Chapter 4.2, Problem 45PPS

(a)

To determine

To draw: Five different triangles.

Explanation of Solution

Given information: The given diagram as follows:

  

Graph: The five possible triangles are drawn below,

  

(b)

To determine

To measure: The sum of all the exterior angle of each triangle.

Answer to Problem 45PPS

The required tabular are as follows.

  $\begin{array}{ccccc}Triangle& \angle 1& \angle 2& \angle 3& Sum\\ 1& 85& 155& 120& 360\\ 2& 90& 135& 135& 360\\ 3& 120& 150& 90& 360\\ 4& 110& 140& 110& 360\\ 5& 120& 120& 120& 360\end{array}$

Explanation of Solution

Given information: The figures of triangles are as follows,

  

The sum of all exterior angles of given triangle are given tabular form.

  $\begin{array}{ccccc}Triangle& \angle 1& \angle 2& \angle 3& Sum\\ 1& 85& 155& 120& 360\\ 2& 90& 135& 135& 360\\ 3& 120& 150& 90& 360\\ 4& 110& 140& 110& 360\\ 5& 120& 120& 120& 360\end{array}$

(c)

To determine

To make: A conjecture about the sum of the exterior angles of a triangle.

Answer to Problem 45PPS

The answer is $360°$ .

Explanation of Solution

Given information: A triangle is given.

The sum of the measure of the exterior angles of a triangle is $360°$ .

(d)

To determine

To find: A sum of the exterior angles of a triangle.

Answer to Problem 45PPS

  $\angle 1+\angle 2+\angle 3=360°$

Explanation of Solution

Given information: The conjecture of Part (c)

The conjecture of part (c) written algebraically.

  $\angle 1+\angle 2+\angle 3=360$

(e)

To determine

To find: A sum of the exterior angles of a triangle.

Answer to Problem 45PPS

  $\angle 1+\angle 2+\angle 3=360°$

Explanation of Solution

Given information: A triangle is given.

By using Exterior Angle theorem,

  $\begin{array}{l}\angle 1=\angle B+\angle C\\ \angle 2=\angle A+\angle C\\ \angle 3=\angle A+\angle C\end{array}$

Now, add all the sides,

  $\angle 1+\angle 2+\angle 3=2\left(\angle A+\angle B+\angle C\right)$

Now, apply the Triangle Angle Sum Theorem,

As we know that $\angle A+\angle B+\angle C=180$

Therefore,

  $\begin{array}{l}\angle 1+\angle 2+\angle 3=2\left(180\right)\\ \angle 1+\angle 2+\angle 3=360\end{array}$

Hence, proved.