Question

warm up
complete each exercise.

1. what angles are supplementary to ∠aeb?
2. name a pair of vertical angles.
3. if the measure of ∠aed is 70°, then what is the measure of ∠ceb?
4. what is the measure of ∠ced if the measure of ∠aed is 70°?
5. if two angles are supplementary, are they sometimes, always, or never congruent?

Explanation:

Step1: Recall supplementary - angle definition

Supplementary angles add up to 180°. $\angle AEB+\angle AED = 180^{\circ}$ and $\angle AEB+\angle BEC=180^{\circ}$. So the angles supplementary to $\angle AEB$ are $\angle AED$ and $\angle BEC$.

Step2: Recall vertical - angle definition

Vertical angles are opposite angles formed by two intersecting lines. $\angle AED$ and $\angle BEC$ are vertical angles, and $\angle AEB$ and $\angle DEC$ are vertical angles. We can choose $\angle AED$ and $\angle BEC$.

Step3: Use vertical - angle property

Vertical angles are congruent. Since $\angle AED$ and $\angle BEC$ are vertical angles and $m\angle AED = 70^{\circ}$, then $m\angle BEC=70^{\circ}$.

Step4: Use linear - pair property

$\angle AED$ and $\angle CED$ form a linear - pair. A linear - pair of angles add up to 180°. If $m\angle AED = 70^{\circ}$, then $m\angle CED=180 - 70=110^{\circ}$.

Step5: Analyze supplementary and congruent angles

Two supplementary angles are congruent when each is a right - angle (90°). In general, supplementary angles are not congruent. So two supplementary angles are sometimes congruent.

Answer:

1. $\angle AED$ and $\angle BEC$
2. $\angle AED$ and $\angle BEC$ (or $\angle AEB$ and $\angle DEC$)
3. $70^{\circ}$
4. $110^{\circ}$
5. Sometimes