Chapter 12.1, Problem 53E

To determine

To prove: the triangle sum theorem using the diagram and proof should be different.

Answer to Problem 53E

The triangular sum theory is proved.

Explanation of Solution

Given:

  

Concept used:

The corresponding angles of transversal are equal.

The alternative interior angle of transversal is equal.

Sum of the three interior angles of triangle is always ${180}^0$ .

Linear pair or supplementary angle made by two adjacent angle that is form by intersection of two lines and their sum is ${180}^0$ .

Calculation:

The corresponding angles of transversal are equal.

  $\angle 1\cong \angle 5.$ ........ $\left(i\right)$

The alternative interior angle of transversal is equal.

  $\angle 2\cong \angle 4.$ ........... $\left(ii\right)$

According to the linear pair or supplementary angle sum is ${180}^0$ .

  $m\angle 3+m\angle 4+m\angle 5=180.$

By putting or substituting the angles from equation $\left(i\right)$ and $\left(ii\right)$ :

  $m\angle 1+m\angle 2+m\angle 3=180.$

Hence,triangular sum theory is proved.