more properties of lines common core geometry homework
1. a parallelogram is a four - sided figure that has two pairs of parallel, opposite sides. parallelogram abcd is shown below. if another line was drawn parallel to $overline{cd}$ passing through a, explain why it must fall on $overline{ab}$.
2. in the following diagram which is not drawn to scale, why cant both $overline{ab}$ and $overline{ac}$ be perpendicular to $overline{bc}$?
3. in exercise #2(c) from the class work we saw that a segment connecting two points on opposite sides of a given line must intersect that line. is it always true that a line connecting two points on the same side of given line will be parallel to that given line?

Question

Accepted Answer

# Explanation:
## Step1: Recall parallelogram property
In parallelogram $ABCD$, $AB\parallel CD$ by definition of parallelogram.
## Step2: Apply parallel - line uniqueness
Through a point $A$ outside the line $CD$, there is exactly one line parallel to $CD$. Since $AB$ is already parallel to $CD$ and we are drawing a line parallel to $CD$ through $A$, this new line must be the same as $AB$.

# Answer:
Because in a parallelogram $ABCD$, $AB\parallel CD$ and through a point outside a line there is exactly one line parallel to the given line.

# Explanation:
## Step1: Recall perpendicular - line property
In a plane, through a given point on a line, there is exactly one line perpendicular to the given line.
## Step2: Analyze points $B$ and $C$
Points $B$ and $C$ are on line $BC$. If we consider the point $B$ on $BC$, there is a unique perpendicular line to $BC$ at $B$ and if we consider point $C$ on $BC$, there is a unique perpendicular line to $BC$ at $C$. $AB$ and $AC$ cannot both be perpendicular to $BC$ because if $AB\perp BC$ at $B$ and $AC\perp BC$ at $C$, it would violate the uniqueness of the perpendicular line through a point on a line.

# Answer:
Because through a point on a line, there is exactly one line perpendicular to the given line, and $AB$ and $AC$ cannot both satisfy the perpendicular - to - $BC$ condition simultaneously as it would violate the uniqueness of the perpendicular.

# Explanation:
## Step1: Consider counter - example
No, it is not always true. We can consider a triangle. For example, in $\triangle ABC$, points $A$ and $B$ are on the same side of the line containing side $BC$, but the line $AB$ is not parallel to the line containing side $BC$. The lines will intersect if extended.

# Answer:
No. A counter - example is a triangle where two points on the same side of one of its sides are connected, and the resulting line is not parallel to that side.

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QUESTION IMAGE

Question

Explanation:

Step1: Recall parallelogram property

Step2: Apply parallel - line uniqueness

Step1: Recall perpendicular - line property

Step2: Analyze points \(B\) and \(C\)

Step1: Consider counter - example

Answer: