Question

Two straight lines are cut by a transversal. If the measures of two co-interior angles are \[\left( {2a} \right)^\circ \] and \[\left( {3a - 10} \right)^\circ \], find the value of \['a'\].

Answer 1

Hint: Here, in the given question, we have been given that a transversal is cutting two straight lines and measures of two co-interior angles are given in terms of \[a\]. And we are asked to find the value of the \[a\]. We will use the property of interior angles to get the value of \[a\].

Complete step-by-step solution:
Given, two straight lines are cut by a transversal
Assumption: Let us assume that the two straight lines are parallel to each other.
Measures of two co-interior angles are \[\left( {2a} \right)^\circ \] and \[\left( {3a - 10} \right)^\circ \]

Now, using a property of co-interior angles, which states that, if two parallel lines are cut by a transversal, then the two angles on the same side of a transversal will be supplementary or we can say that, the sum of two angles will be \[180^\circ \]
Therefore, the sum of measures of two co-interior angles given will be \[180^\circ \]
\[ \Rightarrow \left( {2a} \right)^\circ + \left( {3a - 10} \right)^\circ = 180^\circ \]
Simplifying it, we get,
\[ 5a - 10 = 180 \]
\[ \Rightarrow a = 38 \]
Hence, the value of \['a'\] is \[38\].

Additional information: Also, the converse of the property holds true. It means if the co-interior angles are supplementary, then the two lines must be parallel.

Note: Here, in the question, it is not clearly mentioned that the two straight lines are parallel or not. So we must assume that the two lines are parallel. Because the property we used in the solution holds true in case of parallel lines only. Otherwise, it would not have been possible to solve for \[a\].