# ABCD is a trep. with AB//DC and AD = BC. If AD is parallel to CE and angle A is 70 degrees.Find all the angles ? Diagram is given below.

Feb 7, 2017

See explanation.

#### Explanation:

Given:

(1) trapezoid $A B C D$
(2) $\overline{A B} \text{ || } \overline{C D}$
(3) $\overline{A D} \cong \overline{B C}$
(4) $\overline{A D} \text{ || } \overline{C E}$
(5) "m"angleA=70°

We can immediately obtain

(6) "m"angle D = 110°

because $\angle A$ and $\angle D$ are co-interior angles, thus their sum is 180°.

(2) & (4) tell us that

(7) $A E C D$ is a parallelogram

and thus

(8) $\overline{A D} \cong \overline{E C}$

From (3) and (8) we get

(9) $\overline{B C} \cong \overline{E C}$
(10) $\triangle E C B \cong \triangle B C E$

and so

(11) $\angle C E B \cong \angle B$

From (4), we get

(12) $\angle A \cong \angle C E B$

This, together with (11) gives

(13) $\angle A \cong \angle B$
(14) "m"angle B =70°

Just as $\angle A$ and $\angle D$ are co-interior angles, $\angle B$ and $\angle B C D$ are co-interior angles, so

(15) "m"angle BCD =110°.

From (11) and (14), we get

(16) "m"angle CEB = 70°

and since $\angle A E C$ and $\angle C E B$ are supplementary,

(17) "m"angle AEC = 110°

$\angle A E C$ and $\angle E C D$ are co-interior, so

(18) "m"angle ECD = 70°

Finally, we use (15) and (18) to get

(19) "m"angle BCE = 40°.