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 angles? (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 = 18^{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{1}{3}) (2) (5\frac{1}{2}) (3) (8\frac{1}{3}) (4) (10\frac{1}{2})

Explanation:

Step1: Recall vertical - angle definition

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

Step2: Analyze other options

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

Step1: Use angle - bisector property

Since $\overrightarrow{FH}$ bisects $\angle EFI$ and $m\angle IFG = 38^{\circ}$, and $\angle EFI+\angle IFG = 180^{\circ}$ (linear - pair of angles), and $\angle EFI = 2\angle HFI$. First, find $\angle EFI=180 - 38=142^{\circ}$. Then, since $\overrightarrow{FH}$ bisects $\angle EFI$, $\angle HFI=\frac{1}{2}\angle EFI = 71^{\circ}$. And $\angle HFG=\angle HFI+\angle IFG$.

Step2: Calculate $\angle HFG$

Substitute the values: $\angle HFG=71 + 38=109^{\circ}$

Step1: Recall the definition of a bisector

If $\overline{MN}$ bisects $\overline{OP}$, then the point of intersection $Q$ is the mid - point of $\overline{OP}$, so $OQ = PQ$ (i.e., $\overline{OQ}\cong\overline{PQ}$) and $Q$ is the mid - point of $\overline{OP}$. But just because $\overline{MN}$ bisects $\overline{OP}$ does not mean $\overline{MN}$ is perpendicular to $\overline{OP}$.

Step2: Analyze each option

• Option (1): $\overline{MN}$ being perpendicular to $\overline{OP}$ is not a necessary condition for $\overline{MN}$ to bisect $\overline{OP}$.
• Option (2): $Q$ is the mid - point of $\overline{OP}$ by the definition of a bisector.
• Option (3): $\overline{OQ}\cong\overline{PQ}$ since $Q$ is the mid - point of $\overline{OP}$.
• Option (4): $\angle OQM\cong\angle PQN$ because they are vertical angles.

Step1: Use 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, $\angle EHF=\angle IHG$ (by the property of linear pairs and substitution). They are congruent.

Step2: Analyze each option

• Option (1): They are not supplementary to each other in the sense of $\angle EHF+\angle IHG = 180^{\circ}$.
• Option (2): They are not complementary ($\angle EHF+\angle IHG

eq90^{\circ}$).

• Option (3): They are congruent as shown above.
• Option (4): There is no information to suggest they are right angles.

Answer:

(2) $\angle CMF$ and $\angle DME$