Parallel lines are cut by a transversal such that the alternate interior angles have measures of \[3x + 17\] and \[x + 53\] degrees. How do you find the value of \[x\]?

Hint: To solve the question, we will first draw a rough figure. Then we will use the property that when two parallel lines are cut by transversal the alternate interior angles are equal. Using this property, we will equate the given two angles. From this equation we will find the measure of \[x\].

Complete step by step answer:

From the figure we can see that the given angles \[3x + 17,x + 53\] are alternate interior angles. Now, as we know that the alternate interior angles are equal. So, we will equate the given two angles. So, we have;
\[ \Rightarrow 3x + {17^ \circ } = x + {53^ \circ }\]
On shifting we get;
\[ \Rightarrow 3x - x = {53^ \circ } - {17^ \circ }\]
On subtracting we get;
\[ \Rightarrow 2x = {36^ \circ }\]
Dividing both sides by two we get;
\[ \Rightarrow \dfrac{{2x}}{2} = \dfrac{{{{36}^ \circ }}}{2}\]
Solving we get;
\[ \therefore x = {18^ \circ }\]

Hence \[x = {18^ \circ }\].

Additional information: In geometry, a transversal is a line that intersects two or more than two lines. When two or more lines are cut by the transversal, the angles at the same relative positions are called corresponding angles. The pair of angles inside the two lines on one side of the transversal are called the consecutive interior angles.

Note: We can also solve this question by using the concept of corresponding angles.

From the figure angle \[1\] can be determined using the linear pair concept.
So, we have;
\[\angle 1 + 3x + {17^ \circ } = {180^ \circ }\]
On shifting we get;
\[ \Rightarrow \angle 1 = {180^ \circ } - 3x - {17^ \circ }\]
Similarly, by the linear pair concept;
\[\angle 2 + x + {53^ \circ } = {180^ \circ }\]
On shifting we get;
\[ \Rightarrow \angle 2 = {180^ \circ } - x - {53^ \circ }\]
Now \[\angle 2\], \[\angle 1\] will be equal because they are corresponding angles. So, equating we get;
\[ \Rightarrow {180^ \circ } - 3x - {17^ \circ } = {180^ \circ } - x - {53^ \circ }\]
On solving we get;
\[ \Rightarrow {53^ \circ } - {17^ \circ } = 2x\]
\[ \Rightarrow 2x = {36^ \circ }\]
Dividing by two we get;
\[ \Rightarrow x = {18^ \circ }\]