Question

in the figure below, (mangle1 = 6x^{circ}) and (mangle2=(x - 8)^{circ}). find the angle measures.

Explanation:

Step1: Note angle - relationship

Since $\angle1$ and $\angle2$ are complementary (the angle formed by the two rays is a right - angle, so $m\angle1 + m\angle2=90^{\circ}$).
$6x+(x - 8)=90$

Step2: Simplify the left - hand side

Combine like terms: $6x+x-8 = 90$, which gives $7x-8 = 90$.

Step3: Solve for $x$

Add 8 to both sides of the equation: $7x-8 + 8=90 + 8$, so $7x=98$.
Then divide both sides by 7: $x=\frac{98}{7}=14$.

Step4: Find $m\angle1$

Substitute $x = 14$ into the expression for $m\angle1$: $m\angle1=6x=6\times14 = 84^{\circ}$.

Step5: Find $m\angle2$

Substitute $x = 14$ into the expression for $m\angle2$: $m\angle2=x - 8=14 - 8=6^{\circ}$.

Answer:

$m\angle1 = 84^{\circ}$
$m\angle2 = 6^{\circ}$