1. LESSON 6–4
Rectangles

2. 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
Lesson Menu

3. 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

4. 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

5. 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

6. 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

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

8. Mathematical Processes G.1(D), G.1(F)
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)
TEKS

9. 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

10. rectangle
Vocabulary

11. Concept 1

12. 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

13. 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

14. 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

15. 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

16. 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

17. 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 x – 2 = 90 Substitution
11x + 2 = 90 Add like terms.
Example 2

18. 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

19. 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

20. Concept 2

21. 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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. LESSON 6–4
Rectangles