Chapter 1.5, Problem 4PSA

a.

To determine

To Find: The value of $\angle XTZ$

Answer to Problem 4PSA

  $\angle XTZ=30°$

Explanation of Solution

Given:

  

  $\overrightarrow{TZ}\text{bisects}\angle XTY$

Also, $\angle XTY=60°$

Concept Used:

Bisector bisect the angle in two equal parts of equal measure.

Calculation:

As here $\angle XTY=60°$

  $\begin{array}{l}\Rightarrow \angle XTZ=\angle YTZ=\frac{1}{2}\left(\angle XTY\right)=\frac{1}{2}\times 60°=30°\\ \therefore \angle XTZ=30°\end{array}$

Conclusion:

  $\angle XTZ=30°$

b.

To determine

To Evaluate: The value of $\angle XTZ$

Answer to Problem 4PSA

  $\angle XTZ=24°25^{\prime }$

Explanation of Solution

Given:

  

  $\overrightarrow{TZ}\text{bisects}\angle XTY$

Also, $\angle XTY=48°50^{\prime }$

Concept Used:

Bisector bisect the angle in two equal parts of equal measure.

Calculation:

As here $\angle XTY=48°50^{\prime }$

  $\begin{array}{l}\Rightarrow \angle XTZ=\angle YTZ=\frac{1}{2}\left(\angle XTY\right)=\frac{1}{2}\times 48°50^{\prime }=24°25^{\prime }\\ \therefore \angle XTZ=24°25^{\prime }\end{array}$

Conclusion:

  $\angle XTZ=24°25^{\prime }$

c.

To determine

To Find: The value of $\angle XTZ$

Answer to Problem 4PSA

  $\angle XTZ=\left(18\frac{1}{4}\right)°$

Explanation of Solution

Given:

  

  $\overrightarrow{TZ}\text{bisects}\angle XTY$

Also, $\angle XTY=\left(36\frac{1}{2}\right)°$

Concept Used:

Bisector bisect the angle in two equal parts of equal measure.

Calculation:

As here $\angle XTY=\left(36\frac{1}{2}\right)°$

  $\begin{array}{l}\Rightarrow \angle XTZ=\angle YTZ=\frac{1}{2}\left(\angle XTY\right)=\frac{1}{2}\times \left(36\frac{1}{2}\right)°=\frac{1}{2}\times \left(36+\frac{1}{2}\right)°=18°+\left(\frac{1}{4}\right)°\\ \therefore \angle XTZ=\left(18\frac{1}{4}\right)°\end{array}$

Conclusion:

  $\angle XTZ=\left(18\frac{1}{4}\right)°$

d.

To determine

To Evaluate: The value of $\angle XTZ$

Answer to Problem 4PSA

  $\angle XTZ=42°40^{\prime }$

Explanation of Solution

Given:

  

  $\overrightarrow{TZ}\text{bisects}\angle XTY$

Also, $\angle XTY=85°74^{\prime }$

Concept Used:

Bisector bisect the angle in two equal parts of equal measure.

Calculation:

As here $\angle XTY=85°74^{\prime }$

  $\begin{array}{l}\Rightarrow \angle XTZ=\angle YTZ\\ =\frac{1}{2}\left(\angle XTY\right)\\ =\frac{1}{2}\times 85°74^{\prime }\\ =42.5°37^{\prime }\\ =42°+0.5°+37^{\prime }\\ =42°+0.5\times 6^{\prime }+37^{\prime }\\ =42°+3^{\prime }+37^{\prime }\\ =42°+40^{\prime }\\ \therefore \angle XTZ=42°40^{\prime }\end{array}$

Conclusion:

  $\angle XTZ=42°40^{\prime }$