Did you notice the quadrilateral that is formed at the intersection of 2 train tracks? What is it called? What are its characteristics? Let's take a look at the train tracks, why are train tracks 2 parallel tracks? For the train to not derail, there must be 2 tracks that always maintain the same distance apart. This is the definition of parallel lines that never meet because the distance between them is always equal. This is the definition of parallel lines that never meet because the distance between them is always equal. At the moment when 2 train tracks meet, a quadrilateral is formed between them, which has 2 pairs of opposite sides parallel, which is the parallelogram

Corresponding angles between parallels: the line that crosses 2 parallel lines forms around each intersection point with each line 4 angles. Any pair of angles that are in the same position around the intersection points are called corresponding angles. When the lines are parallel, the corresponding angles are also equal

Alternate interior angles between parallels: each angle around an upper intersection point with the vertex to the corresponding angle around a second intersection point forms a pair of alternate angles. A hallmark: it is possible to look for angles in the shape of a Z in the cut of the straight lines. When the lines are parallel, the cut creates equal alternate angles.

Consecutive interior angles between parallels: any angle around an upper intersection point with the adjacent angle corresponding to that side around a second intersection point. The sum of the unilateral angles between parallels is 180^{o}

$AB ǁ DC$ (by definition of the parallelogram). Therefore: angle $A2 = C1$ (alternate angles between equal parallels)

$AD ǁ BC$ (by the definition of parallels). Therefore: $A1 = C2$ (alternate angles between equal parallels)

$AC = AC$ (Common side) therefore it can be concluded that: $ΔADC≅ΔCBA$ (according to the congruence theorem: angle, side, angle) For this: $AB = DC, AD = BC$ congruent equals)

This congruence leads us to the following property:

The opposite angles at the vertex in a parallelogram are equal.

$ΔADC≅ΔCBA$ (proven in the previous sentence)

Therefore:

Angles$B = D$ (corresponding angles in equal congruent triangles)

And also $C1 + C2 = A1 + A1$ (sum of equal angles)

Therefore: Angles $\sphericalangle A=\sphericalangle C$ (sum of angles)

Calculation of the Area of a Parallelogram Using Trigonometry

It is possible to calculate the area of a parallelogram even without height, using trigonometry: by multiplying 2 adjacent sides by the sine of the angle between them.

Sometimes, the fact that the diagonals divide the parallelogram into $4$equilateral triangles, allows us through the use of the halves of the diagonals and the sine of the angle between them to find the area of the parallelogram. It is enough to find a single triangle and multiply it by $4$.

Parallelogram Verification

What are the necessary conditions to prove that a quadrilateral is a parallelogram?

Definition: A quadrilateral that has 2 pairs of opposite sides parallel is called a parallelogram.

What are the additional theorems that allow us to determine without information that the opposite sides are parallel that the quadrilateral is a parallelogram?

Test your knowledge

Question 1

Find the area of the parallelogram based on the data in the figure:

Is it possible to determine if this quadrilateral is a parallelogram?

Solution:

The definition of a parallelogram is a quadrilateral with two pairs of parallel sides.

In this case, the quadrilateral is not a parallelogram because two adjacent angles on the same side do not add up to $180^o$ degrees

Answer:

No

Exercise 4

Assignment

Given the quadrilateral $ABCD$

$AB=20$

$CD=20$

$BD=8$

$AC=8$

Is it possible to determine if this quadrilateral is a parallelogram?

Solution

In fact, this quadrilateral is a parallelogram because if in a quadrilateral two pairs of opposite sides are of the same length, then the quadrilateral is a parallelogram.

Answer

Yes

Test your knowledge

Question 1

Calculate the area of the parallelogram using the data in the figure:

Is it possible to determine if this quadrilateral is a parallelogram?

Solution

This quadrilateral is not a parallelogram because a quadrilateral whose diagonals intersect is a parallelogram. In this case, the diagonals do not intersect each other.

Answer

No

Do you know what the answer is?

Question 1

Find the area of the parallelogram based on the data in the figure:

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 #2

Below is a quadrilateral:

Is it possible that it is a parallelogram?

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:

$AO=OC=8$

$DO=OB=10$

Therefore, the quadrilateral is actually a parallelogram.

Answer

Yes

Exercise #3

Below is a quadrilateral:

Is it possible that it is a parallelogram?

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:

$D=B=60$

$A=C=120$

Therefore, the quadrilateral is actually a parallelogram.

Answer

Yes

Exercise #4

AB = DC.=

Is the shape below a parallelogram?

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:

$AB=DC=8$

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

We replace$x=8$for each of the sides:

$AD=2\times8+9$

$AD=16+9$

$AD=25$

$BC=8+5$

$BC=13$

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

$25\ne13$

Therefore, the quadrilateral is not a parallelogram.

Answer

No

Exercise #5

Below is a quadrilateral:

Given $∢B+∢C=180$

Is it possible that it is a parallelogram?

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:

$D=B=140$

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:

$B+C=180$

We replace the existing data:

$140+4x=180$

$4x=180-140$

$4x=40$

Divide by 4:

$\frac{4x}{4}=\frac{40}{4}$

$x=10$

Now we replace X:

$C=4\times10=40$

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.