Parallelogram

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

Vertically opposite angles: 2 straight lines that cross each other to form 4 angles at their meeting point. The 2 non-adjacent angles are called vertices.

Important to know: vertically opposite angles are equal.    

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

• Bisector: divides the angle into 2 equal parts.

So, what are the properties of this special quadrilateral called a parallelogram? Get a brief summary

• $AB ǁ DC$ (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:

$ΔADC≅ΔCBA$ (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)

Let's demonstrate that:

• $AO=CO$
• $BO=DO$

To do this, we will superimpose the triangles: $ΔAOB$ with $ΔCOD$

• $AB = DC$ Opposite sides are equal in a parallelogram
• $AB ǁ DC$ by the definition of the parallelogram 

Therefore:

• Angles $B1 = D1$
• Angles $A1 = C1$

According to the theorem alternate angles between equal parallels, therefore:

$OBAOB ≅ ΔCOD$ (according to the congruence theorem: angle, side, angle)

From the congruence, it can be deduced:

• $AO=CO$
• $BO=DO$

According to corresponding sides in equal congruent triangles

We will check in the following exercise if we understood the properties of the parallelogram:

Find the following values in the parallelogram:

• $x$
• $y$
• $t$
• $k$
• $\alpha$
• $\beta$

Observing the properties of the parallelogram:

• The opposite sides are equal therefore
  • $Y=7$
  • $X=5$
• The diagonals intersect therefore
  • $k=4.5$
  • $t=4$
• The alternate angles between parallels are equal therefore
  • $β=50°$
  • $α=30°$

The calculation of the perimeter of a parallelogram is twice the sum of 2 adjacent sides, therefore

$24cm=7\times2+5\times2$

To calculate the area of a parallelogram we will draw a line from one of the vertices perpendicular to the opposite side.

Area of the parallelogram = base x height

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 $4$.

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?

According to the figure

• $AB=DC$
• $AD=BC$
• $AC = AC$ This is a common side

It can be concluded:

• $ΔABC ≅ ΔCDA$ According to the congruence theorem: side, side, side

Therefore:

• $\sphericalangle BAC=\sphericalangle ACD$
• $\sphericalangle ACB=\sphericalangle DAC$

According to the theorem corresponding angles in congruent triangles are equal

Therefore:

$AB ǁ DC$

• $AD ǁ BC$ [when alternate angles are equal - the lines are parallel]

Therefore, $ABCD$ is a parallelogram (2 pairs of opposite sides are parallel)

We will mark:

• Angles $α = B = D$
• Angles $β=A=C$
• The sum of the angles in a quadrilateral is $360^o$ Therefore, the equation $2α+2β=360^o$ is obtained
  Divide the equation by 2 and obtain: $180^o=β+α$

Therefore

• $AB ǁ DC$
• $AD ǁ BC$ {when the sum of the angles on one side is $180^o$ then they are parallel lines}
• $ABCD$ is a parallelogram (when there are 2 pairs of opposite sides parallel it is a parallelogram)

When given:

• $AO=CO$
• $BO=DO$

And the angle trapped between them:

• $\sphericalangle AOB=\sphericalangle DOC$ (Vertically opposite angles are equal)

It can be concluded that: $ΔABO≅ΔCOD$ (according to the congruence theorem: side, angle, side)

Therefore:

• $AB = CD$ (corresponding sides in congruent triangles are equal)

In the same way, we will superimpose $ΔBOC$ with $ΔAOD$

According to the data:

• $BO=DO$
• $AO=CO$
• $\sphericalangle BOC=\sphericalangle AOD$ (Vertically opposite angles are equal)

Therefore:

• $ΔBOC ≅ ΔDOA$ (according to the congruence theorem: side, angle, side)
• And we obtain: $AD = BC$ (corresponding sides in congruent triangles are equal)

Therefore: $ABCD$ is a parallelogram (2 pairs of opposite sides equal, the quadrilateral is a parallelogram)

When the data is:

• $AB=DC$
• $AB ǁ DC$

Then:

• $\sphericalangle BAC=\sphericalangle ACD$
• $\sphericalangle ABD=\sphericalangle BDC$
  According to the theorem alternate angles between equal parallels

Therefore:

• $ΔABO≅ΔCDO$ (according to the congruence theorem: angle, side, angle)

From the congruence:

• $AO=CO$
• $BO = DO$ (corresponding sides in equal congruent triangles)

Therefore: $ABCD$ is a parallelogram (a quadrilateral where the diagonals cross is a parallelogram)

Assignment

Given the quadrilateral $ABCD$

Given that:

$\sphericalangle D=95^o$

$\sphericalangle C=85^o$

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

Solution

In fact, this quadrilateral is a parallelogram because two angles are adjacent

the same side, complementary to: $180^o$

Answer

Yes

Assignment

Given the quadrilateral $ABCD$

Given that:

$\sphericalangle A=100^o$

$\sphericalangle C=80^o$

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

Solution

In fact, this quadrilateral is a parallelogram because two angles are adjacent

the same side, complementary to: $180^o$

Answer

Yes

Assignment

Given the quadrilateral $ABCD$

such that:

$∢A=100°$,$$

And... $∢C=70°$

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

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

Assignment

Given the quadrilateral $ABCD$

$AF=4$

$FD=6$

$BF=2$

$FC=3$

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