Question

in the diagram below, $overline{dg}perpoverline{df}$. use the diagram for questions 1 - 7.
name the vertex of $angle2$.
give another name for $angle3$.
classify $angle5$.
classify $angle cde$.
if $mangle5 = 42^{circ}$ and $mangle1 = 117^{circ}$, find $mangle cdf$.
if $mangle3 = 73^{circ}$, find $mangle fde$.
in the diagram below, $overline{bc}$ bisects $angle fbe$. use the diagram for questions 8 - 10.
if $mangle abf=(7x + 20)^{circ}$, $mangle fbc=(2x - 5)^{circ}$, and $mangle abc = 159^{circ}$, find the value of $x$.
if $mangle dbc=(12x - 3)^{circ}$, $mangle dbe=(5x + 12)^{circ}$, and $mangle ebc=(3x + 13)^{circ}$, find $mangle ebc$.
if $mangle fbc=(10x - 9)^{circ}$, $mangle cbe=(4x + 15)^{circ}$, find $mangle fbe$.

Explanation:

Step1: Recall vertex definition

The vertex of an angle is the common - point of the two rays that form the angle. For $\angle2$, the vertex is $D$.

Step2: Rename $\angle3$

Since the angle is formed by rays $\overrightarrow{DG}$ and $\overrightarrow{DE}$, another name for $\angle3$ is $\angle GDE$.

Step3: Classify $\angle5$

$\angle5$ is less than $90^{\circ}$, so it is an acute angle.

Step4: Classify $\angle CDE$

$\angle CDE$ is greater than $90^{\circ}$ and less than $180^{\circ}$, so it is an obtuse angle.

Step5: Find $m\angle CDF$

Since $\angle CDF=\angle5 + \angle1$, and $m\angle5 = 42^{\circ}$, $m\angle1 = 117^{\circ}$, then $m\angle CDF=42^{\circ}+117^{\circ}=159^{\circ}$.

Step6: Find $m\angle FDE$

We know that $\overrightarrow{DG}\perp\overrightarrow{DF}$, so $\angle FDG = 90^{\circ}$. If $m\angle3 = 73^{\circ}$, and $\angle FDE+\angle3=\angle FDG$, then $m\angle FDE=90^{\circ}-73^{\circ}=17^{\circ}$.

Step7: Solve for $x$ when $\overline{BC}$ bisects $\angle FBE$

Since $\overline{BC}$ bisects $\angle FBE$, then $m\angle ABF + m\angle FBC=m\angle ABC$. So $(7x + 20)+(2x - 5)=159$. Combine like - terms: $9x+15 = 159$. Subtract 15 from both sides: $9x=159 - 15=144$. Divide both sides by 9: $x = 16$.

Step8: Find $m\angle EBC$

Since $m\angle DBC=m\angle DBE + m\angle EBC$, then $(12x - 3)=(5x + 12)+(3x + 13)$. Combine like - terms: $12x-3=8x + 25$. Subtract $8x$ from both sides: $4x-3=25$. Add 3 to both sides: $4x=28$. Divide both sides by 4: $x = 7$. Then $m\angle EBC=3x + 13=3\times7+13=34^{\circ}$.

Step9: Find $m\angle FBE$

Since $\overline{BC}$ bisects $\angle FBE$, then $m\angle FBC=m\angle CBE$. So $10x-9 = 4x + 15$. Subtract $4x$ from both sides: $6x-9=15$. Add 9 to both sides: $6x=24$. Divide both sides by 6: $x = 4$. Then $m\angle FBE=(10x - 9)+(4x + 15)=14x + 6=14\times4+6=62^{\circ}$.

Answer:

1. D
2. $\angle GDE$
3. Acute angle
4. Obtuse angle
5. $159^{\circ}$
6. $17^{\circ}$
7. $x = 16$
8. $m\angle EBC = 34^{\circ}$
9. $m\angle FBE = 62^{\circ}$