Question

1. in the diagram below ∠bad is a straight angle.

a. describe the angle relationship.
b. write an equation for the angle relationship and solve for x
c. check your solution
d. find the measure of angle ∠cad

1. in the diagram below ∠bfc is a straight angle. write an equation and solve for x to find the measure of ∠dfe.

Explanation:

Step1: (For 1a) Identify angle relationship

$\angle BAC$, $\angle CAD$, and $\angle BAD$ form a straight angle, so they are supplementary (sum to $180^\circ$).

Step2: (For 1b) Set up angle sum equation

$x + 132 = 180$

Step3: (For 1b) Solve for $x$

$x = 180 - 132$
$x = 48$

Step4: (For 1c) Verify the solution

Substitute $x=48$ into the equation: $48 + 132 = 180$, which is true.

Step5: (For 1d) Find $\angle CAD$

$\angle CAD$ is given as $132^\circ$.

Step6: (For 2) Set up straight angle equation

$3x + x + 2x = 180$

Step7: (For 2) Simplify and solve for $x$

$6x = 180$
$x = \frac{180}{6} = 30$

Step8: (For 2) Find $\angle DFE$

$\angle DFE = x = 30^\circ$

Answer:

1a. The angles are supplementary (sum to $180^\circ$).
1b. $x = 48$
1c. $48 + 132 = 180$, which confirms the solution is correct.
1d. $\angle CAD = 132^\circ$

1. $x = 30$, $\angle DFE = 30^\circ$