fluency
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?

n-gen math® geometry – unit 3 – euclidean triangle geometry – lesson 3 (v10.15.24)
emathinstruction, red hook, ny 12571, © 2023

Question

fluency
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?

n-gen math® geometry – unit 3 – euclidean triangle geometry – lesson 3 (v10.15.24)
emathinstruction, red hook, ny 12571, © 2023

Accepted Answer

### 1(a)
# Answer:
$\angle STV = \angle UTV$; Reason: Definition of angle bisector (a bisector divides an angle into two equal angles)
# Explanation:
## Step1: Recall angle bisector definition
An angle bisector is a ray that divides an angle into two congruent (equal) angles.
## Step2: Apply to given situation
Since $\overline{TV}$ bisects $\angle STU$, it splits $\angle STU$ into two angles: $\angle STV$ and $\angle UTV$. By the definition of an angle bisector, these two angles must be equal. So, $\angle STV = \angle UTV$.

### 1(b)
# Answer:
$\angle SRV = \angle SRU = \angle VRU = \angle TRU = 90^\circ$; Reason: Definition of perpendicular lines (perpendicular lines form right angles)
# Explanation:
## Step1: Recall perpendicular lines definition
Perpendicular lines are two lines that intersect at a right angle ($90^\circ$).
## Step2: Apply to given situation
Given $\overline{SU} \perp \overline{TV}$, their intersection at point $R$ forms right angles. So, all the angles formed at the intersection ($\angle SRV$, $\angle SRU$, $\angle VRU$, $\angle TRU$) are $90^\circ$ because perpendicular lines intersect to form right angles.

### 1(c)
# Answer:
$SR = RU$; Reason: Definition of midpoint (a midpoint divides a segment into two equal segments)
# Explanation:
## Step1: Recall midpoint definition
A midpoint of a segment is a point that divides the segment into two congruent (equal - length) segments.
## Step2: Apply to given situation
Since $R$ is the midpoint of $\overline{SU}$, it divides $\overline{SU}$ into two segments: $\overline{SR}$ and $\overline{RU}$. By the definition of a midpoint, $SR = RU$.

### 2(a)
# Answer:
$GH = HF$; Reason: Definition of a median (a median of a triangle connects a vertex to the midpoint of the opposite side)
# Explanation:
## Step1: Recall median definition
In a triangle, a median is a segment from a vertex to the midpoint of the opposite side. The midpoint divides the opposite side into two equal - length segments.
## Step2: Apply to given situation
Given $\overline{EH}$ is the median drawn from $E$ to $\overline{GF}$ in $	riangle EGF$, $H$ is the midpoint of $\overline{GF}$. By the definition of a midpoint (which is part of the median definition), $GH = HF$.

### 2(b)
# Answer:
$\angle GIE = 90^\circ$ (or $\overline{GI} \perp \overline{EF}$); Reason: Definition of an altitude (an altitude of a triangle is a perpendicular segment from a vertex to the opposite side)
# Explanation:
## Step1: Recall altitude definition
An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. This means the altitude forms a right angle with the opposite side.
## Step2: Apply to given situation
Given $\overline{GI}$ is the altitude drawn from $G$ to $\overline{EF}$, $\overline{GI}$ is perpendicular to $\overline{EF}$ at point $I$. So, $\angle GIE = 90^\circ$ (or $\overline{GI} \perp \overline{EF}$) because an altitude is perpendicular to the opposite side.

### 3(a)
# Answer:
$\angle AED$ and $\angle BED$ are supplementary ( $\angle AED+\angle BED = 180^\circ$); Reason: Linear pair postulate (a linear pair of angles is supplementary)
# Explanation:
## Step1: Recall linear pair postulate
A linear pair of angles is formed when two adjacent angles form a straight line. The sum of the measures of angles in a linear pair is $180^\circ$ (supplementary).
## Step2: Apply to given situation
$\angle AED$ and $\angle BED$ form a linear pair because they are adjacent angles that lie on the straight line $\overline{AB}$. By the linear pair postulate, $\angle AED+\angle BED = 180^\circ$, so they are supplementary.

### 3(b)
# Answer:
$\angle AED=\angle BEC$; Reason: Vertical angles theorem (vertical angles are congruent)
# Explanation:
## Step1: Recall vertical angles theorem
Vertical angles are the non - adjacent angles formed by the intersection of two lines. Vertical angles are congruent (have equal measures).
## Step2: Apply to given situation
$\overline{CD}$ and $\overline{AB}$ intersect at point $E$. $\angle AED$ and $\angle BEC$ are vertical angles (they are non - adjacent and formed by the intersection of the two lines). By the vertical angles theorem, $\angle AED=\angle BEC$.

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QUESTION IMAGE

Question

Explanation:

1(a)

Step1: Recall angle bisector definition

Step2: Apply to given situation

1(b)

Step1: Recall perpendicular lines definition

Step2: Apply to given situation

1(c)

Step1: Recall midpoint definition

Step2: Apply to given situation

2(a)

Answer: