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. 5-Minute Check 1

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. 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 5-Minute Check 3

Use the Slope Formula to determine if R(2, 3), S(–1, 2), T(–1, –2), U(2, –2) are the vertices of a parallelogram. A. yes B. no 5-Minute Check 4

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

Recognize and apply properties of rectangles. You used properties of parallelograms and determined whether quadrilaterals were parallelograms. Recognize and apply properties of rectangles. Determine whether parallelograms are rectangles. Then/Now

rectangle Vocabulary

Concept 1

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 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 Example 2

Concept 2

A. Since opp. sides are ||, STUR must be a rectangle. 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. Example 3

Rectangles and Coordinate Geometry Quadrilateral JKLM has vertices J(–2, 3), K(1, 4), L(3, –2), and M(0, –3). Determine whether JKLM is a rectangle using the Distance Formula. Step 1 Use the Distance Formula to determine whether JKLM is a parallelogram by determining if opposite sides are congruent. Example 4

Rectangles and Coordinate Geometry Since opposite sides of a quadrilateral have the same measure, they are congruent. So, quadrilateral JKLM is a parallelogram. Example 4

Step 2 Determine whether the diagonals of JKLM are congruent. Rectangles and Coordinate Geometry Step 2 Determine whether the diagonals of JKLM are congruent. Answer: Since the diagonals have the same measure, they are congruent. So JKLM is a rectangle. Example 4

Quadrilateral WXYZ has vertices W(–2, 1), X(–1, 3), Y(3, 1), and Z(2, –1). Determine whether WXYZ is a rectangle by using the Distance Formula. A. yes B. no C. cannot be determined Example 4

Quadrilateral WXYZ has vertices W(–2, 1), X(–1, 3), Y(3, 1), and Z(2, –1). What are the lengths of diagonals WY and XZ? A. B. 4 C. 5 D. 25 Example 4