Examples with solutions for Characteristics and Proofs of a Parallelogram: Is this a parallelogram?

Exercise #1

In front of you the next quadrilateral:

Is it possible that it is a parallelogram?

AAABBBCCCDDD12070

Step-by-Step Solution

To determine if the quadrilateral is a parallelogram, we need to verify the properties of the angles. A key property of parallelograms is that consecutive angles are supplementary, meaning their sum equals 180 180^\circ .

The problem provides the measures of two consecutive angles: B=70 \angle B = 70^\circ and C=120 \angle C = 120^\circ .

Next, let's calculate the sum of these angles:
B+C=70+120=190 \angle B + \angle C = 70^\circ + 120^\circ = 190^\circ

The sum of B \angle B and C \angle C is 190 190^\circ , which is not equal to 180 180^\circ .

This indicates that the quadrilateral cannot be a parallelogram because two consecutive angles do not add up to 180 180^\circ .

Therefore, the given quadrilateral is not a parallelogram.
Thus, the correct answer is No.

Answer

No

Exercise #2

Below is a quadrilateral:

Is it possible that it is a parallelogram?

AAABBBCCCDDDOOO108810

Step-by-Step Solution

According to the properties of the parallelogram: the diagonals intersect each other.

From the data in the drawing, it follows that diagonal AC and diagonal BD are divided into two equal parts, that is, the diagonals intersect each other:

AO=OC=8 AO=OC=8

DO=OB=10 DO=OB=10

Therefore, the quadrilateral is actually a parallelogram.

Answer

Yes

Exercise #3

Below is a quadrilateral:

Is it possible that it is a parallelogram?

AAABBBCCCDDD1206012060

Step-by-Step Solution

Let's review the property: a quadrilateral in which two pairs of opposite angles are equal is a parallelogram.

From the data in the drawing, it follows that:

D=B=60 D=B=60

A=C=120 A=C=120

Therefore, the quadrilateral is actually a parallelogram.

Answer

Yes

Exercise #4

Below is a quadrilateral:

Is it possible that it is a parallelogram?

AAABBBCCCDDD711811

Step-by-Step Solution

According to the properties of a parallelogram, any two opposite sides will be equal to each other.

From the data, it can be observed that only one pair of opposite sides are equal and therefore the quadrilateral is not a parallelogram.

Answer

No

Exercise #5

AB = DC.=

Is the shape below a parallelogram?

AAABBBCCCDDDx+582x+9x

Step-by-Step Solution

In a parallelogram, we know that each pair of opposite sides are equal to each other.

The data shows that only one pair of sides are equal to each other:

AB=DC=8 AB=DC=8

Now we try to see that the additional pair of sides are equal to each other.

We replacex=8 x=8 for each of the sides:

AD=2×8+9 AD=2\times8+9

AD=16+9 AD=16+9

AD=25 AD=25

BC=8+5 BC=8+5

BC=13 BC=13

That is, we find that the pair of opposite sides are not equal to each other:

2513 25\ne13

Therefore, the quadrilateral is not a parallelogram.

Answer

No

Exercise #6

Below is a quadrilateral:

Given B+C=180 ∢B+∢C=180

Is it possible that it is a parallelogram?

AAABBBCCCDDD4x14040140

Step-by-Step Solution

Remember that in a parallelogram each pair of opposite angles are equal to each other.

The data shows that only one pair of angles are equal to each other:

D=B=140 D=B=140

Therefore, we will now find angle C and see if it is equal to angle A, that is, if angle C is equal to 40:

Let's remember that a pair of angles on the same side are equal to 180 degrees, therefore:

B+C=180 B+C=180

We replace the existing data:

140+4x=180 140+4x=180

4x=180140 4x=180-140

4x=40 4x=40

Divide by 4:

4x4=404 \frac{4x}{4}=\frac{40}{4}

x=10 x=10

Now we replace X:

C=4×10=40 C=4\times10=40

That is, we found that angles A and C are equal to each other and that the quadrilateral is a parallelogram since each pair of opposite angles are equal to each other.

Answer

Yes

Exercise #7

Look at the quadrilateral below.

AO = OC

Is it a parallelogram?

AAABBBCCCDDDOOO5x+49x+110x3x-2

Step-by-Step Solution

Let's pay attention to the diagonals, remember that in a parallelogram the diagonals intersect each other.

Therefore, we will find AO, OC, BO, DO and check if they are equal and intersect each other.

We refer to the figure:

AO=OC AO=OC

9x+1=10x 9x+1=10x

We place like terms:

1=10x9x 1=10x-9x

1=x 1=x

We replace:

AO=9×1+1=10 AO=9\times1+1=10

OC=10×1=10 OC=10\times1=10

Now we know that indeedAO=OC AO=OC

Now we establish that X=1 and see if BO is equal to OD:

BO=3x2 BO=3x-2

BO=3×12= BO=3\times1-2=

BO=32=1 BO=3-2=1

OD=5x+4 OD=5x+4

OD=5×1+4 OD=5\times1+4

OD=5+4=9 OD=5+4=9

Now we find that: BOOD BO\ne OD

Since the diagonals do not intersect each other, the quadrilateral is not a parallelogram.

Answer

No