Let's analyze each statement about the parallelogram [tex]\(PQRS\)[/tex] one by one.
### 1. Perimeter [tex]\(= 188\)[/tex]
The perimeter of a parallelogram is given by the formula:
[tex]\[ \text{Perimeter} = 2(\overline{PQ} + \overline{QR}) \][/tex]
Given:
[tex]\[ \overline{PQ} = 6x + 17 \][/tex]
[tex]\[ \overline{RS} = 3x + 35 \][/tex]
[tex]\[ \overline{QR} = 45 \][/tex]
Since [tex]\(\overline{PQ}\)[/tex] and [tex]\(\overline{RS}\)[/tex] are opposite sides in a parallelogram, we know [tex]\( \overline{PQ} = \overline{RS} \)[/tex]:
[tex]\[ 6x + 17 = 3x + 35 \][/tex]
Solving for [tex]\(x\)[/tex]:
[tex]\[ 6x - 3x = 35 - 17 \][/tex]
[tex]\[ 3x = 18 \][/tex]
[tex]\[ x = 6 \][/tex]
Substitute [tex]\(x = 6\)[/tex] back into [tex]\(\overline{PQ} = 6x + 17\)[/tex]:
[tex]\[ \overline{PQ} = 6(6) + 17 \][/tex]
[tex]\[ \overline{PQ} = 36 + 17 \][/tex]
[tex]\[ \overline{PQ} = 53 \][/tex]
Thus, [tex]\(\overline{RS}\)[/tex] should also be [tex]\(53\)[/tex] because [tex]\(\overline{PQ} = \overline{RS}\)[/tex]:
[tex]\[ \overline{RS} = 3(6) + 35 \][/tex]
[tex]\[ \overline{RS} = 18 + 35 \][/tex]
[tex]\[ \overline{RS} = 53 \][/tex]
Now, calculate the perimeter:
[tex]\[ \text{Perimeter} = 2(\overline{PQ} + \overline{QR}) \][/tex]
[tex]\[ \text{Perimeter} = 2(53 + 45) \][/tex]
[tex]\[ \text{Perimeter} = 2 \times 98 \][/tex]
[tex]\[ \text{Perimeter} = 196 \][/tex]
The perimeter is not 188, so this statement is incorrect.
### 2. Opposite angles are congruent
By definition, in a parallelogram, opposite angles are always congruent. So, this statement is correct.
### 3. [tex]\( x = 6 \)[/tex]
We have already calculated that [tex]\( x = 6 \)[/tex]. Therefore, this statement is correct.
### 4. [tex]\(\overline{RS} = 53\)[/tex]
We have calculated that [tex]\(\overline{RS} = 53\)[/tex]. Therefore, this statement is correct.
### 5. The diagonals are perpendicular
There is no information given about the diagonals being perpendicular in a general parallelogram unless it is specified as a rhombus, which it is not. Therefore, this statement is incorrect.
### 6. The parallelogram [tex]\(PQRS\)[/tex] is a square
To be a square, all sides must be equal, and the angles must be right angles (90 degrees). Since [tex]\(\overline{PQ} = 53\)[/tex] and [tex]\(\overline{QR} = 45\)[/tex], the parallelogram cannot be a square since [tex]\(\overline{QR}\)[/tex] is not equal to [tex]\(\overline{PQ}\)[/tex]. Therefore, this statement is incorrect.
### 7. None of these answers are correct
Some of the given statements are correct (statements 2, 3, and 4). Therefore, this statement is incorrect.
In summary, the correct statements are:
- Opposite angles are congruent
- [tex]\( x = 6 \)[/tex]
- [tex]\(\overline{RS} = 53\)[/tex]
