Question

in the figure below, $m\angle 1 = 5x^{\circ}$ and $m\angle 2 = (x - 6)^{\circ}$. find the angle measures.

Explanation:

Step1: Identify angle relationship

The angles form a right angle, so:
$$m\angle 1 + m\angle 2 = 90^\circ$$

Step2: Substitute given expressions

Replace angle measures with $x$ terms:
$$5x + (x - 6) = 90$$

Step3: Simplify and solve for $x$

Combine like terms and isolate $x$:
$$6x - 6 = 90$$
$$6x = 96$$
$$x = 16$$

Step4: Calculate $m\angle 1$

Substitute $x=16$ into $5x$:
$$m\angle 1 = 5\times16 = 80$$

Step5: Calculate $m\angle 2$

Substitute $x=16$ into $x-6$:
$$m\angle 2 = 16 - 6 = 10$$

Answer:

$m\angle 1 = 80^\circ$
$m\angle 2 = 10^\circ$