1. Rectangles LESSON 6–4

2. Lesson Menu Five-Minute Check (over Lesson 6–3) TEKS 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

3. Over Lesson 6–3 5-Minute Check 1 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.

4. Over Lesson 6–3 5-Minute Check 2 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.

5. Over Lesson 6–3 5-Minute Check 5 Given that QRST is a parallelogram, which statement is true? A.m  S = 105 B.m  T = 105 C.QT  ST D.QT  QS ___

6. TEKS Targeted TEKS G.5(A) Investigate patterns to make conjectures about geometric relationships, including angles formed by parallel lines cut by a transversal, criteria required for triangle congruence, special segments of triangles, diagonals of quadrilaterals, interior and exterior angles of polygons, and special segments and angles of circles choosing from a variety of tools. G.6(E) Prove a quadrilateral is a parallelogram, rectangle, square, or rhombus using opposite sides, opposite angles, or diagonals and apply these relationships to solve problems. Mathematical Processes G.1(D), G.1(F)

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

8. Concept 1

9. Example 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.

10. Example 1 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=JLSegment Addition LN + LN=JLSubstitution 2LN=JLSimplify. 2(6.5)=JLSubstitution 13=JLSimplify.

11. Example 1 Use Properties of Rectangles Answer: KM = 13 feet JL = KMDefinition of congruence 13 = KMSubstitution JL  KMIf a is a rectangle, diagonals are .

12. Example 1 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.

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

14. Example 2 Use Properties of Rectangles and Algebra m  SUT + m  SUR=90Angle Addition m  RTU + m  SUR=90Substitution 8x + 4 + 3x – 2=90Substitution 11x + 2=90Add like terms. 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.

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

16. Example 2 A.x = 1 B.x = 3 C.x = 5 D.x = 10 Quadrilateral EFGH is a rectangle. If m  FGE = 6x – 5 and m  HFE = 4x – 5, find x.

17. Concept 2

18. Example 3 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.

19. Example 3 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.

20. Example 3 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.

21. Rectangles LESSON 6–4