Understanding Theorems About Parallelogram and Analyzing Diagonals of a Rectangle and a Rhombus

Reflection Across $A B$
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$R$	$Q$
$S$	$P$
$M$	$M$

Reflection Across

A B

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Image

R

Q

S

P

M

M

Parallelogram Consecutive Angles Theorem
The consecutive angles of a parallelogram are supplementary.

Consider the following parallelogram.

Find the value of

x .

Find the value of

y .

Find the value of

z .

According to the Parallelogram Opposite Sides Theorem opposite sides in a parallelogram are congruent. That means CD≅ AB.

With this information, we can write and solve the following equation. x-8= 20 ⇔ x=28

The Parallelogram Consecutive Angles Theorem states that consecutive angles of a parallelogram are supplementary. That means their angle measures add to 180^(∘).

With this information, we can write and solve the following equation. y+110^(∘)=180^(∘) ⇔ y=70^(∘)

By the Parallelogram Opposite Angles Theorem opposite angles of a parallelogram are congruent.

With this information, we can write and solve an equation. z-21^(∘)= 110^(∘) ⇔ z=131^(∘)

Consider the following parallelogram.

A parallelogram labeled ABCD with both diagonals drawn

Determine the following in $A B C D .$

The length of \overline{A B}

The length of \overline{D B}

The measure of ∠ A B C

By the Parallelogram Opposite Sides Theorem, any pair of opposite sides of a parallelogram are congruent. Therefore, we know that AB≅ DC.

Since AB and DC are congruent we can determine the length of AB. DC=AB=7

According to the Parallelogram Diagonals Theorem, the diagonals of a parallelogram bisect each other. This means they cut each other in two congruent halves which tells us that DE≅ EB.

Since DE≅ EB we know that the length of DB must be twice the length of DE or EB. Therefore, DB equals 2( 4.7)=9.4 units.

According to the Parallelogram Consecutive Angles Theorem, consecutive angles of a parallelogram are supplementary. That means the sum of their angle measures equals 180^(∘).

Since ∠ ABC and ∠ DAB are consecutive angles, we can write the following equation. 101^(∘)+m∠ ABC = 180^(∘) ⇓ m∠ ABC = 79^(∘)

Consider the properties of the following quadrilateral's segments and angles. Which quadrilateral(s) can be classified as a parallelogram?

Before we classify the quadrilaterals, let's recall the definition of a parallelogram.

Parallelogram |- A parallelogram is a quadrilateral that has two pairs of parallel sides.

Let's analyze the four quadrilaterals one at a time.

Quadrilateral A

As we can see, one pair of opposite sides are parallel. However, we only know that the second pair of sides are congruent. That would only be enough information to say this is an isosceles trapezoid.

To determine that it is a parallelogram, additional information would be needed.

Quadrilateral B

For B, we see that each diagonal consists of two congruent segments. Therefore, the diagonals bisect each other. According to the Converse Parallelogram Diagonals Theorem, a quadrilateral is a parallelogram if the diagonals bisect each other. Therefore, B is a parallelogram.

Quadrilateral C

In this quadrilateral, we see that the opposite angles are congruent. According to the Converse Parallelogram Opposite Angles Theorem, if opposite angles of a quadrilateral are congruent, it is a parallelogram. Therefore, C is a parallelogram.

Quadrilateral D

Here, when checking the hatch marks, we see that there is only one pair of congruent sides. A parallelogram is a quadrilateral with two pairs of congruent sides. Therefore, this is not a parallelogram.

Conclusion

Of the four quadrilaterals, B and C are parallelograms.

Which of the following quadrilateral(s) can be classified as squares?

Before we classify the quadrilaterals, let's go over the definition of a square.

Square |- A square is a quadrilateral that has four right angles and four congruent sides.

Let's go analyze the four quadrilaterals one at a time.

Quadrilateral A

This quadrilateral has four congruent sides, which is true for a square. However, we have no information about the quadrilateral's interior angles. They may look like right angles, but they do not have to be. Therefore, we can not claim that this is a square. We can, however, say that it is a rhombus — a parallelogram with four congruent sides.

Quadrilateral B

In this case, we have a quadrilateral with four right angles. However, we do not have information about the side length which means we can not say that this is a square either. We can however, claim that it is a rectangle — a parallelogram with four right angles.

Quadrilateral C

In a square, the diagonals are always congruent. However, this is also true for a rectangle. Since we have no information about the quadrilateral's sides, we can only say for sure that this is a rectangle.

Quadrilateral D

In a square, the diagonal's intersect each other at a right angle. However, that is also true for a rhombus. Therefore, while we know that D has four congruent sides, we do not know if it has four right angles. Therefore, all we can say is that D is a rhombus.

Conclusion

According to the given information and properties of quadrilaterals, none of the given quadrilaterals in the exercise can be classified as a square.

	14 Theory slides
	9 Exercises - Grade E - A
	Each lesson is meant to take 1-2 classroom sessions

Theorems About Parallelograms

Catch-Up and Review

Proof

Proof

Proof

Proof

Completed Proof

Hint

Solution

Value of x

Values of a and b

Proof

Proof

Hint

Solution

Proof

Part I: PQRS Is a Rectangle ⇒PR≅QS​

Part II: PR≅QS​⇒PQRS Is a Rectangle

Proof

Part I: PQRS Is a Rectangle ⇒PR≅QS​

Part II: PR≅QS​⇒PQRS Is a Rectangle

Hint

Solution

Proof

If a Parallelogram Is a Rhombus, Then Its Diagonals Are Perpendicular

If Its Diagonals Are Perpendicular, Then a Parallelogram is a Rhombus

Hint

Solution

Quadrilateral A

Quadrilateral B

Quadrilateral C

Quadrilateral D

Conclusion

Quadrilateral A

Quadrilateral B

Quadrilateral C

Quadrilateral D

Conclusion

Theorems About Parallelograms

Recommended exercises

Value of $x$

Values of $a$ and $b$

Part I: $P Q R S$ Is a Rectangle $\Rightarrow \overline{P R} ≅ \overline{Q S}$

Part II: $\overline{P R} ≅ \overline{Q S} \Rightarrow P Q R S$ Is a Rectangle

Part I: $P Q R S$ Is a Rectangle $\Rightarrow \overline{P R} ≅ \overline{Q S}$

Part II: $\overline{P R} ≅ \overline{Q S} \Rightarrow P Q R S$ Is a Rectangle