Chapter 1, Problem 28RP

To determine

To prove: We need to prove that $\angle OKJ\cong \angle OMN$

Explanation of Solution

Given information: In given figure we have $\angle OMK=50°,\angle OKM=(2x)°\text{ , }\angle OKJ=(5x+5)°$

Formula used: Exterior angle theorem, sum of all angles of a triangle and definition of congruency of angles.

Proof: In the given figure we have $\angle OMK=50°,\angle OKM=(2x)°\text{ ,}\angle OKJ=(5x+5)°$ .We know that sum of all the interior angles of a triangle add up to $180°$ .So in the given triangle $\angle OKM+$ $\angle OKM+\angle KOM=180°$ which means $\angle KOM=180°-50°-(2x)°=130°-(2x)°$ .Also $\angle OKJ$ is an exterior angle and according to exterior angle property “ an exterior angle of a triangle is equal to the sum of its non-adjacent interior angles.” So, we have $\angle OKJ=\angle OMK+\angle KOM\Rightarrow$

  $(5x+5)°=50°+130°-(2x)°\Rightarrow (5x)°+5°=180°-(2x)°\Rightarrow (5x)°+(2x)°=180°-5°\Rightarrow (7x)°=175°$

  $\Rightarrow x=\frac{175°}{7}\Rightarrow x=25°$ . Which gives us values of all angles as $\angle \text{OMK=50}°\text{,}\angle \text{OKM=(2x)}°\text{=50}°$ ,

  $\angle \text{OKJ=(5x+5)}°\text{=(5}\times \text{25+5)}°\text{=(125+5)}°\text{=130}°\text{,}\angle \text{KOM=130}°\text{-(2x)}°\text{=130}°\text{-50}°\text{=80}°\text{.}$ Again $\angle OMN$ is an exterior angle for non-adjacent angles $\angle KOM,\angle OKM.$ So, $\angle OMN=\angle KOM+\angle OKM$

  $\Rightarrow \angle OMN=80°+50°\Rightarrow \angle OMN=130°=\angle OKJ.$ As both $\angle OMN=\angle OKJ=130°$ i.e. have same measurement thus using definition of congruency of angles they can be called congruent angles and can be expressed as $\angle OMN\cong \angle OKJ$ .Which proves the required result.

Conclusion:

  $\angle OKJ\cong \angle OMN$