Question

1. if the lines (overline{fp}) and (overline{ci}) 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 bmc) 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 pqm) 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}{3}) (4) (10\frac{1}{3})

Explanation:

6.

Step1: Recall vertical - angle definition

Vertical angles are formed when two lines intersect. They are opposite each other.
When two lines intersect, vertical angles are congruent. $\angle CMF$ and $\angle DME$ are vertical angles as lines $\overline{CF}$ and $\overline{DE}$ intersect at point $M$.

Step1: Use angle - bisector property

If $\overrightarrow{FH}$ bisects $\angle EFI$ and $m\angle IFG = 31^{\circ}$, and $\angle EFI+\angle IFG = 180^{\circ}$ (since $E$, $F$, $G$ are collinear). Let $m\angle EFI = 2x$ (because of the angle - bisector property, $\angle EFH=\angle HFI = x$). Then $2x + 31^{\circ}=180^{\circ}$, so $2x=180^{\circ}- 31^{\circ}=149^{\circ}$, and $x = 74.5^{\circ}$.

Step2: Calculate $\angle HFG$

$\angle HFG=\angle HFI+\angle IFG$. Since $\angle HFI = 74.5^{\circ}$ and $\angle IFG = 31^{\circ}$, then $\angle HFG=74.5^{\circ}+31^{\circ}=109^{\circ}$

Step1: Recall the definition of a bisector

A line that bisects a line segment divides the line segment into two equal parts. If $\overline{MN}$ bisects $\overline{OP}$ at point $Q$, then $Q$ is the mid - point of $\overline{OP}$, so $\overline{OQ}\cong\overline{PQ}$ and $\angle OQM\cong\angle PQM$ (by the definition of a line bisecting a line segment). But there is no information given that $\overline{MN}$ is perpendicular to $\overline{OP}$.

Answer:

(2)

7.