1. in the diagram below, $overline{su}$ and $overline{tv}$ intersect at point $r$. for each given below, write down what you can conclude from the given. state a reason for your conclusion.
(a) given: $overline{tv}$ bisects $angle stu$.
(b) given: $overline{su} perp overline{tv}$.
(c) given: $r$ is the midpoint of $overline{su}$.
2. in the figure shown below, points $h$ and $i$ lie on sides $overline{gf}$ and $overline{ef}$ respectively. for each given below, write down what you can conclude from the given. state a reason for your conclusion.
(a) given: $overline{eh}$ is the median drawn from $e$ to $overline{gf}$.
(b) given: $overline{gi}$ is the altitude drawn from $g$ to $overline{ef}$.
3. in the diagram shown, $overline{cd}$ and $overline{ab}$ intersect at point $e$.
(a) what can you conclude about the angle pair $angle aed$ and $angle bed$? why?
(b) what can you conclude about the angle pair $angle aed$ and $angle bec$? why?

Question

Accepted Answer

### Problem 1
#### (a)
# Explanation:
## Step1: Recall angle bisector definition
An angle bisector divides an angle into two equal angles. So if $\overline{TV}$ bisects $\angle STU$, then $\angle STV=\angle UTV$.
## Step2: State conclusion and reason
Conclusion: $\angle STV = \angle UTV$
Reason: Definition of angle bisector (a bisector of an angle divides it into two congruent angles).
# Answer: $\angle STV = \angle UTV$ (Reason: Definition of angle bisector)

#### (b)
# Explanation:
## Step1: Recall perpendicular lines definition
Perpendicular lines form right angles (angles of $90^\circ$). So if $\overline{SU}\perp\overline{TV}$, then $\angle SRV=\angle SRU=\angle URV=\angle URT = 90^\circ$ (or relevant right angles at $R$).
## Step2: State conclusion and reason
Conclusion: $\angle SRV = 90^\circ$ (or other right angles at $R$)
Reason: Definition of perpendicular lines (perpendicular lines intersect at right angles).
# Answer: $\angle SRV = 90^\circ$ (Reason: Definition of perpendicular lines)

#### (c)
# Explanation:
## Step1: Recall midpoint definition
A midpoint of a segment divides it into two equal segments. So if $R$ is the midpoint of $\overline{SU}$, then $SR = RU$.
## Step2: State conclusion and reason
Conclusion: $SR = RU$
Reason: Definition of midpoint (a midpoint divides a segment into two congruent segments).
# Answer: $SR = RU$ (Reason: Definition of midpoint)

### Problem 2
#### (a)
# Explanation:
## Step1: Recall median definition
A median of a triangle is a segment from a vertex to the midpoint of the opposite side. So if $\overline{EH}$ is the median from $E$ to $\overline{GF}$, then $H$ is the midpoint of $\overline{GF}$, so $GH = HF$.
## Step2: State conclusion and reason
Conclusion: $GH = HF$
Reason: Definition of median (a median connects a vertex to the midpoint of the opposite side, so it divides the opposite side into two congruent segments).
# Answer: $GH = HF$ (Reason: Definition of median)

#### (b)
# Explanation:
## Step1: Recall altitude definition
An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. So if $\overline{GI}$ is the altitude from $G$ to $\overline{EF}$, then $\overline{GI}\perp\overline{EF}$, so $\angle GIE=\angle GIF = 90^\circ$.
## Step2: State conclusion and reason
Conclusion: $\angle GIE = 90^\circ$
Reason: Definition of altitude (an altitude is perpendicular to the opposite side, forming right angles).
# Answer: $\angle GIE = 90^\circ$ (Reason: Definition of altitude)

### Problem 3
#### (a)
# Explanation:
## Step1: Analyze angle pair $\angle AED$ and $\angle BED$
$\overline{AB}$ is a straight line, so $\angle AEB$ is a straight angle ($180^\circ$). $\angle AED$ and $\angle BED$ are adjacent angles that form $\angle AEB$, so they are supplementary (their sum is $180^\circ$). Also, since $\overline{CD}$ intersects $\overline{AB}$ at $E$, but more directly, they form a linear pair.
## Step2: State conclusion and reason
Conclusion: $\angle AED$ and $\angle BED$ are supplementary (or form a linear pair, sum to $180^\circ$)
Reason: Linear pair postulate (adjacent angles forming a straight line are supplementary).
# Answer: $\angle AED$ and $\angle BED$ are supplementary (Reason: Linear pair postulate)

#### (b)
# Explanation:
## Step1: Analyze angle pair $\angle AED$ and $\angle BEC$
When two lines intersect, vertical angles are congruent. $\overline{CD}$ and $\overline{AB}$ intersect at $E$, so $\angle AED$ and $\angle BEC$ are vertical angles.
## Step2: State conclusion and reason
Conclusion: $\angle AED=\angle BEC$
Reason: Vertical angles theorem (vertical angles are congruent).
# Answer: $\angle AED=\angle BEC$ (Reason: Vertical angles theorem)

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QUESTION IMAGE

Question

Explanation:

Problem 1

(a)

Step1: Recall angle bisector definition

Step2: State conclusion and reason

Step1: Recall perpendicular lines definition

Step2: State conclusion and reason

Step1: Recall midpoint definition

Step2: State conclusion and reason

Answer:

(b)