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Five-Minute Check (over Lesson 6–3)

Then/Now

New Vocabulary

Theorem 6.13: Diagonals of a Rectangle

Example 1: Real-World Example: Use Properties of Rectangles

Example 2: Use Properties of Rectangles and Algebra

Theorem 6.14

Example 3: Real-World Example: Proving Rectangle Relationships

Example 4: Rectangles and Coordinate Geometry

Over Lesson 6–3

A. A

B. B

C. C

D. D

A. Yes, all sides are congruent.

B. Yes, all angles are congruent.

C. Yes, diagonals bisect each other.

D. No, diagonals are not congruent.

Determine whether the quadrilateral is a parallelogram.

Over Lesson 6–3

A. A

B. B

C. C

D. D

A. Yes, both pairs of opposite angles are congruent.

B. Yes, diagonals are congruent.

C. No, all angles are not congruent.

D. No, side lengths are not given.

Determine whether the quadrilateral is a parallelogram.

Over Lesson 6–3

A. yes

B. no

Use the Distance Formula to determine if A(3, 7), B(9, 10), C(10, 6), D(4, 3) are the vertices of a parallelogram.

A. A

B. B

Over Lesson 6–3

A. A

B. B

C. C

D. D

Given that QRST is a parallelogram, which statement is true?

A. mS = 105

B. mT = 105

C. QT ST

D. QT QS

______

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Lesson 6-4 Rectangles (Pg. 419)

TARGETS

• Recognize and apply properties of rectangles.

• Determine whether parallelograms are rectangles.

Content StandardsG-CO.11 Prove geometric theorems. G-CO.12 Make geometric constructions. G-GPE.4 Use coordinates to prove simple geometric theorems algebraically.

Mathematical Practices

1 Make sense of problems and persevere in solving them

2 Reason abstractly and quantitatively.

6 Attend to precision.

You used properties of parallelograms and determined whether quadrilaterals were parallelograms. (Lesson 6–2)

• Recognize and apply properties of rectangles.

• Determine whether parallelograms are rectangles.

• Rectangle - parallelogram with four right angles

•In a rectangle:

•All angles are right angles

•Opposite sides are parallel and congruent

•Opposite angles are congruent

•Consecutive angles are supplementary

•Diagonals bisect each other

Use Properties of Rectangles

CONSTRUCTION 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.

Use Properties of Rectangles

Since JKLM is a rectangle, it is a parallelogram. The diagonals of a parallelogram bisect each other, so LN = JN.

JN + LN = JL Segment Addition

LN + LN = JL Substitution

2LN = JL Simplify.

2(6.5) = JL Substitution

13 = JL Simplify.

Use Properties of Rectangles

Answer: KM = 13 feet

JL = KM Definition of congruence

13 = KM Substitution

JL KM If a is a rectangle,diagonals are .

A. A

B. B

C. C

D. D

A. 3 feet

B. 7.5 feet

C. 9 feet

D. 12 feet

Quadrilateral EFGH is a rectangle. If GH = 6 feet and FH = 15 feet, find GJ.

Use Properties of Rectangles and Algebra

Quadrilateral RSTU is a rectangle. If mRTU = 8x + 4 and mSUR = 3x – 2, find x.

Use Properties of Rectangles and Algebra

mSUT + mSUR = 90 Angle Addition

mRTU + mSUR = 90 Substitution

8x + 4 + 3x – 2 = 90 Substitution

11x + 2 = 90 Add like terms.

Since RSTU is a rectangle, it has four right angles. So, mTUR = 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 mRTU = mSUT.

Use Properties of Rectangles and Algebra

Answer: x = 8

11x = 88 Subtract 2 from eachside.

x = 8 Divide each side by 11.

A. A

B. B

C. C

D. D

A. x = 1

B. x = 3

C. x = 5

D. x = 10

Quadrilateral EFGH is a rectangle. If mFGE = 6x – 5 and mHFE = 4x – 5, find x.

Proving Rectangle Relationships

ART Some artists stretch their own canvas over wooden frames. This allows them to customize the size of a canvas. In order to ensure that the frame is rectangular before stretching the canvas, an artist measures the sides and the diagonals of the frame. If AB = 12 inches, BC = 35 inches, CD = 12 inches, DA = 35 inches, BD = 37 inches, and AC = 37 inches, explain how an artist can be sure that the frame is rectangular.

Proving Rectangle Relationships

Since AB = CD, DA = BC, and AC = BD, AB CD, DA BC, and AC BD.

Answer: Because AB CD and DA BC, ABCD is a parallelogram. Since AC and BD are congruent diagonals in parallelogram ABCD, it is a rectangle.

A. A

B. B

C. C

D. D

Max is building a swimming pool in his backyard. He measures the length and width of the pool so that opposite sides are parallel. He also measures the diagonals of the pool to make sure that they are congruent. How does he know that the measure of each corner is 90?

A. Since opp. sides are ||, STUR must be a rectangle.

B. Since opp. sides are , STUR must be a rectangle.

C. Since diagonals of the are , STUR must be a rectangle.

D. STUR is not a rectangle.

• Homework p 422 14-18 even, 26, 27, 29, 31