Question

Find the measure of \[\angle {\text{P}}\] and \[\angle {\text{S}}\], if \[\overline {{\text{SP}}} ||\overline {{\text{RQ}}} \] in figure (if you find \[{\text{m}}\angle {\text{R}}\], is there more than one method to find \[{\text{m}}\angle {\text{P}}\]?)

Answer 1

Hint: First we will use the property that the line segment SP is parallel to RQ, the adjacent angles are supplementary, that is, the sum of the angles \[\angle {\text{S}}\] and \[\angle {\text{R}}\] is \[180^\circ \] to find the value of \[\angle {\text{S}}\]. Then we will use the another property that the sum of all four angles of a quadrilateral is 360 degrees and substitute the values of \[\angle {\text{R}}\], \[\angle {\text{S}}\] and \[\angle {\text{Q}}\] in the above equation to find the required value.

Complete step by step solution: We are given that the angle \[\angle {\text{Q}}\] is 130 degrees and angle \[\angle {\text{R}}\] is 90 degrees.

Since we know that the line segment SP is parallel to RQ, the adjacent angles are supplementary, that is, the sum of the angles \[\angle {\text{S}}\] and \[\angle {\text{R}}\] is \[180^\circ \].

Then, we have
\[ \Rightarrow \angle {\text{S}} + \angle {\text{R}} = 180^\circ \]

Substituting the value of \[\angle {\text{R}}\] in the above equation, we get
\[ \Rightarrow \angle {\text{S}} + 90^\circ = 180^\circ \]

Subtracting the above equation by \[90^\circ \] on both sides, we get
\[
   \Rightarrow \angle {\text{S}} + 90^\circ - 90^\circ = 180^\circ - 90^\circ \\
   \Rightarrow \angle {\text{S}} = 90^\circ \\
 \]

We also know that the sum of all four angles of a quadrilateral is 360 degrees, we get
\[ \Rightarrow \angle {\text{S}} + \angle {\text{R}} + \angle {\text{P}} + \angle {\text{Q}} = 360^\circ \]

Substituting the values of \[\angle {\text{R}}\], \[\angle {\text{S}}\] and \[\angle {\text{Q}}\] in the above equation, we get
\[
   \Rightarrow 90^\circ + 90^\circ + \angle {\text{P}} + 130^\circ = 360^\circ \\
   \Rightarrow \angle {\text{P}} + 310^\circ = 360^\circ \\
 \]

Subtracting the above equation by \[310^\circ \] on both sides, we get
\[
   \Rightarrow \angle {\text{P}} + 310^\circ - 310^\circ = 360^\circ - 310^\circ \\
   \Rightarrow \angle {\text{P}} = 50^\circ \\
 \]

Note: In solving these types of questions, you should know that a trapezium is a type of quadrilateral, which has only two parallel sides and the other two sides are non-parallel.
We can also this problem by another method, as we know that the adjacent interior angles are supplementary, so the sum of angles \[\angle {\text{Q}}\] and \[\angle {\text{P}}\] is equal to \[180^\circ \].
Then, we have
\[ \Rightarrow \angle {\text{Q}} + \angle {\text{P = 180}}^\circ \]

Substituting the value of the angle \[\angle {\text{Q}}\] from the diagram in the above equation, we get
\[ \Rightarrow 130^\circ + \angle {\text{P = 180}}^\circ \]

Subtracting the above equation by \[130^\circ \] on both sides, we get
\[
   \Rightarrow 130^\circ + \angle {\text{P}} - 130^\circ {\text{ = 180}}^\circ - 130^\circ \\
   \Rightarrow \angle {\text{P = 50}}^\circ \\
 \]