1. what of the following statements is not always true? (1) all vertical - angle pairs are congruent angle pairs (2) all linear - angle pairs are supplementary angle pairs (3) all supplementary - angle pairs are linear - angle pairs (4) all angles supplementary to the same angle are congruent
2. in the diagram shown, $ \overline{ae} $ and $ \overline{cf} $ intersect at b. which of the following pairs of angles is a linear pair? (1) $ \angle dbc $ and $ \angle cbe $ (2) $ \angle dbf $ and $ \angle dbc $ (3) $ \angle abe $ and $ \angle cbe $ (4) $ \angle dbf $ and $ \angle fbe $
3. it is known that $ \angle efg $ and $ \angle gfh $ are a linear - angle pair. if $ m\angle efg = 146^{circ} $, then which of the following is the measure of $ \angle gfh $? (1) $ 34^{circ} $ (2) $ 73^{circ} $ (3) $ 106^{circ} $ (4) $ 146^{circ} $
4. if $ \angle e=\angle k $, $ m\angle e = 124^{circ} $, and $ m\angle k = 4x - 8 $, then which of the following is the value of x? (1) 7 (2) 12 (3) 24 (4) 33
5. if $ \overline{hi} $ and $ \overline{jk} $ intersect at point l and $ m\angle kij = 48^{circ} $, then which of the following is the measure of $ \angle hlj $ (hint: draw a picture and label correctly.) (1) $ 42^{circ} $ (2) $ 48^{circ} $ (3) $ 96^{circ} $ (4) $ 132^{circ} $

Question

Accepted Answer

# Explanation:
## Step1: Recall angle - pair properties
Vertical angles are always congruent, linear - angle pairs are supplementary, and angles supplementary to the same angle are congruent. Supplementary angle pairs are not always linear - angle pairs. So the statement that is not always true is "all supplementary angle pairs are linear - angle pairs".
## Step2: Identify linear - angle pairs
A linear - angle pair is a pair of adjacent angles whose non - common sides are opposite rays. In the given diagram, $\angle ABE$ and $\angle CBE$ are a linear - angle pair.
## Step3: Use the property of linear - angle pairs
If $\angle EFG$ and $\angle GFH$ are a linear - angle pair, then $m\angle EFG + m\angle GFH=180^{\circ}$. Given $m\angle EFG = 146^{\circ}$, then $m\angle GFH=180^{\circ}-146^{\circ}=34^{\circ}$.
## Step4: Solve for $x$ using angle equality
If $\angle E=\angle K$, and $m\angle E = 124^{\circ}$, $m\angle K = 4x - 8$. Then $4x-8 = 124$. Add 8 to both sides: $4x=124 + 8=132$. Divide both sides by 4: $x = 33$.
## Step5: Use the property of vertical angles
If $\overline{HI}$ and $\overline{JK}$ intersect at point $L$ and $m\angle KIL = 48^{\circ}$, then $\angle HIL$ and $\angle KIL$ are a linear - angle pair. So $m\angle HIL=180^{\circ}-48^{\circ}=132^{\circ}$.

# Answer:
1. (c)
2. (3)
3. (1)
4. (There is an error in the options provided, the correct value of $x$ is 33)
5. (4)

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QUESTION IMAGE

Question

Explanation:

Step1: Recall angle - pair properties

Step2: Identify linear - angle pairs

Step3: Use the property of linear - angle pairs

Step4: Solve for $x$ using angle equality

Step5: Use the property of vertical angles

Answer: