4. in quadrilateral klmn shown below it is given that $\overline{km}$ is the perpendicular bisector of $\overline{ln}$. list all conclusions you can make from this one given. give reasons for your conclusions.

5. in the diagram shown, $\overline{ab}$ and $\overline{cd}$ intersect at point e.
(a) given: $\overline{ab}$ and $\overline{cd}$ bisect each other.
what can you conclude? why?
(b) is there a pair of angles that you can conclude are congruent based on the diagram and the fact that $\overline{ab}$ and $\overline{cd}$ intersect at point e? if so, which pair and why?

6. in parallelogram abcd shown below, diagonal $\overline{bd}$ has been drawn. points g and f lie on $\overline{bd}$ and point e lies on $\overline{cd}$.
given: $\overline{ge}$ is a median drawn from point g to $\overline{cd}$, $\overline{af}$ is an altitude drawn from point a to $\overline{bd}$, and $\overline{gc}$ bisects $\angle bcd$.
give all conclusions that can be immediately inferred from the givens. state a reason for each conclusion.

Question

4. in quadrilateral klmn shown below it is given that $\overline{km}$ is the perpendicular bisector of $\overline{ln}$. list all conclusions you can make from this one given. give reasons for your conclusions.

5. in the diagram shown, $\overline{ab}$ and $\overline{cd}$ intersect at point e.
   (a) given: $\overline{ab}$ and $\overline{cd}$ bisect each other.
       what can you conclude? why?
   (b) is there a pair of angles that you can conclude are congruent based on the diagram and the fact that $\overline{ab}$ and $\overline{cd}$ intersect at point e? if so, which pair and why?

6. in parallelogram abcd shown below, diagonal $\overline{bd}$ has been drawn. points g and f lie on $\overline{bd}$ and point e lies on $\overline{cd}$.
   given: $\overline{ge}$ is a median drawn from point g to $\overline{cd}$, $\overline{af}$ is an altitude drawn from point a to $\overline{bd}$, and $\overline{gc}$ bisects $\angle bcd$.
   give all conclusions that can be immediately inferred from the givens. state a reason for each conclusion.

Accepted Answer

### Problem 4
# Explanation:
## Step1: Analyze Perpendicular Bisector
Since $\overline{KM}$ is the perpendicular bisector of $\overline{LN}$, by definition, $O$ (the intersection point) is the midpoint of $\overline{LN}$, so $LO = ON$. Also, $\overline{KM}\perp\overline{LN}$, so $\angle LOK=\angle LOM=\angle NOK=\angle NOM = 90^\circ$.
## Step2: Identify Congruent Triangles
Triangles $\triangle KLO$ and $\triangle KNO$ are congruent (SAS: $LO = ON$, $\angle LOK=\angle NOK$, $KO = KO$). Similarly, $\triangle MLO$ and $\triangle MNO$ are congruent (SAS: $LO = ON$, $\angle LOM=\angle NOM$, $MO = MO$). Also, $KL = KN$ and $ML = MN$ (corresponding parts of congruent triangles).
# Answer:
- $LO = ON$ (Definition of midpoint from perpendicular bisector).
- $\overline{KM}\perp\overline{LN}$ (Definition of perpendicular bisector).
- $\triangle KLO\cong\triangle KNO$, $\triangle MLO\cong\triangle MNO$ (SAS congruence).
- $KL = KN$, $ML = MN$ (CPCTC).

### Problem 5(a)
# Explanation:
## Step1: Analyze Bisecting Segments
Given $\overline{AB}$ and $\overline{CD}$ bisect each other, so $AE = EB$ and $CE = ED$.
## Step2: Prove Triangle Congruence
In $\triangle AEC$ and $\triangle BED$, $AE = EB$, $CE = ED$, and $\angle AEC=\angle BED$ (vertical angles). By SAS congruence, $\triangle AEC\cong\triangle BED$. Then, $AC = BD$ and $\angle A=\angle B$, $\angle C=\angle D$ (CPCTC), so $AC\parallel BD$ (alternate interior angles are equal). Thus, quadrilateral $ACDB$ is a parallelogram (one pair of sides equal and parallel, or opposite sides equal).
# Answer:
- $\triangle AEC\cong\triangle BED$ (SAS: $AE = EB$, $CE = ED$, $\angle AEC=\angle BED$).
- $AC = BD$, $AC\parallel BD$ (CPCTC and alternate interior angles).
- Quadrilateral $ACDB$ is a parallelogram (opposite sides equal and parallel).

### Problem 5(b)
# Explanation:
## Step1: Identify Vertical Angles
When $\overline{AB}$ and $\overline{CD}$ intersect at $E$, $\angle AEC$ and $\angle BED$ are vertical angles, and $\angle AED$ and $\angle BEC$ are vertical angles.
## Step2: Apply Vertical Angles Theorem
By the Vertical Angles Theorem, vertical angles are congruent. So $\angle AEC\cong\angle BED$ and $\angle AED\cong\angle BEC$.
# Answer:
- $\angle AEC$ and $\angle BED$ (or $\angle AED$ and $\angle BEC$) are congruent (Vertical Angles Theorem).

### Problem 6
# Explanation:
## Step1: Analyze Median $GE$
Since $GE$ is a median to $\overline{CD}$, $E$ is the midpoint of $\overline{CD}$, so $DE = EC$.
## Step2: Analyze Altitude $AF$
$AF$ is an altitude to $\overline{BD}$, so $\angle AFD=\angle AFB = 90^\circ$.
## Step3: Analyze Angle Bisector $GC$
$GC$ bisects $\angle BCD$, so $\angle BCG=\angle DCG$.
## Step4: Use Parallelogram Properties
In parallelogram $ABCD$, $AB\parallel CD$, $AD\parallel BC$, $AB = CD$, $AD = BC$.
## Step5: Identify Congruent Triangles/Segments
From $DE = EC$ (median), $\angle AFD = 90^\circ$ (altitude), and $\angle BCG=\angle DCG$ (angle bisector), we can also note relationships like $AD = BC$ (parallelogram), and possible congruences in triangles involving these segments/angles.
# Answer:
- $DE = EC$ (Definition of median: $GE$ is median to $CD$).
- $\angle AFD=\angle AFB = 90^\circ$ (Definition of altitude: $AF$ is altitude to $BD$).
- $\angle BCG=\angle DCG$ (Definition of angle bisector: $GC$ bisects $\angle BCD$).
- $AB\parallel CD$, $AD\parallel BC$, $AB = CD$, $AD = BC$ (Properties of parallelogram $ABCD$).

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QUESTION IMAGE

Question

Explanation:

Problem 4

Step1: Analyze Perpendicular Bisector

Step2: Identify Congruent Triangles

Step1: Analyze Bisecting Segments

Step2: Prove Triangle Congruence

Step1: Identify Vertical Angles

Step2: Apply Vertical Angles Theorem

Answer:

Problem 5(a)