We have the following situation:
We need to find the degree measure of each angle, A, and C since we already know that angle B = 90 degrees.
To answer this question, we need to remember that the sum of the interior angles of a triangle is equal to 180 degrees.
Then we can write the next equation to find the value of x as follows:
[tex]\begin{gathered} m\angle A=(5x+6)^{\circ} \\ m\angle C=(4x-6)^{\circ} \end{gathered}[/tex]
[tex](5x+6)^{\circ}+(4x-6)^{\circ}+90^{\circ}=180^{\circ}[/tex]
Now, we need to add the like terms as follows:
[tex]\begin{gathered} (5x+4x+6-6+90)^{\circ}=180^{\circ} \\ (9x+90)^{\circ}=180^{\circ} \end{gathered}[/tex]
Now, we can subtract 90 from both sides of the equation:
[tex]\begin{gathered} (9x+90-90)^{\circ}=(180-90)^{\circ} \\ (9x)^{\circ}=90^{\circ} \end{gathered}[/tex]
If we divide both sides by 9, we finally have for x:
[tex]\begin{gathered} \frac{9x}{9}=\frac{90}{9} \\ x=10 \end{gathered}[/tex]
Therefore, the value for x = 10.
If we substitute the value of x into the corresponding expressions for angles A and C, then we have:
Finding the measure of angle A
[tex]\begin{gathered} x=10^{}\Rightarrow m\angle A=(5x+6)^{\circ} \\ m\angle A=(5(10)+6)^{\circ}=(50+6)^{\circ}=56^{\circ} \\ m\angle A=56^{\circ} \end{gathered}[/tex]
Finding the measure of angle C
We can proceed in a similar way here. Then we have:
[tex]\begin{gathered} x=10^{}\Rightarrow m\angle C=(4x-6)^{\circ} \\ m\angle C=(4(10)-6)^{\circ}=(40-6)^{\circ}=34^{\circ} \\ m\angle C=34^{\circ} \end{gathered}[/tex]
Therefore, in summary, we can say that:
[tex]\begin{gathered} m\angle A=56^{\circ} \\ m\angle B=90^{\circ} \\ m\angle C=34^{\circ} \end{gathered}[/tex]
[We already knew that the measure of angle B is 90 degrees (right angle).
We can also check that the sum of all the angles is equal to 180 degrees:
[tex]\begin{gathered} 56^{\circ}+90^{\circ}+34^{\circ}=180^{\circ} \\ 180^{\circ}=180^{\circ} \\ \end{gathered}[/tex]
.]
