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) $overrightarrow{eg}$ and $overrightarrow{ge}$ (2) $overrightarrow{ge}$ and $overrightarrow{gf}$ (3) $overrightarrow{ef}$ and $overrightarrow{fg}$ (4) $overrightarrow{ef}$ and $overrightarrow{fe}$ 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: Analyze geometric - object identity

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

Step2: Recall circle definition

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

Step3: Set up an 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 $x+(x + 12)=28$. Simplifying gives $2x+12 = 28$, then $2x=16$, and $x = 8$.

Step4: Use angle - addition property

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

Step5: Solve for supplementary angles

Let the measure of the smaller angle be $x$. Then the measure of the larger angle is $2x + 12$. Since the two angles are supplementary, $x+(2x + 12)=180$. Combining like terms gives $3x+12 = 180$, then $3x=168$, and $x = 56$. The larger angle is $2x + 12=2\times56+12=112 + 12=124^{\circ}$.

Answer:

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