R Chandra Solutions for Chapter: Lines and Angles, Exercise 1: TEST YOUR SKILLS

In the figure, lines $\mathrm{P}\mathrm{Q}\mathrm{}$and $\mathrm{}\mathrm{R}\mathrm{S}$ intersect each other at a point $\mathrm{O}$. If $\mathrm{\angle }\mathrm{P}\mathrm{O}\mathrm{R}:\mathrm{\angle }\mathrm{R}\mathrm{O}\mathrm{Q}=5:7,$ find all the angles.

In the figure, $\mathrm{O}\mathrm{S}$ stands on a line $\mathrm{P}\mathrm{O}\mathrm{Q}$. Ray $\mathrm{O}\mathrm{R}$ and ray $\mathrm{O}\mathrm{T}$ are angle bisectors of $\mathrm{\angle }\mathrm{P}\mathrm{O}\mathrm{S}\hspace{0.17em}\hspace{0.17em}$and $\hspace{0.17em}\hspace{0.17em}\mathrm{\angle }\mathrm{S}\mathrm{O}\mathrm{Q},$ respectively. If $\mathrm{\angle }\mathrm{P}\mathrm{O}\mathrm{S}\hspace{0.17em}=\mathrm{x},\mathrm{}$find $\mathrm{\angle }\mathrm{R}\mathrm{O}\mathrm{T}.$

In the figure, $\mathrm{O}\mathrm{P},\mathrm{}\mathrm{O}\mathrm{Q},\mathrm{}\mathrm{O}\mathrm{R}\mathrm{}$and $\mathrm{}\mathrm{O}\mathrm{S}$ are four rays. Prove that:

$\mathrm{\angle }\mathrm{P}\mathrm{O}\mathrm{Q}\hspace{0.17em}+\hspace{0.17em}\mathrm{\angle }\mathrm{Q}\mathrm{O}\mathrm{R}+\mathrm{\angle }\mathrm{S}\mathrm{O}\mathrm{R}+\mathrm{\angle }\mathrm{P}\mathrm{O}\mathrm{S}=360°$.

Two adjacent angles on a straight line are in the ratio $2\mathrm{}:3$. Find the measures of each one of these angles.

In the following figure, if $\mathrm{A}\mathrm{O}\mathrm{B}$ is a straight line then find the measures of $\mathrm{\angle }\mathrm{A}\mathrm{O}\mathrm{C}$ and  $\mathrm{\angle }\mathrm{B}\mathrm{O}\mathrm{C}.$

If two lines intersect each other, then show that the vertically opposite angles are equal.