Rhombuses, Rectangles, and Squares Section 6.4

Objectives Use properties of special types of parallelograms.

Key Vocabulary Rhombus Rectangle Square

Theorems Corollaries 6.10 Diagonals of a Rhombus Rhombus Corollary Rectangle Corollary Square Corollary 6.10 Diagonals of a Rhombus 6.13 Diagonals of a Rectangle

Special Parallelograms In this section we will study 3 special types of parallelograms. Rhombus Rectangle Square

Rhombuses Or Rhombi What makes a quadrilateral a rhombus?

Rhombuses Or Rhombi A rhombus is an equilateral parallelogram. Four congruent sides.

Rhombus Corollary If a quadrilateral has four congruent sides, then it is a rhombus. A B C D

Rhombi Since rhombi are parallelograms, they have all the properties of a parallelogram. Opposite sides are || and ≅. Opposite s are ≅. Consecutive s are supplementary. Diagonals bisect each other. In addition, rhombi have one other property, which is a theorem.

Theorem 6.10 The diagonals of a rhombus are perpendicular. If ABCD is a rhombus then AC BD. B C D A

Properties of a RHOMBUS 1. Two pairs of parallel sides. 2. All sides are congruent. 3. Diagonals are NOT congruent D B 4. Diagonals bisect each other 5. Diagonals form a right angle C 6. Consecutive angles are supplementary. m A B + = 180° C D

Example 1 The diagonals of rhombus WXYZ intersect at V. If WX = 8x – 5 and WZ = 6x + 3, find x.

Example 1 WX  WZ By definition, all sides of a rhombus are congruent. WX = WZ Definition of congruence 8x – 5 = 6x + 3 Substitution 2x – 5 = 3 Subtract 6x from each side. 2x = 8 Add 5 to each side. x = 4 Divide each side by 4. Answer: x = 4

Your Turn: ABCD is a rhombus. If BC = 4x – 5 and CD = 2x + 7, find x. A. x = 1 B. x = 3 C. x = 4 D. x = 6

Example 2 Use rhombus LMNP to find the value of y if N

Example 2 Answer: The value of y can be 12 or –12. N The diagonals of a rhombus are perpendicular. N Substitution Add 54 to each side. Take the square root of each side. Answer: The value of y can be 12 or –12.

Your Turn: Use rhombus ABCD and the given information to find the value of each variable. Answer: 8 or –8

Rectangles What makes a quadrilateral a rectangle?

Rectangles A rectangle is an equiangular parallelogram. All angles are congruent. What must each angle be then? Four right angles.

Rectangle Corollary If a quadrilateral has four right angles, then it is a rectangle.

Rectangles Since rectangles are parallelograms, they have all their properties: Opposite sides are || and ≅. Opposite s are ≅. Consecutive s are supplementary. Diagonals bisect each other. In addition, rectangles have their own special property, which leads us to our next theorem.

Theorem 6.11 Diagonals of a Rectangle are congruent.

Review: Properties of Rectangles All four s are right s (def. of rectangle). Opposite sides are || and ≅ (prop. of parallelogram). Opposite s are ≅ (prop. of parallelogram). Consecutive s are supplementary (prop. of parallelogram). Diagonals bisect each other (prop. of parallelogram). Diagonals are ≅ (theorem 6.11).

Example 3: Quadrilateral RSTU is a rectangle. If and find x.

Example 3: The diagonals of a rectangle are congruent, Definition of congruent segments Substitution Subtract 6x from each side. Add 4 to each side. Answer: 8

Your Turn: Quadrilateral EFGH is a rectangle. If and find x. Answer: 5

Example 4 A rectangular garden gate is reinforced with diagonal braces to prevent it from sagging. If JK = 12 feet, and LN = 6.5 feet, find KM.

Example 4 Since JKLM is a rectangle, it is a parallelogram. The diagonals of a parallelogram bisect each other, so LN = JN. LN = 6.5 feet JN + LN = JL Segment Addition LN + LN = JL Substitution 2LN = JL Simplify. 2(6.5) = JL Substitution 13 = JL Simplify.

Example 4 JL  KM If a is a rectangle, diagonals are . JL = KM Definition of congruence 13 = KM Substitution Answer: KM = 13 feet

Your Turn: Quadrilateral EFGH is a rectangle. If GH = 6 feet and FH = 15 feet, find GJ. A. 3 feet B. 7.5 feet C. 9 feet D. 12 feet

Example 5 Quadrilateral RSTU is a rectangle. If mRTU = 8x + 4 and mSUR = 3x – 2, find x.

Example 5 Since RSTU is a rectangle, it has four right angles. So, mTUR = 90. The diagonals of a rectangle bisect each other and are congruent, so PT  PU. Since triangle PTU is isosceles, the base angles are congruent so RTU  SUT and mRTU = mSUT. mRTU = 8x + 4 mSUR = 3x – 2 mSUT + mSUR = 90 Angle Addition mRTU + mSUR = 90 Substitution 8x + 4 + 3x – 2 = 90 Substitution 11x + 2 = 90 Add like terms.

Example 5 11x = 88 Subtract 2 from each side. x = 8 Divide each side by 11. Answer: x = 8

Your Turn: Quadrilateral EFGH is a rectangle. If mFGE = 6x – 5 and mHFE = 4x – 5, find x. A. x = 1 B. x = 3 C. x = 5 D. x = 10

Example 6a: Quadrilateral LMNP is a rectangle. Find x.

Example 6a: Answer: 10 Angle Addition Theorem Substitution Simplify. Subtract 10 from each side. Divide each side by 8. Answer: 10

Example 6b: Quadrilateral LMNP is a rectangle. Find y.

Example 6b: Since a rectangle is a parallelogram, opposite sides are parallel. So, alternate interior angles are congruent. Alternate Interior Angles Theorem Substitution Simplify. Subtract 2 from each side. Divide each side by 6. Answer: 5

Your Turn: Quadrilateral EFGH is a rectangle. a. Find x. b. Find y. Answer: 7 Answer: 11

Squares What makes a quadrilateral a square?

Definition: Square A square is a parallelogram with four congruent sides and four right angles.

Squares A square is a regular parallelogram. All angles are congruent All sides are congruent

Square Corollary If a quadrilateral has four congruent sides and four right angles, then it is a square.

Venn Diagram Shows the relationships between some members of the parallelogram family.

Properties of a SQUARE 1. Two pairs of parallel sides. A B 2. All sides are congruent. 3. All angles are right. 4. Diagonals are congruent 5. Diagonals bisect each other D C 6. Diagonals form a right angle 7. Opposite angles are congruent. 8. Consecutive angles are supplementary. m A B + = 180° C D

EX. 7 5X+5 D G DEFG is a square DG = 5X + 5 EF = 7X – 19 Find the value for X and the lenght of the side. E F 7X – 19 Since all sides are congruent: Now since all sides are congruent, we need to find the length of just one side: 5X + 5 = 7X – 19 -5X -5X DG = 5X + 5 5 = 2X – 19 +19 +19 = 5( ) + 5 12 = 60 + 5 24 = 2X 2 2 = 65 X=12 The length of the side is 65.

Your Turn: 9X – 3 K H HIJK is a square KH =9X – 3 IJ = 6X + 24 Find the value for X and the lenght of the side. J I 6X + 24 Since all sides are congruent: Now since all sides are congruent, we need to find the length of just one side: 9X – 3 = 6X + 24 -6X -6X KH = 9X – 3 3X – 3 = 24 +3 +3 = 9( ) – 3 9 = 81 – 3 3x = 27 3 3 = 78 X=9 The length of the side is 78.

Parallelogram, Rectangle, Rhombus, and Square Summary of Properties Parallelogram, Rectangle, Rhombus, and Square

Quadrilateral Relationships 1. Opposite sides parallel. 2. Opposite sides congruent. 3. Opposite angles are congruent. 4. Consecutive ∠s are supplementary. 5. Diagonals bisect each other. 1. Has 4 Congruent sides. 2. Diagonals are perpendicular. 1. Has 4 right angles. 2. Diagonals are congruent. 1. 4 congruent sides 2. 4 congruent (right) ∠s

x Characteristics Parallelogram Rectangle Rhombus Square Both pairs of opposite sides parallel x Diagonals are congruent Both pairs of opposite sides congruent At least one right angle Both pairs of opposite angles congruent Diagonals are perpendicular All sides are congruent Consecutive angles congruent Diagonals bisect each other Consecutive angles supplementary

Review Problems

Example 1 In the diagram, ABCD is a rectangle. b. Find mA, mB, mC, and mD. Find AD and AB. a. SOLUTION By definition, a rectangle is a parallelogram, so ABCD is a parallelogram. Because opposite sides of a parallelogram are congruent, AD = BC = 5 and AB = DC = 8. a. By definition, a rectangle has four right angles, so mA = mB = mC = mD = 90°. b.

Your Turn: In the diagram, PQRS is a rhombus. Find QR, RS, and SP. ANSWER QR = 6, RS = 6, SP = 6

Example 2 Use the information in the diagram to name the special quadrilateral. SOLUTION The quadrilateral has four right angles. So, by the Rectangle Corollary, the quadrilateral is a rectangle. Because all of the sides are not the same length, you know that the quadrilateral is not a square.

Your Turn: Use the information in the diagram to name the special quadrilateral. 1. ANSWER rhombus 2. ANSWER square

Example 3 ABCD is a rhombus. Find the value of x. SOLUTION By Theorem 6.10, the diagonals of a rhombus are perpendicular. Therefore, BEC is a right angle, so ∆BEC is a right triangle. By the Corollary to the Triangle Sum Theorem, the acute angles of a right triangle are complementary. So, x = 90 – 60 = 30.

Example 4 You nail four pieces of wood together to build a four-sided frame, as shown. What is the shape of the frame? a. The diagonals measure 7 ft 4 in. and 7 ft 2 in. Is the frame a rectangle? b. SOLUTION The frame is a parallelogram because both pairs of opposite sides are congruent. a. b. The frame is not a rectangle because the diagonals are not congruent.

Your Turn: Find the value of x. rhombus ABCD 1. ANSWER 90 rectangle EFGH 2. ANSWER 12 square JKLM 3. ANSWER 45

Match the properties of a quadrilateral The diagonals are congruent Both pairs of opposite sides are congruent Both pairs of opposite sides are parallel All angles are congruent All sides are congruent Parallelogram Rectangle Rhombus Square B,D A,B,C,D A,B,C,D B,D C,D

Decide if the statement is sometimes, always, or never true. A rhombus is equilateral. 2. The diagonals of a rectangle are ⊥. 3. The opposite angles of a rhombus are supplementary. 4. A square is a rectangle. 5. The diagonals of a rectangle bisect each other. 6. The consecutive angles of a square are supplementary. Always Sometimes Sometimes Always Always Always

Joke Time Why did the geometry student get so excited after they finished a jigsaw puzzle in only 6 months? Because on the box it said from 2-4 years. Why did the geometry student climb the chain-link fence? To see what was on the other side. How did the geometry student die drinking milk? The cow fell on them.

Assignment Pg. 328 - 330 #1 – 31 odd