Question

∠3 is a complement of ∠4, and m∠3 = 46°. find m∠4.
∠7 is a supplement of ∠8, and m∠7 = 109°. find m∠8.
∠uvw and ∠xyz are complementary angles, m∠uvw=(x - 10)°, and m∠xyz=(4x - 10)°.

Explanation:

Step1: Recall complementary - angle property

Complementary angles add up to 90°. Given $\angle3$ is a complement of $\angle4$ and $m\angle3 = 46^{\circ}$, then $m\angle4=90^{\circ}-m\angle3$.
$m\angle4 = 90^{\circ}-46^{\circ}=44^{\circ}$

Step2: Recall supplementary - angle property

Supplementary angles add up to 180°. Given $\angle7$ is a supplement of $\angle8$ and $m\angle7 = 109^{\circ}$, then $m\angle8=180^{\circ}-m\angle7$.
$m\angle8 = 180^{\circ}-109^{\circ}=71^{\circ}$

Step3: Use complementary - angle relationship for $\angle UVW$ and $\angle XYZ$

Since $\angle UVW$ and $\angle XYZ$ are complementary, $m\angle UVW + m\angle XYZ=90^{\circ}$.
Substitute $m\angle UVW=(x - 10)^{\circ}$ and $m\angle XYZ=(4x - 10)^{\circ}$ into the equation:
$(x - 10)+(4x - 10)=90$.
Combine like - terms: $x+4x-10 - 10 = 90$, $5x-20 = 90$.
Add 20 to both sides: $5x=90 + 20=110$.
Divide both sides by 5: $x = 22$.
Find $m\angle UVW=x - 10=22-10 = 12^{\circ}$ and $m\angle XYZ=4x - 10=4\times22-10=88 - 10 = 78^{\circ}$.

Answer:

$m\angle4 = 44^{\circ}$
$m\angle8 = 71^{\circ}$
$m\angle UVW = 12^{\circ}$, $m\angle XYZ = 78^{\circ}$