Question

1. using the diagram below, identify a set of complementary and a set of supplementary angles. then find m∠dfe, m∠bfc, and m∠bfe set of complementary angles: set of supplementary angles:

Explanation:

Step1: Define complementary angles

Complementary angles sum to $90^{\circ}$. Since $\angle DFC = 90^{\circ}$ and $\angle DFC=\angle DFE+\angle CFE$, $\angle DFE$ and $\angle CFE$ are complementary. Given $\angle DFE = 27^{\circ}$.

Step2: Find $\angle BFC$

It is directly given as $39^{\circ}$ in the diagram.

Step3: Find $\angle BFE$

$\angle BFE=\angle BFC+\angle CFE$. Since $\angle CFE = 90^{\circ}-\angle DFE=90 - 27=63^{\circ}$ and $\angle BFC = 39^{\circ}$, then $\angle BFE=39 + 90=129^{\circ}$. Supplementary angles sum to $180^{\circ}$, and $\angle AFB+\angle BFE = 180^{\circ}$ (linear - pair), so $\angle AFB$ and $\angle BFE$ are supplementary.

Answer:

Set of Complementary angles: $\angle DFE$ and $\angle CFE$; $m\angle DFE = 27^{\circ}$, $m\angle BFC=39^{\circ}$, $m\angle BFE = 129^{\circ}$
Set of Supplementary angles: $\angle AFB$ and $\angle BFE$