Question

geometry
unit 1 test review
lessons 1.5 - 1.7: angle addition postulate and angle relationships
use the diagram to the right to answer the questions.

1. give another name for ∠2.
2. name the sides of ∠cbe.
3. classify ∠3.
4. classify ∠abe.
5. classify ∠abc.
6. if m∠abd = 46 and m∠dbc = 24, what is the measure of m∠abc?
7. if m∠def = 117°, find the value of x.
8. if m∠pqs = 16°, m∠sqr=(9x + 17)°, and m∠pqr=(12x - 6)°, find m∠pqr.

Explanation:

Step1: Recall angle - naming rules

An angle can be named by its vertex letter when there is only one angle at that vertex, or by three - letter naming where the vertex letter is in the middle. For $\angle2$, another name is $\angle ABD$ since $B$ is the vertex and $A$ and $D$ are points on the rays forming the angle.

Step2: Identify angle sides

The sides of $\angle CBE$ are rays $\overrightarrow{BC}$ and $\overrightarrow{BE}$.

Step3: Classify $\angle3$

If $\angle3$ is less than $90^{\circ}$, it is an acute angle.

Step4: Classify $\angle ABE$

If $\angle ABE = 180^{\circ}$, it is a straight angle.

Step5: Classify $\angle ABC$

If $\angle ABC$ is between $0^{\circ}$ and $90^{\circ}$, it is an acute angle.

Step6: Use angle - addition postulate

By the angle - addition postulate, $m\angle ABC=m\angle ABD + m\angle DBC$. So $m\angle ABC=46 + 24=70$.

Step7: Use angle - addition postulate for $\angle DEF$

We know that $m\angle DEG+m\angle GEF=m\angle DEF$. So $(12x + 1)+(5x-3)=117$. Combine like terms: $12x+5x+1 - 3=117$, $17x-2 = 117$. Add 2 to both sides: $17x=117 + 2=119$. Divide both sides by 17: $x = 7$.

Step8: Use angle - addition postulate for $\angle PQR$

Since $m\angle PQS+m\angle SQR=m\angle PQR$, we have $16+(9x + 17)=12x-6$. Combine like terms: $9x+33 = 12x-6$. Subtract $9x$ from both sides: $33=12x-9x-6$. So $33 = 3x-6$. Add 6 to both sides: $3x=33 + 6=39$. Divide both sides by 3: $x = 13$. Then $m\angle PQR=(12x-6)=(12\times13 - 6)=150$.

Answer:

1. $\angle ABD$
2. Rays $\overrightarrow{BC}$ and $\overrightarrow{BE}$
3. Acute angle (assuming its measure is between $0^{\circ}$ and $90^{\circ}$)
4. Straight angle (assuming its measure is $180^{\circ}$)
5. Acute angle (assuming its measure is between $0^{\circ}$ and $90^{\circ}$)
6. $70$
7. $7$
8. $150$