Chapter 3.2, Problem 25WE

To determine

To calculate: measure of $\angle 1$

Answer to Problem 25WE

  $\angle 1=60°$

Explanation of Solution

Given information: $AS\parallel BT$ , $\angle 4=\angle 5$ and $\overrightarrow{SB}$ bisects $\angle AST$

Calculation:

  

First, we will use the given conditions: $AS\parallel BT$ , $\angle 4=\angle 5$ and $\overrightarrow{SB}$ bisects $\angle AST$

  $\angle 4=\angle 5$ (Given) (i)

As $\overrightarrow{SB}$ bisects $\angle AST$ , $\angle 2=\angle 3$ (ii)

Also, $AS\parallel BT$

Take $BS$ as a transversal.

  $\angle 2=\angle 5$ (iii)

(Alternate interior angles are equal)

From (i), (ii) and (iii),

  $\angle 2=\angle 3=\angle 4=\angle 5$

Let $\angle 2=\angle 3=\angle 4=\angle 5=x$

Now find value of $x$

In $\Delta BST$ ,

  $\angle 3+\angle 4+\angle 5=180°$ (Angle sum property of triangle)

Put $\angle 3=\angle 4=\angle 5=x$

  $\begin{array}{l}x+x+x=180°\\ \Rightarrow 3x=180°\\ \Rightarrow x=\frac{180°}{3}\\ =60°\end{array}$

Now find value of $\angle 1$

As $\angle 1,\angle 2,\angle 3$ lie on a straight line,

  $\angle 1+\angle 2+\angle 3=180°$

Put $\angle 2=\angle 3=x$

  $\angle 1+x+x=180°$

Put $x=60°$

  $\begin{array}{l}\angle 1+60°+60°=180°\\ \Rightarrow \angle 1+120°=180°\\ \Rightarrow \angle 1=180°-120°=60°\end{array}$

Therefore,

  $\angle 1=60°$