Question

1. ∠uvw and ∠xyz are complementary angles, ( mangle uvw = (x - 10)^circ ), and ( mangle xyz = (4x - 10)^circ ).

Explanation:

Step1: Define complementary angles sum

Complementary angles sum to $90^\circ$, so:
$$(x - 10) + (4x - 10) = 90$$

Step2: Simplify the equation

Combine like terms:
$$5x - 20 = 90$$

Step3: Isolate the variable term

Add 20 to both sides:
$$5x = 110$$

Step4: Solve for $x$

Divide by 5:
$$x = \frac{110}{5} = 22$$

Step5: Calculate each angle's measure

For $m\angle UVW$:
$$m\angle UVW = 22 - 10 = 12^\circ$$
For $m\angle XYZ$:
$$m\angle XYZ = 4(22) - 10 = 78^\circ$$

Answer:

$x=22$, $m\angle UVW=12^\circ$, $m\angle XYZ=78^\circ$