “**if the two sides of a triangle are equal, then the angle opposite to them are also equal**”.

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## Properties Of Isosceles Triangle

Property | Explanation |
---|---|

Congruent Sides | Two sides of an isosceles triangle are congruent to each other. |

Base Definition | The third side, unequal to the others, is referred to as the base. |

Congruent Angles | The angles opposite the congruent sides are themselves congruent. |

## Isosceles Triangle Theorem

The Isosceles Triangle Theorem is a fundamental geometric principle that establishes a relationship between the sides and angles of an isosceles triangle.

The theorem states that if two sides of a triangle are congruent (of equal length), then the angles opposite those sides are also congruent (equal).

Mathematically, the theorem can be expressed as:

**Isosceles Triangle Theorem:**

If line AB ≅ AC in triangle ABC, then ∠B ≅ ∠C.

Here’s a breakdown of the components of the theorem:

**Isosceles Triangle (△ABC):**This is the triangle under consideration, and the theorem specifically focuses on an isosceles triangle, where sides AB and AC are equal.**Congruent Sides:**line AB ≅ AC states that the two sides opposite angles A and B (the base angles) are of equal length.**Congruent Angles:**∠B ≅ ∠C asserts that the angles opposite the congruent sides (angles B and C) are also of equal measure.

**Visual Representation:**

In the diagram, ∠A and ∠B are the base angles, and sides AB and AC are congruent. The Isosceles Triangle Theorem affirms that ∠A and ∠B are equal due to the equality of the corresponding sides.

## Types of Isosceles Triangle

In an isosceles triangle, where two sides are of equal length, the relationships between the angles can vary based on different situations. Let’s explore several scenarios:

### 1. **Equal Base Angles:**

**Situation:**Both base angles are equal.**Example:**In triangle ABC with sides AB = AC, if ∠A = ∠B, then it’s an isosceles triangle.**Relation:**∠A = ∠B(base angles are congruent).

### 2. **Unequal Base Angles:**

**Situation:**The base angles are not equal.**Example:**In triangle XYZ with sides XY = XZ, if ∠X ≠ ∠Y, it’s still an isosceles triangle.**Relation:**∠X ≠∠Y, but sides XY = XZ, making it isosceles.

### 3. **Right Isosceles Triangle:**

**Situation:**One base angle is a right angle (90 degrees).**Example:**In triangle PQR with PQ = PR, if ∠Q = 90°, it’s a right isosceles triangle.**Relation:**∠P = ∠R = 45° (base angles are equal).

### 4. **Obtuse Isosceles Triangle:**

**Situation:**One base angle is obtuse (more than 90°).**Example:**In triangle LMN with LM = LN, if ∠L > 90°, it’s an obtuse isosceles triangle.**Relation:**∠M = ∠N (base angles are congruent).

### 5. **Acute Isosceles Triangle:**

**Situation:**All angles, including base angles, are acute (less than 90°).**Example:**In triangle UVW with UV = UW, if ∠U, ∠V, ∠W are acute, it’s an acute isosceles triangle.**Relation:**∠U = ∠W < 90°(base angles are congruent).

These situations illustrate the versatility of isosceles triangles. While the sides’ equality is a constant, the angles’ relationships can vary, giving rise to different geometric configurations and properties.

## Formulas Related to Isosceles Triangles

### General Formulas:

**Area ((A)):**

A = $\frac{1}{2} \times \text{Base} \times \text{Height} $

- The area of an isosceles triangle can be calculated using the standard triangle area formula. The height is determined by drawing a perpendicular line from the vertex angle to the base, creating two congruent right-angled triangles.

**Perimeter ((P)):**

$ P = \text{Side}_1 + \text{Side}_2 + \text{Base} $

- The perimeter of an isosceles triangle is the sum of its three sides.

### Angle Formulas:

**Base Angles (∠B, ∠C):**

- In an isosceles triangle ABC with sides (AB = AC), the base angles are congruent:

[ ∠B = ∠C] - This symmetry is a key property of isosceles triangles.

**Vertex Angle (∠A):**

- In an isosceles triangle ABC with sides (AB = AC), the vertex angle can be found using the supplementary angle property:

$[ \angle A = 180^\circ – 2 \times \text{Base Angle} ]$ - The vertex angle is determined by subtracting twice the measure of one of the base angles from 180 degrees.

### Special Cases:

**Right Isosceles Triangle:**

- If $(\angle B = 90^\circ)$, then $(\angle C = 90^\circ)$, and the Pythagorean theorem can be applied:

$[ \text{Hypotenuse}^2 = \text{Leg}^2 + \text{Leg}^2 ]$ - This formula is specific to right isosceles triangles and relates the sides using the Pythagorean theorem.

**Law of Cosines:**

- In a general isosceles triangle ABC:

$[ \text{Side}_3^2 = \text{Side}_1^2 + \text{Side}_2^2 – 2 \times \text{Side}_1 \times \text{Side}_2 \times \cos(\angle A) ]$ - The law of cosines provides a relationship between the sides and angles of a triangle, including isosceles triangles.

**Apollonius Theorem:**

- In an isosceles triangle ABC, if D is the midpoint of the base BC:

$[ AB^2 + AC^2 = 2(AD^2 + BD^2) ]$ - Apollonius theorem relates the sides and medians of an isosceles triangle.

These formulas offer a comprehensive toolkit for understanding and solving problems involving isosceles triangles. Depending on the given information and context, different formulas can be applied to analyze and derive various geometric properties.

## 10 Solved Problems On Isosceles Triangle

**Q1. In an isosceles triangle ABC, if $\angle A = 50^\circ $, what is the measure of ( \angle B )?**

**Ans:**

Since an isosceles triangle has two congruent base angles, $\angle B = \angle C $. Therefore, $\angle B = \angle C = \frac{1}{2} \times (180^\circ – \angle A = 65^\circ $.

**Q2. If the base angles of an isosceles triangle are $\angle X = 80^\circ$, what is the measure of the vertex angle $\angle Y$?**

**Ans:**

Since the base angles are congruent, $\angle Y = \angle X = 80^\circ$.

**Q3.**In an isosceles triangle ABC with AB = AC = 12cm, and $\angle A = 40^\circ$, find the area of the triangle.

**Ans:**

Using the formula for the area of a triangle A = $\frac{1}{2} \times \text{Base} \times \text{Height}$, the area is A = $\frac{1}{2} \times 12 \times 12 \times \tan(40^\circ$.

**Q4.**If an isosceles triangle has a perimeter of 30 units, and one side is 10 units, what are the lengths of the other two sides?

**Ans:**

Let the lengths of the congruent sides be ( x ). Since the perimeter is the sum of the three sides, ( 10 + x + x = 30 hence x=10.

**Q5.**In an isosceles right triangle, if one leg is 5 cm, what is the length of the hypotenuse?

**Ans:**

In a right isosceles triangle, the legs are congruent, so the hypotenuse is $\sqrt{2}$ times the length of one leg. Thus, the hypotenuse is $5 \sqrt{2}$cm.

**Q6.**If $\angle A$ is the vertex angle of an isosceles triangle ABC, what is the measure of $\angle C $?

**Answer:**

The base angles are congruent, so $angle C = $\angle B$.

**Q7. **

In an isosceles triangle, the lengths of the equal sides are 8 cm each. If the vertex angle is $\angle A = 60^\circ$, what is the area of the triangle?

**Answer:**

Use the formula A = $\frac{1}{2} \times \text{Base} \times \text{Height}$, and find the height using trigonometry.

**Q8. **

If the base angles of an isosceles triangle are $\angle P = 45^\circ$, what is the measure of the vertex angle $\angle Q$?

**Answer:**

Since the base angles are congruent, $\angle Q = \angle P = 45^\circ$.

**Q9.**

In an isosceles triangle, if the base is 16 cm and one base angle is $ \angle X = 30^\circ$, find the length of the equal sides.

**Answer:**

Use trigonometry to find the length of the equal sides using $\tan(30^\circ) = \frac{\text{Height}}{16}$.

**Q10. **

In a right isosceles triangle, if the hypotenuse is 10 cm, what is the length of each leg?

**Answer:**

Since it is a right isosceles triangle, the legs are congruent. Use the Pythagorean theorem to find the length of each leg.