# Parallelogram Classification

## Rectangles, rhombuses, and squares are specific parallelograms.

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Parallelograms - Rhombuses, Rectagngles and Squares

What if you were designing a patio for you backyard? You decide to mark it off using your tape measure. Two sides are 21 feet long and two sides are 28 feet long. Explain how you would only use the tape measure to make your patio a rectangle. After completing this Concept, you'll be able to answer this question using properties of special parallelograms.

### Guidance

Rectangles, rhombuses (the plural is also rhombi) and squares are all more specific versions of parallelograms.

Rectangle Theorem: A quadrilateral is a rectangle if and only if it has four right (congruent) angles.

Rhombus Theorem: A quadrilateral is a rhombus if and only if it has four congruent sides.

Square Theorem: A quadrilateral is a square if and only if it has four right angles and four congruent sides.

From the Square Theorem, we can also conclude that a square is a rectangle and a rhombus.

Recall that diagonals in a parallelogram bisect each other. Therefore, the diagonals of a rectangle, square and rhombus also bisect each other. The diagonals of these parallelograms also have additional properties.

##### Investigation: Drawing a Rectangle

Tools Needed: pencil, paper, protractor, ruler

1. Draw two lines on either side of your ruler, to ensure they are parallel. Make these lines 3 inches long.
2. Remove the ruler and mark two \begin{align*}90^\circ\end{align*} angles, 2.5 inches apart on the bottom line drawn in Step 1. Then, draw the angles to intersect the top line. This will ensure that all four angles are \begin{align*}90^\circ\end{align*}. Depending on your ruler, the sides should be 2.5 inches and 1 inch.
3. Draw in the diagonals and measure them. What do you discover?

Theorem: A parallelogram is a rectangle if and only if the diagonals are congruent.

##### Investigation: Drawing a Rhombus

Tools Needed: pencil, paper, protractor, ruler

1. Draw two lines on either side of your ruler, to ensure they are parallel. Make these lines 3 inches long.
2. Remove the ruler and mark a \begin{align*}50^\circ\end{align*} angle, at the left end of the bottom line drawn in Step 1. Draw the other side of the angle and make sure it intersects the top line. Measure the length of this side.
3. The measure of the diagonal (red) side should be about 1.3 inches (if your ruler is 1 inch wide). Mark this length on the bottom line and the top line from the point of intersection with the \begin{align*}50^\circ\end{align*} angle. Draw in the fourth side. It will connect the two endpoints of these lengths.
4. By the way we drew this parallelogram; it is a rhombus because all four sides are 1.3 inches long. Draw in the diagonals.

Measure the angles created by the diagonals: the angles at their point of intersection and the angles created by the sides and each diagonal. You should find the measure of 12 angles total. What do you discover?

Theorem: A parallelogram is a rhombus if and only if the diagonals are perpendicular.

Theorem: A parallelogram is a rhombus if and only if the diagonals bisect each angle.

We know that a square is a rhombus and a rectangle. So, the diagonals of a square have the properties of a rhombus and a rectangle.

#### Example A

What type of parallelogram are the ones below?

a)

b)

a) All sides are congruent and one angle is \begin{align*}135^\circ\end{align*}, meaning that the angles are not congruent. By the Rhombus Theorem, this is a rhombus.

b) This quadrilateral has four congruent angles and all the sides are not congruent. By the Rectangle Theorem, this is a rectangle.

#### Example B

Is a rhombus SOMETIMES, ALWAYS, or NEVER a square? Explain your reasoning.

A rhombus has four congruent sides, while a square has four congruent sides and angles. Therefore, a rhombus is only a square when it also has congruent angles. So, a rhombus is SOMETIMES a square.

#### Example C

List everything you know about the square \begin{align*}SQRE\end{align*}.

A square has all the properties of a parallelogram, rectangle and rhombus.

Properties of Parallelograms Properties of Rhombuses Properties of Rectangles
• \begin{align*}\overline{SQ} \ || \ \overline{ER}\end{align*}
• \begin{align*}\overline{SQ} \cong \overline{ER} \cong \overline{SE} \cong \overline{QR}\end{align*}
• \begin{align*}\angle SER \cong \angle SQR \cong \angle QSE \cong \angle QRE\end{align*}
• \begin{align*}\overline{SE} \ || \ \overline{QR}\end{align*}
• \begin{align*}\overline{SR} \perp \overline{QE}\end{align*}
• \begin{align*}\overline{SR} \cong \overline{QE}\end{align*}
• \begin{align*}\overline{SQ} \cong \overline{ER}\end{align*}
• \begin{align*}\angle SEQ \cong \angle QER \cong \angle SQE \cong \angle EQR\end{align*}
• \begin{align*}\overline{SA} \cong \overline{AR} \cong \overline{QA} \cong \overline{AE}\end{align*}
• \begin{align*}\overline{SE} \cong \overline{QR}\end{align*}
• \begin{align*}\angle QSR \cong \angle RSE \cong \angle QRS \cong \angle SRE\end{align*}
• \begin{align*}\overline{SA} \cong \overline{AR}\end{align*}
• \begin{align*}\overline{QA} \cong \overline{AE}\end{align*}
• \begin{align*}\angle SER \cong \angle SQR\end{align*}
• \begin{align*}\angle QSE \cong \angle QRE\end{align*}

Watch this video for help with the Examples above.

#### Concept Problem Revisited

In order for the patio to be a rectangle, first the opposite sides must be congruent. So, two sides are 21ft and two are 28 ft. To ensure that the parallelogram is a rectangle without measuring the angles, the diagonals must be equal. You can find the length of the diagonals by using the Pythagorean Theorem.

\begin{align*}d^2&=21^2+28^2=441+784=1225\\ d& = \sqrt{1225}=35 \ ft\end{align*}

### Vocabulary

A parallelogram is a quadrilateral with two pairs of parallel sides.

A quadrilateral is a rectangle if and only if it has four right (congruent) angles:

A quadrilateral is a rhombus if and only if it has four congruent sides:

A quadrilateral is a square if and only if it has four right angles and four congruent sides.

### Guided Practice

1. Is a rectangle SOMETIMES, ALWAYS, or NEVER a parallelogram? Explain why.

2. Is a rhombus SOMETIMES, ALWAYS, or NEVER equiangular? Explain why.

3. Is a quadrilateral SOMETIMES, ALWAYS, or NEVER a pentagon? Explain why.

1. A rectangle has two sets of parallel sides, so it is ALWAYS a parallelogram.

2. Any quadrilateral, including a rhombus, is only equiangular if all its angles are \begin{align*}90^\circ\end{align*}. This means a rhombus is SOMETIMES equiangular, only when it is a square.

3. A quadrilateral has four sides, so it will NEVER be a pentagon with five sides.

### Practice

1. \begin{align*}RACE\end{align*} is a rectangle. Find:
1. \begin{align*}RG\end{align*}
2. \begin{align*}AE\end{align*}
3. \begin{align*}AC\end{align*}
4. \begin{align*}EC\end{align*}
5. \begin{align*}m \angle RAC\end{align*}
2. \begin{align*}DIAM\end{align*} is a rhombus. Find:
1. \begin{align*}MA\end{align*}
2. \begin{align*}MI\end{align*}
3. \begin{align*}DA\end{align*}
4. \begin{align*}m \angle DIA\end{align*}
5. \begin{align*}m \angle MOA\end{align*}
3. Draw a square and label it \begin{align*}CUBE\end{align*}. Mark the point of intersection of the diagonals \begin{align*}Y\end{align*}. Find:
1. \begin{align*}m \angle UCE\end{align*}
2. \begin{align*}m \angle EYB\end{align*}
3. \begin{align*}m \angle UBY\end{align*}
4. \begin{align*}m \angle UEB\end{align*}

For questions 4-12, determine if the quadrilateral is a parallelogram, rectangle, rhombus, square or none. Explain your reasoning.

For problems 13-15, find the value of each variable in the figures.

For questions 16-19 determine if the following are ALWAYS, SOMETIMES, or NEVER true. Explain your reasoning.

1. A rectangle is a rhombus.
2. A square is a parallelogram.
3. A parallelogram is regular.
4. A square is a rectangle.

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