19. if x=3 yes or no
4x+2
5x+2
6x-1
x+11
20. if y=10
(5y+10)° 12y°
(10y+20)° (7y-10)°
21.
22.
8
10 10
8
23.
130°
130°
24. if a=4
2a+10 4a-1
3a+3 5a+1

Question

Accepted Answer

### Problem 19: Check if the figure is a parallelogram when $ x = 3 $
# Explanation:
## Step 1: Recall the property of a parallelogram
In a parallelogram, opposite sides are equal in length. So we need to check if $ 5x + 2 = 6x - 1 $ and $ 4x + 2 = x + 11 $ when $ x = 3 $.

## Step 2: Check the first pair of sides ($ 5x + 2 $ and $ 6x - 1 $)
Substitute $ x = 3 $ into $ 5x + 2 $:
$ 5(3) + 2 = 15 + 2 = 17 $
Substitute $ x = 3 $ into $ 6x - 1 $:
$ 6(3) - 1 = 18 - 1 = 17 $
So $ 5x + 2 = 6x - 1 $ when $ x = 3 $.

## Step 3: Check the second pair of sides ($ 4x + 2 $ and $ x + 11 $)
Substitute $ x = 3 $ into $ 4x + 2 $:
$ 4(3) + 2 = 12 + 2 = 14 $
Substitute $ x = 3 $ into $ x + 11 $:
$ 3 + 11 = 14 $
So $ 4x + 2 = x + 11 $ when $ x = 3 $.

Since both pairs of opposite sides are equal, the figure is a parallelogram.
# Answer: Yes

### Problem 20: Check if the figure is a parallelogram when $ y = 10 $
# Explanation:
## Step 1: Recall the property of a parallelogram
In a parallelogram, opposite angles are equal, and consecutive angles are supplementary (sum to $ 180^\circ $). Let's first find the measure of each angle by substituting $ y = 10 $.

## Step 2: Calculate each angle
- For $ (5y + 10)^\circ $: Substitute $ y = 10 $, we get $ 5(10) + 10 = 50 + 10 = 60^\circ $
- For $ 12y^\circ $: Substitute $ y = 10 $, we get $ 12(10) = 120^\circ $
- For $ (10y + 20)^\circ $: Substitute $ y = 10 $, we get $ 10(10) + 20 = 100 + 20 = 120^\circ $
- For $ (7y - 10)^\circ $: Substitute $ y = 10 $, we get $ 7(10) - 10 = 70 - 10 = 60^\circ $

## Step 3: Check opposite angles and consecutive angles
- Opposite angles: $ 60^\circ $ (from $ 5y + 10 $) and $ 60^\circ $ (from $ 7y - 10 $) are equal; $ 120^\circ $ (from $ 12y $) and $ 120^\circ $ (from $ 10y + 20 $) are equal.
- Consecutive angles: $ 60^\circ + 120^\circ = 180^\circ $, so they are supplementary.

Since opposite angles are equal and consecutive angles are supplementary, the figure is a parallelogram.
# Answer: Yes

### Problem 21: Determine if the figure is a parallelogram
# Explanation:
## Step 1: Recall the property of a parallelogram (based on sides)
The figure has arrows on the sides, indicating that opposite sides are parallel. In a quadrilateral, if both pairs of opposite sides are parallel, then it is a parallelogram (by the definition of a parallelogram).

Since the diagram shows both pairs of opposite sides with parallel arrows, the figure satisfies the definition of a parallelogram.
# Answer: Yes

### Problem 22: Determine if the figure is a parallelogram
# Explanation:
## Step 1: Recall the property of a parallelogram
In a parallelogram, opposite sides are equal in length. The given figure has sides $ 8, 10, 8, 10 $. So the opposite sides: one pair is $ 8 $ and $ 8 $, the other pair is $ 10 $ and $ 10 $.

Since both pairs of opposite sides are equal in length, the figure satisfies the property of a parallelogram.
# Answer: Yes

### Problem 23: Determine if the figure is a parallelogram
# Explanation:
## Step 1: Recall the property of a parallelogram (based on angles)
In a parallelogram, opposite angles are equal, and consecutive angles are supplementary. The given figure has two angles of $ 130^\circ $. Let the other two angles be $ \angle A $ and $ \angle B $.

The sum of the interior angles of a quadrilateral is $ 360^\circ $. So $ 130^\circ + 130^\circ + \angle A + \angle B = 360^\circ $, which simplifies to $ \angle A + \angle B = 100^\circ $. But for a parallelogram, opposite angles should be equal, so if it's a parallelogram, the other two angles should also be equal (i.e., $ \angle A = \angle B $), so each would be $ 50^\circ $. However, the diagram only shows two $ 130^\circ $ angles and no information about the other two being equal or the sides. Wait, actually, in a parallelogram, if two adjacent angles are equal? No, wait, no—wait, the figure is a parallelogram? Wait, no, wait, maybe I misread. Wait, the figure is a quadrilateral with two $ 130^\circ $ angles. Wait, no, in a parallelogram, opposite angles are equal. If two angles are $ 130^\circ $, the other two should also be equal (let's say $ x $), so $ 2\times130 + 2x = 360 $, so $ 260 + 2x = 360 $, so $ 2x = 100 $, $ x = 50 $. But the diagram doesn't show the other two angles as equal. Wait, maybe the figure is a parallelogram? Wait, no, the problem is to check if it's a parallelogram. Wait, maybe the diagram is a parallelogram with two $ 130^\circ $ angles (opposite angles) and the other two opposite angles equal. Wait, the key is: in a parallelogram, opposite angles are equal. If two angles are $ 130^\circ $ (opposite), then the other two are equal (and $ 50^\circ $ each). But the diagram shows two $ 130^\circ $ angles—maybe they are opposite? If the figure is a parallelogram, opposite angles are equal. So if two angles are $ 130^\circ $ (opposite), then it's a parallelogram? Wait, no, the problem is to determine. Wait, maybe the figure is a parallelogram. Wait, the answer: Let's think again. The sum of interior angles of a quadrilateral is $ 360^\circ $. If two angles are $ 130^\circ $ (opposite), then the other two are equal (since in a parallelogram opposite angles are equal). So $ 130 + 130 + x + x = 360 $, $ 260 + 2x = 360 $, $ 2x = 100 $, $ x = 50 $. But the diagram shows two $ 130^\circ $ angles—maybe the figure is a parallelogram. Wait, maybe the answer is yes? Wait, no, maybe I made a mistake. Wait, the problem is to check if it's a parallelogram. Let's recall: a quadrilateral with both pairs of opposite angles equal is a parallelogram. If two angles are $ 130^\circ $ (opposite), and the other two are equal (opposite), then it's a parallelogram. So if the diagram shows two $ 130^\circ $ angles (opposite), then yes. So the answer is yes.
# Answer: Yes

### Problem 24: Check if the figure is a parallelogram when $ a = 4 $
# Explanation:
## Step 1: Recall the property of a parallelogram (diagonals)
In a parallelogram, the diagonals bisect each other, which means the lengths of the segments of one diagonal are equal, and the lengths of the segments of the other diagonal are equal. So we need to check if $ 2a + 10 = 5a + 1 $ and $ 3a + 3 = 4a - 1 $ when $ a = 4 $.

## Step 2: Check the first diagonal ($ 2a + 10 $ and $ 5a + 1 $)
Substitute $ a = 4 $ into $ 2a + 10 $:
$ 2(4) + 10 = 8 + 10 = 18 $
Substitute $ a = 4 $ into $ 5a + 1 $:
$ 5(4) + 1 = 20 + 1 = 21 $
Since $ 18
eq 21 $, the segments of the first diagonal are not equal.

## Step 3: Check the second diagonal ($ 3a + 3 $ and $ 4a - 1 $)
Substitute $ a = 4 $ into $ 3a + 3 $:
$ 3(4) + 3 = 12 + 3 = 15 $
Substitute $ a = 4 $ into $ 4a - 1 $:
$ 4(4) - 1 = 16 - 1 = 15 $
The segments of the second diagonal are equal, but the segments of the first diagonal are not. Since the diagonals of a parallelogram must bisect each other (both pairs of segments equal), the figure is not a parallelogram.
# Answer: No

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QUESTION IMAGE

Question

Explanation:

Problem 19: Check if the figure is a parallelogram when \( x = 3 \)

Step 1: Recall the property of a parallelogram

Step 2: Check the first pair of sides (\( 5x + 2 \) and \( 6x - 1 \))

Step 3: Check the second pair of sides (\( 4x + 2 \) and \( x + 11 \))

Step 1: Recall the property of a parallelogram

Step 2: Calculate each angle

Step 3: Check opposite angles and consecutive angles

Step 1: Recall the property of a parallelogram (based on sides)

Answer:

Problem 20: Check if the figure is a parallelogram when \( y = 10 \)