When we have atriangle, we can identify that it is anisosceles if at least one of the following conditions is met:

1) If the triangle has two equal angles - The triangle is isosceles. 2) If in the triangle the height also bisects the angle of the vertex - The triangle is isosceles. 3) If in the triangle the height is also the median - The triangle is isosceles. 4) If in the triangle the median is also the bisector - The triangle is isosceles.

Before we talk about how to identify an isosceles triangle, let's remember that it is a triangle with two sides (or edges) of the same length - This means that the base angles are also equal. Moreover, in an isosceles triangle, the median of the base, the bisector, and the height are the same, that is, they coincide.

Let's see it illustrated

These magnificent properties of the isosceles triangle cannot prove by themselves that it is an isosceles triangle. So, how can we prove that our triangle is isosceles?

If at least one of the following conditions is met: 1) If our triangle has two equal angles - The triangle is isosceles. This derives from the fact that the sides opposite to equal angles are also equal, therefore, if the angles are equal, the sides are too.

2) If in the triangle the height also bisects the vertex angle - The triangle is isosceles. 3) If in the triangle the height is also the median - The triangle is isosceles. 4) If in the triangle the median is also the angle bisector - The triangle is isosceles. In fact, we can summarize guidelines $2$ and $4$ and write a single condition: If two of these coincide - the median, the height, and the bisector - The triangle is isosceles.

Great, now you know how to identify isosceles triangles easily and quickly.

If you are interested in learning more about other angle topics, you can enter one of the following articles:

Sum of the interior angles of a polygon

Angles in regular hexagons and octagons

Measure of an angle of a regular polygon

Sum of the exterior angles of a polygon

Exterior angle of a triangle

Relationships between angles and sides of the triangle

The relationship between the lengths of the sides of a triangle

In the blog ofTutorelayou will find a variety of articles about mathematics.

Examples and exercises with solutions for identifying an isosceles triangle

Exercise #1

What kid of triangle is given in the drawing?

Video Solution

Step-by-Step Solution

The measure of angle C is 90°, therefore it is a right angle.

If one of the angles of the triangle is right, it is a right triangle.

Answer

Right triangle

Exercise #2

What kind of triangle is given in the drawing?

Video Solution

Step-by-Step Solution

As all the angles of a triangle are less than 90° and the sum of the angles of a triangle equals 180°:

$70+70+40=180$

The triangle is isosceles.

Answer

Isosceles triangle

Exercise #3

What kid of triangle is the following

Video Solution

Step-by-Step Solution

Given that in an obtuse triangle it is enough for one of the angles to be greater than 90°, and in the given triangle we have an angle C greater than 90°,

$C=107$

Furthermore, the sum of the angles of the given triangle is 180 degrees so it is indeed a triangle:

$107+34+39=180$

The triangle is obtuse.

Answer

Obtuse Triangle

Exercise #4

What kind of triangle is given in the drawing?

Video Solution

Step-by-Step Solution

Given that sides AB and AC are both equal to 9, which means that the legs of the triangle are equal and the base BC is equal to 5,

Therefore, the triangle is isosceles.

Answer

Isosceles triangle

Exercise #5

Which kind of triangle is given in the drawing?

Video Solution

Step-by-Step Solution

As we know that sides AB, BC, and CA are all equal to 6,

All are equal to each other and, therefore, the triangle is equilateral.

Answer

Equilateral triangle

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