Certainly! Let's go through the problem step by step.
1. Sum of Angles in a Triangle:
It's a fundamental property of triangles that the sum of their interior angles is always [tex]\( 180^\circ \)[/tex]. We can express this as:
[tex]\[
m \angle A + m \angle B + m \angle C = 180^\circ
\][/tex]
2. Given Equation:
We are given an equation related to one of the angles in the triangle:
[tex]\[
7x = 84
\][/tex]
3. Solving for [tex]\( x \)[/tex]:
To find [tex]\( x \)[/tex], we use the division property of equality. We divide both sides of the equation by 7:
[tex]\[
x = \frac{84}{7}
\][/tex]
Simplifying this gives:
[tex]\[
x = 12
\][/tex]
4. Finding the Measure of Angle [tex]\( A \)[/tex]:
We are told to use the substitution property to find the measure of angle [tex]\( A \)[/tex]. According to the problem, we need to calculate:
[tex]\[
m \angle A = 6x
\][/tex]
Substituting [tex]\( x = 12 \)[/tex] into the expression:
[tex]\[
m \angle A = 6 \times 12
\][/tex]
Simplifying this gives:
[tex]\[
m \angle A = 72^\circ
\][/tex]
So, by following these steps, we find that the measure of angle [tex]\( A \)[/tex] is [tex]\( 72^\circ \)[/tex], and the sum of the angles in the triangle is indeed [tex]\( 180^\circ \)[/tex].
