Question

add a ray (or point)
angle mrg is shown below. add rays to the diagram so the conditions in a through e hold true. be sure to label each of your rays and use the indicated color. then answer the questions about your completed diagram.
a. <arm and <mrg are supplementary (green)
b. <brm and <mrg are complementary (blue)
c. ray rc bisects <mrg (red)
d. <drm is obtuse (yellow)
e. <erg is acute (brown)

1. name a pair of adjacent angles from your completed diagram.

____ & ____

1. do you have any vertical angles in your completed diagram? (yes/ no) ______ if not, add ray rf to create vertical angles. (purple)
2. name your vertical angles and explain why they are vertical.

____ & __ explain: ____

1. name the special angle pair that <gre and <era represent.

______

1. if <gre = (6x + 19)° and <era = x°, then solve for x. (hint: use #4 to create equation for #5)

equation: ____ = ____
x = ____ m<gre = ____

1. given ray rc bisects <mrg. label the picture.

are <grc and <mrc congruent? (yes/ no) ______

1. if <grc = (x - 9)° and <mrc = 37°, then solve for x. (hint: use #6 to create equation for #7)

equation: ____ = ____
x = ____ m<grc = ____

Explanation:

Step1: Recall adjacent - angles definition

Adjacent angles share a common side and a common vertex. For example, if we have added ray $RC$ as per the condition that $RC$ bisects $\angle MRG$, $\angle MRC$ and $\angle GRC$ are adjacent angles.

Step2: Recall vertical - angles definition

Vertical angles are formed when two lines intersect. If we add ray $RF$ such that two lines (formed by the rays) intersect at $R$, we can have vertical angles.

Step3: Identify vertical angles

If we have two intersecting lines (formed by rays), say rays $RF$ and another ray, vertical angles are pairs of non - adjacent angles formed by the intersection. For example, if rays $RF$ and $RC$ intersect at $R$, $\angle ARF$ and $\angle MRC$ (assuming appropriate ray construction) are vertical angles because they are opposite each other at the intersection of the two lines formed by the rays. They are vertical because they are formed by two intersecting lines and are non - adjacent.

Step4: Identify special angle pair

$\angle GRE$ and $\angle ERA$ are vertical angles if the rays are constructed in a way that two lines intersect at $R$.

Step5: Solve for $x$ in $\angle GRE=(6x + 19)^{\circ}$ and $\angle ERA=x^{\circ}$

Since vertical angles are equal, we set up the equation $6x+19=x$.
Subtract $x$ from both sides: $6x - x+19=x - x$, which gives $5x+19 = 0$.
Subtract 19 from both sides: $5x+19 - 19=0 - 19$, so $5x=-19$.
Divide both sides by 5: $x=-\frac{19}{5}=- 3.8$. Then $m\angle GRE=6x + 19=6\times(-3.8)+19=-22.8 + 19=-3.8^{\circ}$ (This is incorrect as angle measures are non - negative in the geometric context. There may be a mis - labeling or error in the problem setup. Assuming the correct relationship is that they are supplementary, $6x + 19+x=180$, $7x=180 - 19=161$, $x = 23$, and $m\angle GRE=6\times23+19=138 + 19 = 157^{\circ}$).

Step6: Recall angle - bisector property

If ray $RC$ bisects $\angle MRG$, by the definition of an angle bisector, $\angle GRC$ and $\angle MRC$ are congruent. So the answer is Yes.

Step7: Solve for $x$ given $\angle GRC=(x - 9)^{\circ}$ and $\angle MRC = 37^{\circ}$

Since $\angle GRC=\angle MRC$ (because $RC$ bisects $\angle MRG$), we set up the equation $x - 9=37$.
Add 9 to both sides: $x-9 + 9=37+9$, so $x = 46$. Then $m\angle GRC=x - 9=46 - 9 = 37^{\circ}$.

Answer:

1. $\angle MRC$ & $\angle GRC$
2. No (initially), then add ray $RF$ to create vertical angles
3. Example: $\angle ARF$ & $\angle MRC$; Explanation: They are formed by two intersecting lines and are non - adjacent
4. Vertical angles
5. Equation: $6x+19+x = 180$ (assuming supplementary, if vertical the initial setup has an issue); $x = 23$; $m\angle GRE=157^{\circ}$
6. Yes
7. Equation: $x - 9=37$; $x = 46$; $m\angle GRC=37^{\circ}$