Splash Screen

Five-Minute Check (over Lesson 6–3) CCSS 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

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

G.CO.11 Prove theorems about parallelograms. Content Standards G.CO.11 Prove theorems about parallelograms. G.GPE.4 Use coordinates to prove simple geometric theorems algebraically. Mathematical Practices 3 Construct viable arguments and critique the reasoning of others. 5 Use appropriate tools strategically. CCSS

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 Rhombii or Rhombuses Squares Vocabulary

Concept 3

Concept 1

Concept 2

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

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

rhombus square Vocabulary

Concept 1

Concept 1

Concept

Concept

Concept

Concept

HOMEWORK PGS. 426-428 #’S 5,6,14-19 & 23 PGS. 435-436 Sarah Robinson

Use Properties of a Rhombus A. The diagonals of rhombus WXYZ intersect at V. If mWZX = 39.5, find mZYX. Example 1A

mWZY + mZYX = 180 Consecutive Interior Angles Theorem Use Properties of a Rhombus Since WXYZ is a rhombus, diagonal ZX bisects WZY. Therefore, mWZY = 2mWZX. So, mWZY = 2(39.5) or 79. Since WXYZ is a rhombus, WZ║XY, and ZY is a transversal. mWZY + mZYX = 180 Consecutive Interior Angles Theorem 79 + mZYX = 180 Substitution mZYX = 101 Subtract 79 from both sides. Answer: mZYX = 101 Example 1A

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

WX  WZ By definition, all sides of a rhombus are congruent. Use Properties of a Rhombus 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 Example 1B

A. ABCD is a rhombus. Find mCDB if mABC = 126. A. mCDB = 126 B. mCDB = 63 C. mCDB = 54 D. mCDB = 27 Example 1A

B. 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 1B

Understand Plot the vertices on a coordinate plane. Classify Quadrilaterals Using Coordinate Geometry Determine whether parallelogram ABCD is a rhombus, a rectangle, or a square for A(–2, –1), B(–1, 3), C(3, 2), and D(2, –2). List all that apply. Explain. Understand Plot the vertices on a coordinate plane. Example 4

Classify Quadrilaterals Using Coordinate Geometry It appears from the graph that the parallelogram is a rhombus, rectangle, and a square. Plan If the diagonals are perpendicular, then ABCD is either a rhombus or a square. The diagonals of a rectangle are congruent. If the diagonals are congruent and perpendicular, then ABCD is a square. Solve Use the Distance Formula to compare the lengths of the diagonals. Example 4

Use slope to determine whether the diagonals are perpendicular. Classify Quadrilaterals Using Coordinate Geometry Use slope to determine whether the diagonals are perpendicular. Example 4

Answer: ABCD is a rhombus, a rectangle, and a square. Classify Quadrilaterals Using Coordinate Geometry Since the slope of is the negative reciprocal of the slope of the diagonals are perpendicular. The lengths of and are the same, so the diagonals are congruent. Answer: ABCD is a rhombus, a rectangle, and a square. Check You can verify ABCD is a square by using the Distance Formula to show that all four sides are congruent and by using the Slope Formula to show consecutive sides are perpendicular. Example 4

End of the Lesson