Question

in the figure below, (mangle1 = 4x^{circ}) and (mangle2=(x - 5)^{circ}). find the angle measures.

Explanation:

Step1: Set up equation

Since $\angle1$ and $\angle2$ are complementary (they form a right - angle), we have $m\angle1 + m\angle2=90^{\circ}$. Substituting the given expressions, we get $4x+(x - 5)=90$.

Step2: Simplify the left - hand side

Combining like terms in $4x+(x - 5)=90$, we have $(4x+x)-5 = 90$, which simplifies to $5x-5 = 90$.

Step3: Solve for $x$

Add 5 to both sides of the equation $5x-5 = 90$: $5x-5 + 5=90 + 5$, so $5x=95$. Then divide both sides by 5: $x=\frac{95}{5}=19$.

Step4: Find $m\angle1$

Substitute $x = 19$ into the expression for $m\angle1$: $m\angle1=4x=4\times19 = 76^{\circ}$.

Step5: Find $m\angle2$

Substitute $x = 19$ into the expression for $m\angle2$: $m\angle2=x - 5=19-5 = 14^{\circ}$.

Answer:

$m\angle1 = 76^{\circ}$
$m\angle2 = 14^{\circ}$