Question

1. in the figure, \\(\overrightarrow{yw}\\) bisects \\(\angle xwz\\). fill in the following blanks. (lesson 8.2) (2 points)

a. \\(m\angle xwy = m\angle \underline{\qquad}\\)

b. if \\(xy = 9\text{ cm}\\), then \\(zy = \underline{\qquad}\\)

c. solve for \\(x\\). once you solve for \\(x\\), take a look at the four student solutions below. only one of the student solutions shown below is correct. circle the student whose work is correct.

\

$$\begin{array}{|l|l|l|l|} \\hline \\textbf{joshs solution:} & \\textbf{alys solution:} & \\textbf{karis solution:} & \\textbf{vivians solution:} \\\\ 7x - 20 - 2x - 5 = 180 & 7x - 20 = 2x - 5 & 7x - 20 + 2x - 5 = 90 & 7x - 20 = 2x - 5 \\\\ 5x - 25 = 180 & 5x - 20 = -5 & 9x - 25 = 90 & 9x = -25 \\\\ 5x = 205 & 5x = 15 & 9x = 115 & x = -\\frac{25}{9} \\\\ x = \\frac{205}{5} & x = \\frac{15}{5} & x = \\frac{115}{9} & x = 2.78 \\\\ x = 41 & x = 3 & x = 12.78 & \\\\ \\hline \\end{array}$$

Explanation:

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<reasoning>

Identify bisected angles and segment lengths

\[

$$\begin{aligned} &\text{Given: } \overrightarrow{YW} \text{ bisects } \angle XWZ \implies m\angle XWY = m\angle ZWY\\ &Y \text{ lies on the bisector of } \angle XWZ \text{ with } YX \perp WX \text{ and } YZ \perp WZ\\ &\text{By the Angle Bisector Theorem: } YX = YZ\\ &\text{Given: } XY = 9\text{ cm} \implies ZY = 9\text{ cm} \end{aligned}$$

\]

Set up and solve the equation for x

\[

$$\begin{aligned} &WX = WZ \implies 7x - 20 = 2x - 5\\ &5x - 20 = -5\\ &5x = 15\\ &x = 3 \end{aligned}$$

\]

Identify the correct student solution

\[

$$\begin{aligned} &\text{Aly's solution steps:}\\ &7x - 20 = 2x - 5\\ &5x - 20 = -5\\ &5x = 15\\ &x = \frac{15}{5} = 3\\ &\text{Therefore, Aly's solution is correct.} \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

a. \(m\angle XWY = m\angle\) <blank>\(ZWY\)</blank>

Question 2

b. If \(XY = 9\text{ cm}\), then \(ZY =\) <blank>\(9\text{ cm}\)</blank>

Question 3

c. Solve for \(x\). Once you solve for \(x\), take a look at the four student solutions below. Only one of the student solutions shown below is correct. Circle the student whose work is correct.
The student whose work is correct is <blank>Aly</blank>.
</answer>

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Answer:

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<reasoning>

Identify bisected angles and segment lengths

\[

$$\begin{aligned} &\text{Given: } \overrightarrow{YW} \text{ bisects } \angle XWZ \implies m\angle XWY = m\angle ZWY\\ &Y \text{ lies on the bisector of } \angle XWZ \text{ with } YX \perp WX \text{ and } YZ \perp WZ\\ &\text{By the Angle Bisector Theorem: } YX = YZ\\ &\text{Given: } XY = 9\text{ cm} \implies ZY = 9\text{ cm} \end{aligned}$$

\]

Set up and solve the equation for x

\[

$$\begin{aligned} &WX = WZ \implies 7x - 20 = 2x - 5\\ &5x - 20 = -5\\ &5x = 15\\ &x = 3 \end{aligned}$$

\]

Identify the correct student solution

\[

$$\begin{aligned} &\text{Aly's solution steps:}\\ &7x - 20 = 2x - 5\\ &5x - 20 = -5\\ &5x = 15\\ &x = \frac{15}{5} = 3\\ &\text{Therefore, Aly's solution is correct.} \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

a. \(m\angle XWY = m\angle\) <blank>\(ZWY\)</blank>

Question 2

b. If \(XY = 9\text{ cm}\), then \(ZY =\) <blank>\(9\text{ cm}\)</blank>

Question 3

c. Solve for \(x\). Once you solve for \(x\), take a look at the four student solutions below. Only one of the student solutions shown below is correct. Circle the student whose work is correct.
The student whose work is correct is <blank>Aly</blank>.
</answer>

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