Isosceles triangle theorem states that the angles opposite to equal sides of an isosceles triangle are also equal or vice versa. School going students who are learning the concept of geometry and measurement can get to know the complete details of the theorem on the isosceles triangle in the below-mentioned sections.

## Theorem on Isosceles Triangle

The theorem of the isosceles triangle states that the equal sides of the isosceles triangle are produced beyond their common vertex to two different points such that the distance from the vertex to the points are equal and straight lines joining points and extremities of the base are equal. The complete proof for this statement is given here:

In ∆XYZ is an isosceles triangle with XY = XZ

XY and XZ are produced to A and B such that XA = XB. B, Y and A, Z are joined.

The above statement can be written as XY, XZ are the equal sides of the isosceles triangle XYZ and those equal sides are produced beyond the common vertex X to the points A and B such that XA and XB are equal. Show that BY is equal to AZ.

**To prove:** BY = AZ.

**Proof:**

Statement | Reason |
---|---|

In ∆XBY and ∆XAZ, XY = XZ XB = XA ∠BXY = ∠AXZ |
Given Given Vertically opposite angles |

∆XBY ≅ ∆XAZ | SAS Criterion |

BY = AZ | CPCTC |

Hence proved.

**Do Check:**

- Problems on Properties of Isosceles Triangles
- Point on the Bisector of an Angle
- Straight Lines Joining the Extremities of the Base of an Isosceles Triangle

### Problems on Isosceles Triangle Theorem

**Problem 1:**

What is the value of x?

**Solution:**

We know that angles opposite to the equal sides of an isosceles triangle are equal.

So, another base angle is 64°.

We also know that vertical angles are congruent in an isosceles triangle.

So, the remaining angle in the first triangle = 2x

The sum of the interior angles of a triangle is equal to 180 degrees

64 + 64 + 2x = 180

128 + 2x = 180

2x = 52

x = 26°

**Problem 2:**

Find the length of the altitude of an isosceles triangle whose side lengths are 3 cm, 5 cm and 5 cm.

**Solution:**

Given congruent sides are 5 cm and base is 3 cm

Let the congruent side be a and base is b

The formula to find the altitude of the isosceles triangle h = √(a² – \(\frac { b² }{ 4 } \))

= √(5² – \(\frac { 3² }{ 4 } \))

= \(\frac { √91 }{ 2 } \)

Therefore, the altitude of the triangle is \(\frac { √91 }{ 2 } \) cm

### Frequently Asked Question’s

**1. Are median, altitude is drawn on the base from the opposite vertex angle the same for an isosceles triangle?**

Yes, the median and altitude of an isosceles triangle are the same if they are drawn from vertex angle to the base opposite to it.

**2. Write the theorems related to the isosceles triangle?**

The two important and basic theorems on the isosceles triangle are Angles opposite to the equal sides of an isosceles triangle are also equal. Another one is the sides opposite to the equal angles of the triangle are equal.

**3. State one isosceles triangle theorem.**

If two points on the base of an isosceles triangle are equidistant from the extremities of the base, then they are equidistant from the vertex.