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 Lesson Menu

A B C D Determine whether the quadrilateral is a parallelogram. A. Yes, all sides are congruent. B. Yes, all angles are congruent. C. Yes, diagonals bisect each other. D. No, diagonals are not congruent. A B C D 5-Minute Check 1

A B C D Determine whether the quadrilateral is a parallelogram. 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. A B C D 5-Minute Check 2

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. yes B. no A B 5-Minute Check 3

A B C D Given that QRST is a parallelogram, which statement is true? A. mS = 105 B. mT = 105 C. QT  ST D. QT  QS ___ A B C D 5-Minute Check 5

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Lesson 6-4 Rectangles (Pg. 419) TARGETS Recognize and apply properties of rectangles. Determine whether parallelograms are rectangles. Then/Now

Content Standards G-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. Then/Now

Recognize and apply properties of rectangles. 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. Then/Now

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 Vocabulary

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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. Example 1

JN + LN = JL Segment Addition LN + LN = JL Substitution 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. Example 1

JL  KM If a is a rectangle, diagonals are . Use Properties of Rectangles JL  KM If a is a rectangle, diagonals are . JL = KM Definition of congruence 13 = KM Substitution Answer: KM = 13 feet Example 1

Quadrilateral EFGH is a rectangle 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 A B C D Example 1

Use Properties of Rectangles and Algebra Quadrilateral RSTU is a rectangle. If mRTU = 8x + 4 and mSUR = 3x – 2, find x. Example 2

mSUT + mSUR = 90 Angle Addition mRTU + mSUR = 90 Substitution Use Properties of Rectangles and Algebra 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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 2

11x = 88 Subtract 2 from each side. x = 8 Divide each side by 11. Use Properties of Rectangles and Algebra 11x = 88 Subtract 2 from each side. x = 8 Divide each side by 11. Answer: x = 8 Example 2

Quadrilateral EFGH is a rectangle 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 A B C D Example 2

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. Example 3

Since AB = CD, DA = BC, and AC = BD, AB  CD, DA  BC, and AC  BD. 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. Example 3

Max is building a swimming pool in his backyard 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. A B C D Example 3

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