Chapter 4.6, Problem 44PPS

(a)

To determine

To sketch: Three different isosceles triangles will be drawn,

Explanation of Solution

Given information: Following figure of the triangle has been given.

  

Sketch: Three possible isosceles triangles are drawn below,

  

(b)

To determine

To find: The measure of each angle that is shown in the diagram.

Explanation of Solution

Given information: Following figure of the triangle has been given.

  

  

Calculation: The measuring of angle is done with the help of protractor are given in following table.

Using the triangles of part (a)

  $\begin{array}{ccccc}Triangle& \angle 1& \angle 3& \angle 4& \angle 5\\ 1& 120& 60& 60& 60\\ 2& 130& 80& 50& 50\\ 3& 150& 120& 30& 30\end{array}$

  $\begin{array}{ccccc}Triangle& \angle 2& \angle 3& \angle 4& \angle 5\\ 1& 120& 60& 60& 60\\ 2& 130& 80& 50& 50\\ 3& 60& 120& 30& 30\end{array}$

(c)

To determine

To explain: The way in which we find the measures of $\angle 3$ , $\angle 4$ and $\angle 5$ by using $\angle 1$ . Also tell for $\angle 2$ .

Explanation of Solution

Given information: Following figure of the triangle has been given.

  

Calculation: Using$\angle 1$:

  $\angle 1$ and $\angle 5$ are a linear pair, so they are supplementary,

Hence, $\angle 5=180-\angle 1$ .

Since the triangle is isosceles, then $\angle 4\cong \angle 5$

In the last, we use Triangle Angle-Sum theorem to find $\angle 3$ ,

  $\angle 3=\angle 4-\angle 5$

Using$\angle 2$ :

  $\angle 2$ and $\angle 3$ are a linear pair, so they are supplementary,

Hence, $\angle 3=180-\angle 2$ .

By using Exterior Angle Theorem, $\angle 2=\angle 4+\angle 5$

Since $\angle 4=\angle 5$ ,

Hence, $\angle 2$ is two times the measure of $\angle 4\&\angle 5$

(d)

To determine

To find: The expression for the measures of $\angle 3$ , $\angle 4$ and $\angle 5$ by using $\angle 1$ and $\angle 2$ .

Explanation of Solution

Given information: Following figure of the triangle has been given.

  

Calculation: Let $\angle 1=x$:

  $\begin{array}{l}\therefore \angle 5=180-x\\ \therefore \angle 4=180-x\end{array}$

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

Similarly, let $\angle 2=x$:

  $\begin{array}{l}\angle 3=180-x\\ \angle 4=\frac{x}{2}\\ \angle 2=\frac{x}{2}\end{array}$