part 1:
1. find the sum of the measures of the interior angles of a convex 70 - gon.
2. the measure of each interior angle of a regular polygon is 172. find the number of sides in the polygon.

part 2:
1. determine whether this quadrilateral is a parallelogram. justify your answer.
for questions 2–4, write true or false.
2. a quadrilateral with two pairs of parallel sides is always a parallelogram.
3. the diagonals of a parallelogram are always perpendicular.
4. the slope of $\overline{ab}$ and $\overline{cd}$ is $\frac{3}{5}$ and the slope of $\overline{bc}$ and $\overline{ad}$ is $-\frac{5}{3}$. $abcd$ is a parallelogram.
5. refer to parallelogram $abcd$. if $ab = 8$ cm, what is the perimeter of the parallelogram?

Question

part 1:
1. find the sum of the measures of the interior angles of a convex 70 - gon.
2. the measure of each interior angle of a regular polygon is 172. find the number of sides in the polygon.

part 2:
1. determine whether this quadrilateral is a parallelogram. justify your answer.
for questions 2–4, write true or false.
2. a quadrilateral with two pairs of parallel sides is always a parallelogram.
3. the diagonals of a parallelogram are always perpendicular.
4. the slope of $\overline{ab}$ and $\overline{cd}$ is $\frac{3}{5}$ and the slope of $\overline{bc}$ and $\overline{ad}$ is $-\frac{5}{3}$. $abcd$ is a parallelogram.
5. refer to parallelogram $abcd$. if $ab = 8$ cm, what is the perimeter of the parallelogram?

Accepted Answer

### Part 1
#### 1.
# Explanation:
## Step 1: Recall the formula for the sum of interior angles of a polygon
The formula for the sum of the interior angles of a convex polygon with $ n $ sides is $ S=(n - 2)\times180^{\circ} $.
## Step 2: Substitute $ n = 70 $ into the formula
For a 70 - gon, $ n=70 $. So we have $ S=(70 - 2)\times180^{\circ} $.
First, calculate $ 70-2 = 68 $. Then, $ 68\times180^{\circ}=12240^{\circ} $.
# Answer:
$ 12240^{\circ} $

#### 2.
# Explanation:
## Step 1: Recall the formula for the measure of each interior angle of a regular polygon
The measure of each interior angle $ I $ of a regular polygon with $ n $ sides is given by $ I=\frac{(n - 2)\times180^{\circ}}{n} $. We know that $ I = 172^{\circ} $, so we set up the equation $ 172=\frac{(n - 2)\times180}{n} $.
## Step 2: Solve the equation for $ n $
Multiply both sides of the equation by $ n $: $ 172n=(n - 2)\times180 $.
Expand the right - hand side: $ 172n = 180n-360 $.
Subtract $ 172n $ from both sides: $ 0 = 180n-172n - 360 $.
Simplify: $ 0 = 8n-360 $.
Add 360 to both sides: $ 8n=360 $.
Divide both sides by 8: $ n=\frac{360}{8} = 45 $.
# Answer:
45

### Part 2
#### 1.
# Brief Explanations:
To determine if a quadrilateral is a parallelogram, we can use the properties of parallelograms. One of the properties is that if a quadrilateral has one pair of opposite sides that are both congruent and parallel, then it is a parallelogram. From the diagram, we can see that one pair of opposite sides are marked as congruent (the tick marks) and the arrows indicate that they are parallel. So, by the theorem "If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram", this quadrilateral is a parallelogram.
# Answer:
Yes, it is a parallelogram. (Justification: One pair of opposite sides are congruent and parallel, so by the property of parallelograms, the quadrilateral is a parallelogram.)

#### 2.
# Brief Explanations:
By the definition of a parallelogram, a parallelogram is a quadrilateral with two pairs of parallel sides. So if a quadrilateral has two pairs of parallel sides, it satisfies the definition of a parallelogram.
# Answer:
True

#### 3.
# Brief Explanations:
The diagonals of a parallelogram bisect each other, but they are not always perpendicular. For example, in a rectangle (which is a parallelogram), the diagonals are equal in length but not perpendicular. Only in special parallelograms like rhombuses (and squares, which are special rhombuses) are the diagonals perpendicular.
# Answer:
False

#### 4.
# Brief Explanations:
For a quadrilateral to be a parallelogram, both pairs of opposite sides must be parallel. If the slope of $ \overline{AB} $ and $ \overline{CD} $ is $ \frac{3}{5} $, then $ \overline{AB}\parallel\overline{CD} $ (since parallel lines have the same slope). If the slope of $ \overline{BC} $ and $ \overline{AD} $ is $ -\frac{5}{3} $, then $ \overline{BC}\parallel\overline{AD} $ (since parallel lines have the same slope). Since both pairs of opposite sides are parallel, $ ABCD $ is a parallelogram.
# Answer:
True

#### 5.
# Explanation:
## Step 1: Recall the property of a parallelogram
In a parallelogram, opposite sides are equal. So in parallelogram $ ABCD $, $ AB = CD $ and $ AD=BC $.
## Step 2: Identify the lengths of the sides
We know that $ AB = 8\space\text{cm} $ and from the diagram $ AD = 6\space\text{cm} $. So $ CD = 8\space\text{cm} $ and $ BC = 6\space\text{cm} $.
## Step 3: Calculate the perimeter
The perimeter $ P $ of a parallelogram is given by $ P=2(AB + AD) $. Substitute $ AB = 8\space\text{cm} $ and $ AD = 6\space\text{cm} $ into the formula: $ P=2(8 + 6) $.
First, calculate $ 8 + 6=14 $. Then, $ 2\times14 = 28\space\text{cm} $.
# Answer:
$ 28\space\text{cm} $

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QUESTION IMAGE

Question

Explanation:

Part 1

1.

Step 1: Recall the formula for the sum of interior angles of a polygon

Step 2: Substitute \( n = 70 \) into the formula

Step 1: Recall the formula for the measure of each interior angle of a regular polygon

Step 2: Solve the equation for \( n \)

Answer:

2.