Question

in the figure below, (mangle1=(x - 12)^{circ}) and (mangle2 = 5x^{circ}). find the angle measures.

Explanation:

Step1: Set up equation

Since $\angle1$ and $\angle2$ are complementary (the angle formed by the two - ray is a right - angle, so $m\angle1 + m\angle2=90^{\circ}$), we have the equation $(x - 12)+5x=90$.

Step2: Combine like terms

Combining the $x$ terms on the left - hand side gives $x+5x-12 = 90$, which simplifies to $6x-12 = 90$.

Step3: Add 12 to both sides

Adding 12 to both sides of the equation $6x-12 = 90$ gives $6x=90 + 12$, so $6x=102$.

Step4: Solve for x

Dividing both sides of $6x=102$ by 6, we get $x=\frac{102}{6}=17$.

Step5: Find $m\angle1$

Substitute $x = 17$ into the expression for $m\angle1$: $m\angle1=(x - 12)^{\circ}=(17-12)^{\circ}=5^{\circ}$.

Step6: Find $m\angle2$

Substitute $x = 17$ into the expression for $m\angle2$: $m\angle2=5x^{\circ}=5\times17^{\circ}=85^{\circ}$.

Answer:

$m\angle1 = 5^{\circ}$
$m\angle2 = 85^{\circ}$