Chapter 12.1, Problem 6C

To determine

The other three angles formed when only one of the four angles formedby the intersection of two lines is given.

Answer to Problem 6C

The measures of the other three angles $\angle 2,\angle 3,\angle 4$ are ${45}^o,{135}^o,{45}^o$ respectively.

Explanation of Solution

Given Information: Measure of $\angle 1$ is ${45}^o$ in the following figure

  

Concept Used:

• When two lines intersect, the angles opposite each other are called ‘vertical angles’ and they have the same measure.
• If two adjacent angles form a straight angle, then the two angles aresupplementary anglesand the sum of their measures is ${180}^o$ .

In the given figure, $\angle 1$ and $\angle 3$ are vertical angles. So, they should have the same measure. Therefore

  $m\angle 3=m\angle 1=135^o$

Also, in the given figure, $\angle 1$ and $\angle 2$ are supplementary angles as they are adjacent and form a straight angle. So, the sum of their measures should be ${180}^o$ .Therefore

  $m\angle 1+m\angle 2={180}^o$

  $\Rightarrow m\angle 2={180}^o-m\angle 1={180}^o-{135}^o=45^o$

Similarly, $\angle 1$ and $\angle 4$ are supplementary angles. Therefore

  $m\angle 1+m\angle 4={180}^o$

  $\Rightarrow m\angle 4={180}^o-m\angle 1={180}^o-{135}^o=45^o$