Chapter 3.4, Problem 15CE

To determine

To prove: If an angle is exterior to a triangle, then its measure equals the sum of the measures of the two remote interior angles by using the equations:

  $m\angle 1+m\angle 2+m\angle 3=180(1)$ ,

  $m\angle 3+m\angle 4=180(2)$

Explanation of Solution

Given information:

  $\Delta ABC$ and its angles.

Proof:

Theorem 3-12:- If an angle is exterior to a triangle, then its measure equals the sum of the measures of the two remote interior angles.

  

As per Theorem, you need to prove that

  $m\angle 1+m\angle 2=m\angle 4\mathrm{....}(3)$

From equation (1)

  $\begin{array}{l}m\angle 1+m\angle 2+m\angle 3=180\\ m\angle 1+m\angle 2=180-m\angle 3\mathrm{.....}(4)\end{array}$

From equation (2),

  $m\angle 3+m\angle 4=180$

  $m\angle 4=180-m\angle 3\mathrm{....}(5)$

Putting the value of equation (5) in equation (4), you get

  $m\angle 1+m\angle 2=m\angle 4$

Hence, the theorem is proved.

Therefore, the given equation can be used to prove Theorem 3-12.