Question

in the diagram below, dg ⊥ df. use the diagram for questions 1 - 7. 1. name the sides of ∠4. 2. name the vertex of ∠2. 3. give another name for ∠3. 4. classify ∠5. 5. classify ∠cde. 6. if m∠5 = 42° and m∠1 = 117°, find m∠cdf. 7. if m∠3 = 73°, m∠fde. in the diagram below, bc bisects ∠fbe. use the diagram for questions 8 - 10. 8. if m∠abf=(7x + 20)°, m∠fbc=(2x - 5)°, and m∠abc = 159°, find the value of x.

Explanation:

Step1: Recall angle - naming convention

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

Step2: Recall angle - naming by three - point system

An angle can be named using three points with the vertex in the middle. Another name for $\angle3$ is $\angle GDF$ as the rays forming the angle are $\overrightarrow{DG}$ and $\overrightarrow{DF}$ with $D$ as the vertex.

Step3: Classify $\angle5$

An acute angle is an angle whose measure is between $0^{\circ}$ and $90^{\circ}$. Since no information about its measure is given in a way to suggest otherwise, and based on the general appearance in the diagram, $\angle5$ is an acute angle.

Step4: Classify $\angle CDE$

A right - angle is an angle that measures $90^{\circ}$. There is no indication (such as a right - angle symbol) that $\angle CDE = 90^{\circ}$, and based on the general appearance in the diagram, it is an obtuse angle (an angle whose measure is between $90^{\circ}$ and $180^{\circ}$).

Step5: Use angle - addition property

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

Step6: Use angle - measure relationship

Given $m\angle3 = 73^{\circ}$, and since $\angle FDE$ and $\angle3$ are vertical angles (opposite angles formed by two intersecting lines), vertical angles are congruent. So $m\angle FDE = 73^{\circ}$.

Step7: Use angle - bisector property

If $BC$ bisects $\angle FBE$, then $\angle FBC=\angle CBE$. Also, $m\angle ABF+m\angle FBC=m\angle ABC$. So $(7x + 20)+(2x - 5)=159$.

Step8: Solve the equation for $x$

First, combine like terms: $7x+2x+20 - 5 = 159$, which simplifies to $9x+15 = 159$. Then subtract 15 from both sides: $9x=159 - 15=144$. Divide both sides by 9: $x = 16$.

Answer:

1. Sides of $\angle4$: $\overrightarrow{DG}$ and $\overrightarrow{DC}$
2. Vertex of $\angle2$: $D$
3. Another name for $\angle3$: $\angle GDF$
4. Classification of $\angle5$: Acute angle
5. Classification of $\angle CDE$: Obtuse angle
6. $m\angle CDF$: $159^{\circ}$
7. $m\angle FDE$: $73^{\circ}$
8. $x$: $16$