Question

if $overline{de}paralleloverline{bc},angle bcongangle c$, and $mangle ade = 60^{circ}$, prove that the sum of the measures of the interior angles of the trapezoid $bdec$ is $360^{circ}$. statements reasons 1. $overline{de}paralleloverline{bc},angle bcongangle c$, and $mangle ade = 60^{circ}$ 1. 2. $angle adecongangle b$ 2. 3. $mangle b = 60^{circ}$ 3. 4. $mangle b = mangle c$ 4. 5. $mangle c = 60^{circ}$ 5. 6. $angle c$ is supplementary to $angle dec$. 6. 7. $mangle dec + mangle c = 180^{circ}$ 7. 8. $mangle ade + mangle bde = 180^{circ}$ 8. 9. $mangle bde = 120^{circ}$ 9. 10. sum of the measures of the interior angles of $bdec=mangle bde+(mangle dec + mangle c)+mangle b$ 10. 11. sum of the measures of the interior angles of $bdec = 120^{circ}+180^{circ}+60^{circ}=360^{circ}$ 11.

Explanation:

Step1: Given information

Given $\overline{DE}\parallel\overline{BC}$, $\angle B\cong\angle C$, and $m\angle ADE = 60^{\circ}$

Step2: Corresponding - angles property

Since $\overline{DE}\parallel\overline{BC}$, $\angle ADE\cong\angle B$ (corresponding angles are congruent when two parallel lines are cut by a transversal)

Step3: Substitution

If $m\angle ADE = 60^{\circ}$ and $\angle ADE\cong\angle B$, then $m\angle B=60^{\circ}$

Step4: Given equality

Given $\angle B\cong\angle C$, so $m\angle B = m\angle C$

Step5: Substitution

Since $m\angle B = 60^{\circ}$ and $m\angle B = m\angle C$, then $m\angle C = 60^{\circ}$

Step6: Supplementary - angles property

Because $\overline{DE}\parallel\overline{BC}$, $\angle C$ is supplementary to $\angle DEC$ (same - side interior angles are supplementary when two parallel lines are cut by a transversal)

Step7: Definition of supplementary angles

If $\angle C$ is supplementary to $\angle DEC$, then $m\angle DEC + m\angle C=180^{\circ}$

Step8: Linear - pair property

$\angle ADE$ and $\angle BDE$ form a linear pair, so $m\angle ADE + m\angle BDE = 180^{\circ}$

Step9: Subtraction

If $m\angle ADE = 60^{\circ}$ and $m\angle ADE + m\angle BDE = 180^{\circ}$, then $m\angle BDE=180 - 60=120^{\circ}$

Step10: Angle - sum formula for a quadrilateral

The sum of the measures of the interior angles of trapezoid $BDEC$ is $m\angle BDE+(m\angle DEC + m\angle C)+m\angle B$

Step11: Substitution

Substitute $m\angle BDE = 120^{\circ}$, $m\angle DEC + m\angle C = 180^{\circ}$, and $m\angle B = 60^{\circ}$ into the above formula: $120^{\circ}+180^{\circ}+60^{\circ}=360^{\circ}$

Answer:

The sum of the measures of the interior angles of trapezoid $BDEC$ is $360^{\circ}$