Question

1. if the lines $overline{ef}$ and $overline{cd}$ intersect at point $m$, then which of the following would be a pair of vertical (1) $angle fmd$ and $angle dme$ (2) $angle cmf$ and $angle dme$ (3) $angle emc$ and $angle cmf$ (4) $angle cmd$ and $angle emf$ 7. in the diagram below, points $e$, $f$, and $g$ are collinear. if $overline{fh}$ bisects $angle efi$ and $mangle ifg = 38^{circ}$, then which of the following is the measure of $angle hfg$? (1) $71^{circ}$ (2) $98^{circ}$ (3) $109^{circ}$ (4) $142^{circ}$ 8. in the diagram, $overline{mn}$ bisects $overline{op}$. which of the following does not have to be true? (1) $overline{mn}$ is perpendicular to $overline{op}$ (2) $q$ is the mid - point of $overline{op}$ (3) $overline{oq}congoverline{pq}$ (4) $angle oqmcongangle pqn$ 9. if $angle ehf$ and $angle fhi$ form a linear pair and $angle fhi$ and $angle ihg$ also form a linear pair, then which of the following must be true about $angle ehf$ and $angle ihg$? (1) they are supplementary. (2) they are complementary. (3) they are congruent. (4) they are both right angles. 10. in the diagram below, points $k$, $l$, and $m$ are collinear with $kl = 34$ cm and $km = 3(x + 1)$. for what value of $x$ below will $m$ be the mid - point of $overline{kl}$? (1) $4\frac{2}{3}$ (2) $5\frac{1}{2}$ (3) $8\frac{1}{4}$ (4) $10\frac{1}{3}$

Explanation:

Step1: Recall vertical - angles definition

Vertical angles are formed when two lines intersect, and they are opposite each other. When lines $\overline{EF}$ and $\overline{CD}$ intersect at point $M$, $\angle CMF$ and $\angle DME$ are opposite each other.

Step2: Analyze other options

• Option (1): $\angle FMD$ and $\angle DME$ are adjacent angles.
• Option (3): $\angle EMC$ and $\angle CMF$ are adjacent angles.
• Option (4): $\angle CMD$ and $\angle EMF$ are not vertical - angles.

Step3: Recall angle - bisector and angle - addition properties

Since $\overline{FH}$ bisects $\angle EFI$, let $\angle EFH=\angle HFI = y$. Given $\angle IFG = 38^{\circ}$, and $\angle EFG$ is a straight - angle ($180^{\circ}$), so $\angle EFI=180^{\circ}-\angle IFG = 180 - 38=142^{\circ}$. Then $y=\frac{1}{2}\angle EFI = 71^{\circ}$, and $\angle HFG=\angle HFI+\angle IFG=71 + 38=109^{\circ}$.

Step4: Recall line - bisector properties

If $\overline{MN}$ bisects $\overline{OP}$, it means that $Q$ is the mid - point of $\overline{OP}$, so $\overline{OQ}\cong\overline{PQ}$ and $\angle OQM\cong\angle PQN$ (vertically opposite angles). But $\overline{MN}$ does not have to be perpendicular to $\overline{OP}$ just because it bisects it.

Step5: Recall linear - pair properties

If $\angle EHF$ and $\angle FHI$ form a linear pair, then $\angle EHF+\angle FHI = 180^{\circ}$. If $\angle FHI$ and $\angle IHG$ form a linear pair, then $\angle FHI+\angle IHG = 180^{\circ}$. So $\angle EHF=\angle IHG$ (congruent).

Step6: Recall mid - point property

If $M$ is the mid - point of $\overline{KL}$, then $KM=\frac{1}{2}KL$. Given $KL = 34$ cm and $KM = 3(x + 1)$, we set up the equation $3(x + 1)=\frac{34}{2}$. Simplify the right - hand side: $\frac{34}{2}=17$. Then solve the equation $3(x + 1)=17$, $3x+3 = 17$, $3x=17 - 3=14$, $x=\frac{14}{3}=4\frac{2}{3}$.

Answer:

1. (2) $\angle CMF$ and $\angle DME$
2. (3) $109^{\circ}$
3. (1) $\overline{MN}$ is perpendicular to $\overline{OP}$
4. (3) They are congruent.
5. (1) $4\frac{2}{3}$