6.4: Rectangles, Rhombuses and Squares
Learning Objectives
 Define a rectangle, rhombus, and square.
 Determine if a parallelogram is a rectangle, rhombus, or square in the \begin{align*}xy\end{align*} plane.
 Compare the diagonals of a rectangle, rhombus, and square.
Review Queue
 List five examples where you might see a square, rectangle, or rhombus in real life.
 Find the values of \begin{align*}x\end{align*} and \begin{align*}y\end{align*} that would make the quadrilateral a parallelogram.
Know What? You are designing a patio for your backyard and are marking it off with a 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. (You do not need to find any measurements.)
Defining Special Parallelograms
Rectangles, Rhombuses (also called Rhombi) and Squares are all more specific versions of parallelograms, also called special parallelograms. Taking the theorems we learned in the previous two sections, we have three more new theorems.
Rectangle Theorem: A quadrilateral is a rectangle if and only if it has four right (congruent) angles.
\begin{align*}ABCD\end{align*} is a rectangle if and only if \begin{align*}\angle A \cong \angle B \cong \angle C \cong \angle D\end{align*}.
Rhombus Theorem: A quadrilateral is a rhombus if and only if it has four congruent sides.
\begin{align*}ABCD\end{align*} is a rhombus if and only if \begin{align*}\overline{AB} \cong \overline{BC} \cong \overline{CD} \cong \overline{AD}\end{align*}.
Square Theorem: A quadrilateral is a square if and only if it has four right angles and four congruent sides.
\begin{align*}ABCD\end{align*} is a square if and only if \begin{align*}\angle A \cong \angle B \cong \angle C \cong \angle D\end{align*} and \begin{align*}\overline{AB} \cong \overline{BC} \cong \overline{CD} \cong \overline{AD}\end{align*}.
From the Square Theorem, we can also conclude that a square is a rectangle and a rhombus.
Example 1: What type of parallelogram are the ones below?
a)
b)
Solution:
a) All sides are congruent and one angle is \begin{align*}135^\circ\end{align*}, so the angles are not congruent. This is a rhombus.
b) All four congruent angles and the sides are not. This is a rectangle.
Example 2: Is a rhombus SOMETIMES, ALWAYS, or NEVER a square? Explain why.
Solution: A rhombus has four congruent sides and a square has four congruent sides and angles. Therefore, a rhombus is a square when it has congruent angles. This means a rhombus is SOMETIMES a square.
Example 3: Is a rectangle SOMETIMES, ALWAYS, or NEVER a parallelogram? Explain why.
Solution: A rectangle has two sets of parallel sides, so it is ALWAYS a parallelogram.
Diagonals in Special Parallelograms
Recall from previous lessons that the diagonals in a parallelogram bisect each other. Therefore, the diagonals of a rectangle, square and rhombus also bisect each other. They also have additional properties.
Investigation 63: Drawing a Rectangle
Tools Needed: pencil, paper, protractor, ruler
 Like with Investigation 62, draw two lines on either side of your ruler, making them parallel. Make these lines 3 inches long.
 Using the protractor, mark two \begin{align*}90^\circ\end{align*} angles, 2.5 inches apart on the bottom line from Step 1. Extend the sides to intersect the top line.
 Draw in the diagonals and measure. What do you discover?
Theorem 614: A parallelogram is a rectangle if the diagonals are congruent.
\begin{align*}ABCD\end{align*} is parallelogram. If \begin{align*}\overline{AC} \cong \overline{BD}\end{align*}, then \begin{align*}ABCD\end{align*} is also a rectangle.
Investigation 64: Drawing a Rhombus
Tools Needed: pencil, paper, protractor, ruler
1. Like with Investigation 62 and 63, draw two lines on either side of your ruler, 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. Mark the length found in Step 2 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 the sides are equal. Draw in the diagonals.
Measure the angles at the point of intersection of the diagonals (4).
Measure the angles created by the sides and each diagonal (8).
Theorem 615: A parallelogram is a rhombus if the diagonals are perpendicular.
\begin{align*}ABCD\end{align*} is a parallelogram. If \begin{align*}\overline{AC} \perp \overline{BD}\end{align*}, then \begin{align*}ABCD\end{align*} is also a rhombus.
Theorem 616: A parallelogram is a rhombus if the diagonals bisect each angle.
\begin{align*}ABCD\end{align*} is a parallelogram. If \begin{align*}\overline{AC}\end{align*} bisects \begin{align*}\angle BAD\end{align*} and \begin{align*}\angle BCD\end{align*} and \begin{align*}\overline{BD}\end{align*} bisects \begin{align*}\angle ABC \ \angle ADC\end{align*}, then \begin{align*}ABCD\end{align*} is also a rhombus.
The converses of these three theorems are true. There are no theorems about the diagonals of a square. The diagonals of a square have the properties of a rhombus and a rectangle.
Example 4: List everything you know about the square \begin{align*}SQRE\end{align*}.
Solution: A square has all the properties of a parallelogram, rectangle and rhombus.
Properties of a Parallelogram  Properties of a Rhombus  Properties of a Rectangle 












All the bisected angles are \begin{align*}45^\circ\end{align*}.
Parallelograms in the Coordinate Plane
Example 4: Determine what type of parallelogram \begin{align*}TUNE\end{align*} is: \begin{align*}T(0, 10), U(4, 2), N(2, 1)\end{align*}, and \begin{align*}E(6, 7)\end{align*}.
Solution: Let’s see if the diagonals are equal. If they are, then \begin{align*}TUNE\end{align*} is a rectangle.
\begin{align*}EU & = \sqrt{(6 4)^2 + (72)^2} && TN = \sqrt{(0 + 2)^2 +(10 + 1)^2}\\ & = \sqrt{(10)^2 + 5^2} && \quad \ \ = \sqrt{2^2 + 11^2}\\ & = \sqrt{100 + 25} && \quad \ \ = \sqrt{4 + 121}\\ & = \sqrt{125} && \quad \ \ = \sqrt{125}\end{align*}
If the diagonals are also perpendicular, then \begin{align*}TUNE\end{align*} is a square.
\begin{align*}\text{Slope of}\ EU = \frac{7  2}{6  4} = \frac{5}{10} = \frac{1}{2} \quad \text{Slope of}\ TN = \frac{10  7}{0(6)} = \frac{3}{6} = \frac{1}{2}\end{align*}
The slope of \begin{align*}EU \neq\end{align*} slope of \begin{align*}TN\end{align*}, so \begin{align*}TUNE\end{align*} is a rectangle.
Steps to determine if a quadrilateral is a parallelogram, rectangle, rhombus, or square.
1. Graph the four points on graph paper.
2. See if the diagonals bisect each other. (midpoint formula)
Yes: Parallelogram, continue to #2. No: A quadrilateral, done.
3. See if the diagonals are equal. (distance formula)
Yes: Rectangle, skip to #4. No: Could be a rhombus, continue to #3.
4. See if the sides are congruent. (distance formula)
Yes: Rhombus, done. No: Parallelogram, done.
5. See if the diagonals are perpendicular. (find slopes)
Yes: Square, done. No: Rectangle, done.
Know What? Revisited In order for the patio to be a rectangle, the opposite sides must be congruent (see picture). To ensure that the parallelogram is a rectangle without measuring the angles, the diagonals must be equal.
Review Questions
 Questions 13 are similar to #2 in the Review Queue and Example 1.
 Questions 415 are similar to Example 1.
 Questions 1621 are similar to Examples 2 and 3.
 Questions 2225 are similar to Investigations 63 and 64.
 Questions 2629 are similar to Example 4.
 Question 30 is a challenge.

\begin{align*}RACE\end{align*} is a rectangle. Find:
 \begin{align*}RG\end{align*}
 \begin{align*}AE\end{align*}
 \begin{align*}AC\end{align*}
 \begin{align*}EC\end{align*}
 \begin{align*}m \angle RAC\end{align*}

\begin{align*}DIAM\end{align*} is a rhombus. Find:
 \begin{align*}MA\end{align*}
 \begin{align*}MI\end{align*}
 \begin{align*}DA\end{align*}
 \begin{align*}m \angle DIA\end{align*}
 \begin{align*}m \angle MOA\end{align*}

\begin{align*}CUBE\end{align*} is a square. Find:
 \begin{align*}m \angle UCE\end{align*}
 \begin{align*}m \angle EYB\end{align*}
 \begin{align*}m \angle UBY\end{align*}
 \begin{align*}m \angle UEB\end{align*}
For questions 415, determine if the quadrilateral is a parallelogram, rectangle, rhombus, square or none.
For questions 1621 determine if the following are ALWAYS, SOMETIME, or NEVER true. Explain your reasoning.
 A rectangle is a rhombus.
 A square is a parallelogram.
 A parallelogram is regular.
 A square is a rectangle.
 A rhombus is equiangular.
 A quadrilateral is a pentagon.
Construction Draw or construct the following quadrilaterals.
 A quadrilateral with congruent diagonals that is not a rectangle.
 A quadrilateral with perpendicular diagonals that is not a rhombus or square.
 A rhombus with a 6 cm diagonal and an 8 cm diagonal.
 A square with 2 inch sides.
For questions 2629, determine what type of quadrilateral \begin{align*}ABCD\end{align*} is. Use Example 4 and the steps following it to help you.
 \begin{align*}A(2, 4), B(1, 2), C(3, 1), D(4, 3)\end{align*}
 \begin{align*}A(2, 3), B(3, 4), C(2, 1), D(3, 2)\end{align*}
 \begin{align*}A(1, 1), B(7, 1), C(8, 2), D(2, 4)\end{align*}
 \begin{align*}A(10, 4), B(8, 2), C(2, 2), D(4, 8)\end{align*}

Challenge \begin{align*}SRUE\end{align*} is a rectangle and \begin{align*}PRUC\end{align*} is a square.
 What type of quadrilateral is \begin{align*}SPCE\end{align*}?
 If \begin{align*}SR = 20\end{align*} and \begin{align*}RU = 12\end{align*}, find \begin{align*}CE\end{align*}.
 Find \begin{align*}SC\end{align*} and \begin{align*}RC\end{align*} based on the information from part b. Round your answers to the nearest hundredth.
Review Queue Answers
 Possibilities: picture frame, door, baseball diamond, windows, walls, floor tiles, book cover, pages/paper, table/desk top, black/white board, the diamond suit (in a deck of cards).
 \begin{align*}x = 11, \ y = 6\end{align*}
 \begin{align*}x = y = 90^\circ\end{align*}
 \begin{align*}x = 9, \ y = 133^\circ\end{align*}
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