Question

1. in the diagram below, points e, f, and g all lie on the same line. which two geometric objects listed below are not the same? (1) $overline{eg}$ and $overline{ge}$ (2) $overline{ge}$ and $overline{gf}$ (3) $overline{ef}$ and $overline{fg}$ (4) $overline{ef}$ and $overline{fe}$ 2. which of the following is the term we use for the collection of all points that lie a fixed distance away from another point? (1) a line (2) an arc (3) a ray (4) a circle 3. points a, e, and n are collinear with point a between points e and n. if $en = 28$ and $overline{an}$ is 12 units longer than $overline{ea}$, then which of the following is the length of $overline{ea}$? (1) 8 (2) 12 (3) 20 (4) 40 4. in the diagram shown, $mangle qrt=127^{circ}$ and $angle qrs$ is a right angle. which of the following is the measure of $angle srt$? (1) $30^{circ}$ (2) $37^{circ}$ (3) $44^{circ}$ (4) $49^{circ}$ 5. in a supplementary - angle pair, the measure of the larger angle is 12 degrees more than twice the measure of the smaller angle. which of the following is the measure of the larger angle? (1) $112^{circ}$ (2) $118^{circ}$ (3) $124^{circ}$ (4) $136^{circ}$

Explanation:

Step1: Understand ray properties

A ray has a starting - point and extends infinitely in one direction. $\overline{GE}$ starts at $G$ and goes towards $E$, while $\overline{GF}$ starts at $G$ and goes towards $F$. They are different rays.

Step2: Analyze other options

• $\overline{EG}$ and $\overline{GE}$ are opposite - directed rays but on the same line.
• $\overline{EF}$ and $\overline{FG}$ are different segments on the line.
• $\overline{EF}$ and $\overline{FE}$ are opposite - directed segments.

Step3: Recall circle definition

A circle is the set of all points in a plane that are at a fixed distance (radius) from a given point (center).

Step4: Set up equation for collinear points

Let the length of $\overline{EA}=x$. Then the length of $\overline{AN}=x + 12$. Since $EN=EA+AN$ and $EN = 28$, we have the equation $x+(x + 12)=28$. Simplify it: $2x+12 = 28$, $2x=28 - 12=16$, $x = 8$.

Step5: Find angle measure

Since $\angle QRT=\angle QRS+\angle SRT$ and $\angle QRS = 90^{\circ}$, $\angle SRT=\angle QRT-\angle QRS$. So $\angle SRT=127^{\circ}-90^{\circ}=37^{\circ}$.

Step6: Solve for supplementary angles

Let the measure of the smaller angle be $x$. Then the measure of the larger angle is $2x + 12$. Since they are supplementary, $x+(2x + 12)=180$. Combine like terms: $3x+12 = 180$, $3x=180 - 12 = 168$, $x = 56$. The larger angle is $2x+12=2\times56 + 12=112 + 12=124^{\circ}$.

Answer:

1. (2) $\overline{GE}$ and $\overline{GF}$
2. (4) a circle
3. (1) 8
4. (2) $37^{\circ}$
5. (3) $124^{\circ}$