Question

In the figure below, AA’ is parallel to CC’. The measure w of $\angle A'AB$ is equal to 135 degrees and the measure z of $\angle C'CB$ is equal to 147 degrees. Find $\angle ABC$ .

A. 100 degrees
B. 78 degrees
C. 80 degrees

Answer 1

Hint: Before solving the question we should be aware of the concept of transversal of parallel lines and the relation between its angles. The sum of alternate interior angles of a transversal is 180 degrees. Here we have two lines AA’ and CC’ parallel to each other. Here we will draw another line BB’ parallel to both of them.

Complete step by step answer:
We have w = 135 degrees = $\angle A'AB$ and z = 147 degrees = $\angle C'CB$ and AA’ is parallel to CC’. We will draw a line BB’ parallel to both AA’ and CC’ in between.

Here we have $\angle A'AB$ as the alternate interior angle to $\angle ABB'$ when AA’ is parallel to BB’.
We also have $\angle C'CB$ as the alternate interior angle to $\angle B'BC$ when CC’ is parallel to BB’.
It is clear from the figure that
$\angle ABC$= $\angle ABB'$+$\angle B'BC$
Since we know that the sum of alternate interior angle is 180 degrees for a transversal, we can write relations as
$\angle A'AB$ + $\angle ABB'$ = 180 degrees
$\angle C'CB$ + $\angle B'BC$ = 180 degrees
So as we know $\angle A'AB$ = 135 degrees, we have
135 degrees + $\angle ABB'$ = 180 degrees
$\angle ABB'$ = 180 degrees – 135 degrees
$\angle ABB'$ = 45 degrees
As we know $\angle C'CB$ = 147 degrees, we have
$\angle C'CB$ + $\angle B'BC$ = 180 degrees
147 degrees + $\angle B'BC$ = 180 degrees
$\angle B'BC$ = 180 degrees – 147 degrees
$\angle B'BC$ = 33 degrees
We had already formed the relation, $\angle ABC$ = $\angle ABB'$ + $\angle B'BC$ , so we have
$\angle ABC$ = (45+33) degrees
$\angle ABC$ = 78 degrees

So, the correct answer is “Option B”.

Note: The sum of alternate interior angles of a transversal is 180 degrees. The consecutive angles of a transversal are equal. The sum of alternate exterior angles is 180 degrees.

Here c, e and a, f are pairs of alternate interior angles. Here b, e and a, g and d, f and c, h are pairs of corresponding angles. Here d, g and b, h are alternate exterior angles.
c + e = a + f = 180 degrees
d + g = b + h = 180 degrees
d = f and b = e and a = g and c = h