Question

The isosceles trapezoid ABDE is part of an isosceles triangle ACE. Find the measure of the vertex angle of ACE. (See attachment)


A. 130 degrees

B. 60 degrees

C. 65 degrees

D. 50 degrees


I really need an explanation along with the answer, thank you!!

Answer 1

Answer:

We know that $\triangle ACE$ is isosceles, that means $\angle A \cong \angle E$, by definition.

Also, $\angle BDC \cong \angle DBC$, because $BD \parallel AE$.

Then, we have $115\° + \angle BDC = 180\°$, by sumpplementary angles.

$\angle BDC = 180 -115 = 65\° = \angle DBC$

Which means,

$\angle C= 180 - 65 - 65$, by definition.

$\angle C= 50$

Then,

$\angle A + \angle E + 50 = 180\\2\angle A = 180 - 50\\\angle A= \frac{130}{2}=65 = \angle E$

Therefore, the measures of vertex angles are 65 for the base angles of triangle and 50 for the different angle.