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 \(\fbox{\begin\\\ \bf 45^{\circ}\\\end{minispace}}\).
Learn more:
1. A problem to determine the equation of line https://brainly.com/question/1646698
2. A problem on ray https://brainly.com/question/1251787
3. A problem to determine intercepts of a line https://brainly.com/question/1332667
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.
