Let's solve this step-by-step to determine the correct values of [tex]\( x \)[/tex] and [tex]\( \angle A \)[/tex], filling in the blanks with the appropriate mathematical properties:
1. Triangle Sum Theorem: The sum of the angles in a triangle is equal to [tex]\( 180^\circ \)[/tex]. Therefore, [tex]\( m \angle A + m \angle B + m \angle C = 180^\circ \)[/tex].
2. Using the given information, we have the equation:
[tex]\[
(5x)^\circ + 90^\circ + (x + 10)^\circ = 180^\circ
\][/tex]
3. Combine like terms: To solve for [tex]\( x \)[/tex], first combine like terms:
[tex]\[
5x + x + 90 + 10 = 180
\][/tex]
Simplifying, we get:
[tex]\[
6x + 100 = 180
\][/tex]
4. Subtraction Property of Equality: Use this property to isolate the term with [tex]\( x \)[/tex]:
[tex]\[
6x = 180 - 100
\][/tex]
Simplifying, we get:
[tex]\[
6x = 80
\][/tex]
5. Division Property of Equality: Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{80}{6}
\][/tex]
Simplifying, we get:
[tex]\[
x = 13 \frac{1}{3}
\][/tex]
6. Substitution Property: To find the measure of [tex]\( \angle A \)[/tex], substitute [tex]\( x = 13 \frac{1}{3} \)[/tex] into [tex]\( 5x \)[/tex]:
[tex]\[
m \angle A = 5 \left( 13 \frac{1}{3} \right)^\circ
\][/tex]
7. Simplifying the expression: We get:
[tex]\[
m \angle A = 66 \frac{2}{3}^\circ
\][/tex]
Thus, the value of [tex]\( x \)[/tex] is [tex]\( 13 \frac{1}{3} \)[/tex] and the measure of [tex]\( \angle A \)[/tex] is [tex]\( 66 \frac{2}{3}^\circ \)[/tex].
