Question

Triangle ABC is an isosceles right triangle. What is the measure of a base angle? 90 °

Answer 1

The measure of a base angle is 45°

Further explanation

Triangle is a plane figure that has 3 sides and also 3 angles.

The sum of all of the angles of triangle always 180°.

There are several types of triangles such as :

1. Equilateral Triangle → 3 equal sides
2. Isosceles Triangle → 2 equal sides
3. Scalene Triangle → no equal sides

Let us now tackle the problem!

If triangle ABC is an isosceles right triangle , then one of the angles will be 90°. The other two angles ( base angles ) will have the same value.

We could draw this triangle as shown in the attachment.

Let :

the base angle = x

$x + x + 90^o = 180^o$

$2x + 90^o = 180^o$

$2x = 180^o - 90^o$

$2x = 90^o$

$x = 90^o \div 2$

$\large { \boxed { x = 45^o } }$

The measure of each of the base angle will be 45°

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Answer details

Grade: High School

Subject: Mathematics

Chapter: Triangle

Keywords: Triangle , Isosceles , Right , Angle , Base

Answer 2

The measure of the base angle in an isosceles right triangle is $\fbox{\begin\\\ 45^{\circ}\\\end{minispace}}$.

Further explanation:

A triangle is two dimensional figure which is formed when three non-collinear points are joined. It has three sides, three angles and three edges.

On the basis of the angle a triangle is classified into three categories as follows:

1) Acute angled triangle:

If all the angles of a triangle are less than $90^{\circ}$ than the triangle is called an acute angled triangle.

2) Obtuse angled triangle:

If any of the angle of a triangle is greater than $90^{\circ}$ than the triangle is called obtuse angled triangle.

3) Right angled triangle:

If any one of the angle of the triangle is of $90^{\circ}$ than the triangle is called a right angled triangle.

On the basis of the sides of a triangle is classified into three categories as follows:

1) Scalene triangle:

If all the sides of a triangle are unequal or are distinct than the triangle is called a scalene triangle.

2) Equilateral triangle:

If all the sides of triangle are equal in length than the triangle is called an equilateral triangle.

3) Isosceles triangle:

If any two sides of a triangle are equal in length than the triangle is called an isosceles triangle.

In the question it is given that $\triangle \text{ABC}$ is an isosceles right triangle i.e., one of the angle must be of $90^{\circ}$ and at least two side of the triangle must be $90^{\circ}$ because it is an isosceles triangle.

Consider a triangle as $\triangle \text{ABC}$ which is right angled at A and the sides AB and AC are the equal sides.

Figure 1 (attached in the end) shows the $\triangle \text{ABC}$ in which $\angle \text{BAC}=90^{\circ}$ and $\text{AB=AC}$.

In an isosceles triangle the angles opposite to the equal sides of the triangle are equal.

Since, $\triangle \text{ABC}$ is an isosceles triangle and $\text{AB=AC}$ so, the angles opposite to the equal sides are equal.

From figure 1 (attached in the end) it is observed that angle opposite to the side AB is $\angle\text{ACB}$ and the angle opposite to the side AC is $\angle\text{ABC}$.

So, as per the property stated above $\angle\text{ACB}=\angle\text{ABC}$.

Consider the measure of the $\angle\text{ACB}$ as $x^{\circ}$.

Since, $\angle\text{ACB}=\angle\text{ABC}$ so, $\angle\text{ACB}=\angle\text{ABC}=x^{\circ}$.

The measure of all the three angles of $\triangle \text{ABC}$ are $90^{\circ},x^{\circ}\ \text{and} \ x^{\circ}$.

Angle sum property:

Sum of all the interior angles of triangle is always equals to $180^{\circ}$.

Apply the angle sum property for $\triangle \text{ABC}$.

$\begin{aligned}x+x+90&=180\\2x+90&=180\\2x&=90\\x&=45\end{aligned}$

Therefore, the value of $x$ is $45$.  

This implies that the measure of the $\angle\text{ACB}$ and $\angle\text{ABC}$ is of $45^{\circ}$.

Thus, the measure of the base angle in an isosceles right triangle is $\fbox{\begin\\\ \bf 45^{\circ}\\\end{minispace}}$.

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Answer details:

Grade: High school

Subject: Mathematics

Chapter: Triangle

Keywords:Triangle, angles, sides, isosceles triangle, scalene triangle, equilateral triangle, right triangle, isosceles right triangle, angle sum property, equal opposite sides, base angle, 90 degrees.