QuestionIn the diagram, $\overrightarrow{X Y}$ bisects $\angle W X Z$ . Given $m \angle W X Y = 7x - 7$ and $m \angle Y X Z = 5x + 3$ , find:
a. $x$ and $m \angle W X Y$ .
b. $m \angle Y X Z$ .
c. $m \angle W X Z$ .

Studdy Solution

STEP 1

Assumptions1. $\overrightarrow{X Y}$ bisects $\angle W X Z$ . $m \angle W X Y =7x -7$
3. $m \angle Y X Z =5x +3$

STEP 2

Since $\overrightarrow{X Y}$ bisects $\angle W X Z$ , it means that $m \angle W X Y = m \angle Y X Z$ .

STEP 3

Now, we can set up an equation using the given expressions for $m \angle W X Y$ and $m \angle Y X Z$ .
$7x -7 =5x +3$

STEP 4

To solve for $x$ , we first need to get all terms with $x$ on one side of the equation. We can do this by subtracting $x$ from both sides of the equation.
$7x -x =x -x +3 +7$

STEP 5

implify the equation.
$2x =10$

STEP 6

Now, to solve for $x$ , we divide both sides of the equation by2.
$x = \frac{10}{2}$

STEP 7

Calculate the value of $x$ .
$x =5$

STEP 8

Now that we have the value of $x$ , we can find $m \angle W X Y$ by substituting $x =5$ into the expression for $m \angle W X Y$ .
$m \angle W X Y =7x -7$

STEP 9

Substitute $x =5$ into the equation.
$m \angle W X Y =7(5) -7$

STEP 10

Calculate the value of $m \angle W X Y$ .
$m \angle W X Y =35 -7 =28$

STEP 11

Now, we can find $m \angle Y X Z$ by substituting $x =5$ into the expression for $m \angle Y X Z$ .
$m \angle Y X Z =5x +3$

STEP 12

Substitute $x =5$ into the equation.
$m \angle Y X Z =5(5) +$

STEP 13

Calculate the value of $m \angle Y X Z$ .
$m \angle Y X Z =25 +3 =28$

STEP 14

Finally, we can find $m \angle W X Z$ by adding $m \angle W X Y$ and $m \angle Y X Z$ .
$m \angle W X Z = m \angle W X Y + m \angle Y X Z$

STEP 15

Substitute the values of $m \angle W X Y$ and $m \angle Y X Z$ into the equation.
$m \angle W X Z =28 +28$

STEP 16

Calculate the value of $m \angle W X Z$ .
$m \angle W X Z =56$ a. $x =5$ and $m \angle W X Y =28^{\circ}$ b. $m \angle Y X Z =28^{\circ}$ c. $m \angle W X Z =56^{\circ}$

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QuestionIn the diagram, $\overrightarrow{X Y}$ bisects $\angle W X Z$ . Given $m \angle W X Y = 7x - 7$ and $m \angle Y X Z = 5x + 3$ , find:
a. $x$ and $m \angle W X Y$ .
b. $m \angle Y X Z$ .
c. $m \angle W X Z$ .

Studdy Solution

STEP 1

Assumptions1. $\overrightarrow{X Y}$ bisects $\angle W X Z$ . $m \angle W X Y =7x -7$
3. $m \angle Y X Z =5x +3$

STEP 2

Since $\overrightarrow{X Y}$ bisects $\angle W X Z$ , it means that $m \angle W X Y = m \angle Y X Z$ .

STEP 3

Now, we can set up an equation using the given expressions for $m \angle W X Y$ and $m \angle Y X Z$ .
$7x -7 =5x +3$

STEP 4

To solve for $x$ , we first need to get all terms with $x$ on one side of the equation. We can do this by subtracting $x$ from both sides of the equation.
$7x -x =x -x +3 +7$

STEP 5

implify the equation.
$2x =10$

STEP 6

Now, to solve for $x$ , we divide both sides of the equation by2.
$x = \frac{10}{2}$

STEP 7

Calculate the value of $x$ .
$x =5$

STEP 8

Now that we have the value of $x$ , we can find $m \angle W X Y$ by substituting $x =5$ into the expression for $m \angle W X Y$ .
$m \angle W X Y =7x -7$

STEP 9

Substitute $x =5$ into the equation.
$m \angle W X Y =7(5) -7$

STEP 10

Calculate the value of $m \angle W X Y$ .
$m \angle W X Y =35 -7 =28$

STEP 11

Now, we can find $m \angle Y X Z$ by substituting $x =5$ into the expression for $m \angle Y X Z$ .
$m \angle Y X Z =5x +3$

STEP 12

Substitute $x =5$ into the equation.
$m \angle Y X Z =5(5) +$

STEP 13

Calculate the value of $m \angle Y X Z$ .
$m \angle Y X Z =25 +3 =28$

STEP 14

Finally, we can find $m \angle W X Z$ by adding $m \angle W X Y$ and $m \angle Y X Z$ .
$m \angle W X Z = m \angle W X Y + m \angle Y X Z$

STEP 15

Substitute the values of $m \angle W X Y$ and $m \angle Y X Z$ into the equation.
$m \angle W X Z =28 +28$

STEP 16

Calculate the value of $m \angle W X Z$ .
$m \angle W X Z =56$ a. $x =5$ and $m \angle W X Y =28^{\circ}$ b. $m \angle Y X Z =28^{\circ}$ c. $m \angle W X Z =56^{\circ}$

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Studdy Solution

STEP 1

Assumptions1. XYundefined\overrightarrow{X Y}XY bisects ∠WXZ\angle W X Z∠WXZ . m∠WXY=7x−7m \angle W X Y =7x -7m∠WXY=7x−73. m∠YXZ=5x+3m \angle Y X Z =5x +3m∠YXZ=5x+3

STEP 2

Since XYundefined\overrightarrow{X Y}XY bisects ∠WXZ\angle W X Z∠WXZ, it means that m∠WXY=m∠YXZm \angle W X Y = m \angle Y X Zm∠WXY=m∠YXZ.

STEP 3

Now, we can set up an equation using the given expressions for m∠WXYm \angle W X Ym∠WXY and m∠YXZm \angle Y X Zm∠YXZ.7x−7=5x+37x -7 =5x +37x−7=5x+3

STEP 4

To solve for xxx, we first need to get all terms with xxx on one side of the equation. We can do this by subtracting xxx from both sides of the equation.7x−x=x−x+3+77x -x =x -x +3 +77x−x=x−x+3+7

STEP 5

implify the equation.2x=102x =102x=10

STEP 6

Now, to solve for xxx, we divide both sides of the equation by2.x=102x = \frac{10}{2}x=210​

STEP 7

Calculate the value of xxx.x=5x =5x=5

STEP 8

Now that we have the value of xxx, we can find m∠WXYm \angle W X Ym∠WXY by substituting x=5x =5x=5 into the expression for m∠WXYm \angle W X Ym∠WXY.m∠WXY=7x−7m \angle W X Y =7x -7m∠WXY=7x−7

STEP 9

Substitute x=5x =5x=5 into the equation.m∠WXY=7(5)−7m \angle W X Y =7(5) -7m∠WXY=7(5)−7

STEP 10

Calculate the value of m∠WXYm \angle W X Ym∠WXY.m∠WXY=35−7=28m \angle W X Y =35 -7 =28m∠WXY=35−7=28

STEP 11

Now, we can find m∠YXZm \angle Y X Zm∠YXZ by substituting x=5x =5x=5 into the expression for m∠YXZm \angle Y X Zm∠YXZ.m∠YXZ=5x+3m \angle Y X Z =5x +3m∠YXZ=5x+3

STEP 12

Substitute x=5x =5x=5 into the equation.m∠YXZ=5(5)+m \angle Y X Z =5(5) +m∠YXZ=5(5)+

STEP 13

Calculate the value of m∠YXZm \angle Y X Zm∠YXZ.m∠YXZ=25+3=28m \angle Y X Z =25 +3 =28m∠YXZ=25+3=28

STEP 14

Finally, we can find m∠WXZm \angle W X Zm∠WXZ by adding m∠WXYm \angle W X Ym∠WXY and m∠YXZm \angle Y X Zm∠YXZ.m∠WXZ=m∠WXY+m∠YXZm \angle W X Z = m \angle W X Y + m \angle Y X Zm∠WXZ=m∠WXY+m∠YXZ

STEP 15

Substitute the values of m∠WXYm \angle W X Ym∠WXY and m∠YXZm \angle Y X Zm∠YXZ into the equation.m∠WXZ=28+28m \angle W X Z =28 +28m∠WXZ=28+28

STEP 16

Calculate the value of m∠WXZm \angle W X Zm∠WXZ.m∠WXZ=56m \angle W X Z =56m∠WXZ=56a. x=5x =5x=5 and m∠WXY=28∘m \angle W X Y =28^{\circ}m∠WXY=28∘b. m∠YXZ=28∘m \angle Y X Z =28^{\circ}m∠YXZ=28∘c. m∠WXZ=56∘m \angle W X Z =56^{\circ}m∠WXZ=56∘

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Assumptions1. $\overrightarrow{X Y}$ bisects $\angle W X Z$ . $m \angle W X Y =7x -7$
3. $m \angle Y X Z =5x +3$

Since $\overrightarrow{X Y}$ bisects $\angle W X Z$ , it means that $m \angle W X Y = m \angle Y X Z$ .

Now, we can set up an equation using the given expressions for $m \angle W X Y$ and $m \angle Y X Z$ .
$7x -7 =5x +3$

To solve for $x$ , we first need to get all terms with $x$ on one side of the equation. We can do this by subtracting $x$ from both sides of the equation.
$7x -x =x -x +3 +7$

implify the equation.
$2x =10$

Now, to solve for $x$ , we divide both sides of the equation by2.
$x = \frac{10}{2}$

Calculate the value of $x$ .
$x =5$

Now that we have the value of $x$ , we can find $m \angle W X Y$ by substituting $x =5$ into the expression for $m \angle W X Y$ .
$m \angle W X Y =7x -7$

Substitute $x =5$ into the equation.
$m \angle W X Y =7(5) -7$

Calculate the value of $m \angle W X Y$ .
$m \angle W X Y =35 -7 =28$

Now, we can find $m \angle Y X Z$ by substituting $x =5$ into the expression for $m \angle Y X Z$ .
$m \angle Y X Z =5x +3$

Substitute $x =5$ into the equation.
$m \angle Y X Z =5(5) +$

Calculate the value of $m \angle Y X Z$ .
$m \angle Y X Z =25 +3 =28$

Finally, we can find $m \angle W X Z$ by adding $m \angle W X Y$ and $m \angle Y X Z$ .
$m \angle W X Z = m \angle W X Y + m \angle Y X Z$

Substitute the values of $m \angle W X Y$ and $m \angle Y X Z$ into the equation.
$m \angle W X Z =28 +28$

Calculate the value of $m \angle W X Z$ .
$m \angle W X Z =56$ a. $x =5$ and $m \angle W X Y =28^{\circ}$ b. $m \angle Y X Z =28^{\circ}$ c. $m \angle W X Z =56^{\circ}$