Question

in the diagram below, $ad \cong ae$, $ba \parallel ce$, $cb \parallel da$ a $m\angle bcd = 62^\circ$. find $m\angle e$.

step angle reason
1 $m\angle bcd = 62^\circ$ given
$m\angle e = 62^\circ$
select a reason
select a reason
vertical angles
linear pair (or triple)
congruent angles
sum of angles in a triangle
sum of angles in a quadrilateral
base angle of an isosceles triangle
angle bisector definition
corresponding angles
alternate interior angles
alternate exterior angles
same-side interior angles
angle addition postulate
angle subtraction postulate
perpendicular lines form right angles

Explanation:

Step1: Identify parallelogram properties

Since $BA \parallel CE$ and $CB \parallel DA$, $ABCD$ is a parallelogram, so $m\angle BCD = m\angle BAD = 62^\circ$.

Step2: Use parallel line angle relationship

Because $BA \parallel CE$, $\angle BAD$ and $\angle ADE$ are alternate interior angles, so $m\angle ADE = 62^\circ$.

Step3: Use isosceles triangle property

Given $AD \cong AE$, $\triangle ADE$ is isosceles, so base angles $\angle E = \angle ADE$.

Answer:

$m\angle E = 62^\circ$, Reason: Alternate interior angles (and base angle of an isosceles triangle; the key parallel line relationship leading to the congruent angle is alternate interior angles)