Question

What value of \[x\] will make \[CD\parallel EF\] and \[AB\parallel CD\] ?

A. \[{{150}^{\circ }}\]
B. \[{{145}^{\circ }}\]
C. \[{{140}^{\circ }}\]
D. \[{{135}^{\circ }}\]

Answer 1

Hint: In this question we have to find the value of \[x\] hence apply the alternate interior angles theorem in \[\angle ABC\] and \[\angle BCD\] then apply the theorem for consecutive interior angles so that the pair of consecutive interior angles become supplementary and we can find out the value of \[x\] and find out the correct option.

Complete step by step answer:
Angles come in a variety of shapes and sizes in geometry, depending on how they are measured. Acute angle, obtuse angle, right angle, straight angle, reflex angle, and full rotation are the names of basic angles. A geometrical figure formed by uniting two rays at their ends is called an angle.
Eight angles are generated when two parallel lines are intersected by a transversal. The alternate interior angles are the angles that are on the inner side of the parallel lines but on the opposite sides of the transversal. When a transversal crosses two parallel lines, the pair of angles formed on the inner side of the parallel lines but on opposing sides of the transversal is referred to as alternate interior angles. These angles are always in the same proportion. This can also be interpreted in a different way. The alternate internal angles can be used to determine whether or not the supplied lines are parallel. The supplied lines that are crossed by a transversal are said to be parallel if these angles are equal. Many angles are formed when a transversal crosses any two parallel lines, including alternate interior angles, matching angles, alternate exterior angles, and consecutive interior angles. Consecutive interior angles, also known as co-interior angles or same-side interior angles, are created on the inner sides of the transversal.
According to the given figure we can say that \[\angle ABC\] and \[\angle BCD\] are alternate interior angles hence apply Alternate Interior Angles Theorem:
\[\Rightarrow \angle ABC=\angle BCD={{65}^{\circ }}\]
And we have given that \[\angle BCE={{30}^{\circ }}\]
\[\Rightarrow \angle BCE+\angle ECD={{65}^{\circ }}\]
\[\Rightarrow {{30}^{\circ }}+\angle ECD={{65}^{\circ }}\]
\[\Rightarrow \angle ECD={{65}^{\circ }}-{{30}^{\circ }}\]
\[\Rightarrow \angle ECD={{35}^{\circ }}\]
Now in the given figure \[\angle ECD\] and \[\angle FEC\] are consecutive interior angles, and according to the Consecutive Interior Angles Theorem the pair of consecutive interior angles are supplementary
\[\Rightarrow \angle ECD+\angle FEC={{180}^{\circ }}\]
\[\Rightarrow {{35}^{\circ }}+x={{180}^{\circ }}\]
\[\Rightarrow x={{180}^{\circ }}-{{35}^{\circ }}\]
\[\Rightarrow x={{145}^{\circ }}\]

So, the correct answer is “Option B”.

Note:
When two parallel lines cross with a third, the angles that are in the same relative position at each intersection are referred to as comparable angles to each other. The two lines are considered to be parallel if the corresponding angles in the two intersection zones are congruent.