Question

in the figure below, (mangle1 = 7x^{circ}) and (mangle2=(x - 6)^{circ}). find the angle measures.

Explanation:

Step1: Set up equation

Since $\angle1$ and $\angle2$ are complementary (they form a right - angle, so $m\angle1 + m\angle2=90^{\circ}$), we have the equation $7x+(x - 6)=90$.

Step2: Simplify the left - hand side

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

Step3: Isolate the variable term

Add 6 to both sides of the equation: $8x-6 + 6=90 + 6$, resulting in $8x=96$.

Step4: Solve for x

Divide both sides by 8: $x=\frac{96}{8}=12$.

Step5: Find $m\angle1$

Substitute $x = 12$ into the expression for $m\angle1$: $m\angle1=7x=7\times12 = 84^{\circ}$.

Step6: Find $m\angle2$

Substitute $x = 12$ into the expression for $m\angle2$: $m\angle2=x - 6=12-6 = 6^{\circ}$.

Answer:

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