Question

In the figure below, lines XY and MN intersect at O. If \[\angle POY={{90}^{\circ }}\] and \[a:b=2:3\], find c?

Answer 1

Hint: To solve the given problem, we should know some of the properties of geometry. The first property we should know states that the sum of all angles made on a line is 180 degrees or that the angles made on one side of the line by other lines are supplementary. We will use this property and basic addition to find the value of c.

Complete step-by-step solution:
In the above figure, we are given that the \[a:b=2:3\]. Let the constant of proportionality is k, so we can also write it as \[a=2k\And b=3k\]. We are also given measure of one angle as \[\angle POY={{90}^{\circ }}\]. Using the property that angles on one side of a line are supplementary, we can write the equation as
\[a+b+{{90}^{\circ }}={{180}^{\circ }}\]
Substituting the values of a and b, we get
\[\Rightarrow 2k+3k={{90}^{\circ }}\]
Simplifying the equation, we get
\[\Rightarrow k={{18}^{\circ }}\]
Hence, the values of a and b are \[{{36}^{\circ }}\And {{54}^{\circ }}\] respectively.
As \[\angle MOX\And \angle NOY\] are vertically opposite angles, their measure will be equal. Hence, \[\angle MOX=\angle NOY=b={{54}^{\circ }}\].
Sum of all angels should be 360 degrees, using this property we can write the following equation
\[\Rightarrow a+b+{{90}^{\circ }}+b+c={{360}^{\circ }}\]
Substituting the values of a and b, we get
\[\Rightarrow {{234}^{\circ }}+c={{360}^{\circ }}\]
Solving the above equation, we get
\[\Rightarrow c={{126}^{\circ }}\]

Note: To solve these types of questions, we need to know the basic properties of angles. The properties we used here are as follows:
The first property is of the supplementary angles made by a line, and the other property is the measure of the vertically opposite angles. Calculation mistakes while solving these types of questions should be avoided.