Lesson 8-4

Rectangles

5-Minute Check on Lesson 8-35-Minute Check on Lesson 8-35-Minute Check on Lesson 8-35-Minute Check on Lesson 8-3 Transparency 8-4

Click the mouse button or press the Click the mouse button or press the Space Bar to display the answers.Space Bar to display the answers.

Determine whether each quadrilateral is a parallelogram. Justify your answer.

1. 2.

Determine whether the quadrilateral with the given vertices is a parallelogram using the method indicated.

3. A(,), B(,), C(,), D(,) Distance formula4. R(,), S(,), T(,), U(,) Slope formula

5. Which set of statements will prove LMNO a parallelogram?

Standardized Test Practice:

A

C

B

D

LM // NO and LO MN

L M

NO

LM LO and ON MN

LO // MN and LO MN

LO MN and LO ON

5-Minute Check on Lesson 8-35-Minute Check on Lesson 8-35-Minute Check on Lesson 8-35-Minute Check on Lesson 8-3 Transparency 8-4

Click the mouse button or press the Click the mouse button or press the Space Bar to display the answers.Space Bar to display the answers.

Determine whether each quadrilateral is a parallelogram. Justify your answer.

1. 2.

Determine whether the quadrilateral with the given vertices is a parallelogram using the method indicated.

3. A(,), B(,), C(,), D(,) Distance formula4. R(,), S(,), T(,), U(,) Slope formula

5. Which set of statements will prove LMNO a parallelogram?

Standardized Test Practice:

A

C

B

D

LM // NO and LO MN

L M

NO

LM LO and ON MN

LO // MN and LO MN

LO MN and LO ON

Yes, diagonal bisect each other

Yes, opposite angles congruent

Yes, opposite sides equalNo, RS not // UT

Objectives

• Recognize and apply properties of rectangles– A rectangle is a quadrilateral with four right angles

and congruent diagonals

• Determine whether parallelograms are rectangles– If the diagonals of a parallelogram are congruent,

then the parallelogram is a rectangle

Vocabulary

• Rectangle – quadrilateral with four right angles.

Polygon Hierarchy

Polygons

Squares

RhombiRectangles

Parallelograms Kites Trapezoids

IsoscelesTrapezoids

Quadrilaterals

Quadrilateral RSTU is a rectangle. If RT = 6x + 4 and SU = 7x - 4 find x.

The diagonals of a rectangle are congruent, so

Definition of congruent segments

Substitution

Subtract 6x from each side.

Add 4 to each side.

Answer: 8

Answer: 5

Quadrilateral EFGH is a rectangle. If FH = 5x + 4 and GE = 7x – 6, find x.

30°

60°A B

D C

Solve for x and y in the following rectangles

x

y

8

3x - 8

A B

D C

x2y + 8

4y -12

3y

2y

A B

D C

x

3x -9

Hint: Special Right Triangles

Hint: 2 Equations, 2 Variables Substitution

2xA B

D C

x

2x

x

P = 36 feet

Hint: p is perimeter

Quadrilateral LMNP is a rectangle. Find x.

Angle Addition Theorem

Answer: 10

Substitution

Simplify.

Subtract 10 from each side.

Divide each side by 8.

MLP is a right angle, so mMLP = 90°

Quadrilateral LMNP is a rectangle. Find y.

Since a rectangle is a parallelogram, opposite sides are parallel. So, alternate interior angles are congruent.

Alternate Interior Angles Theorem

Divide each side by 6.

Substitution

Subtract 2 from each side.

Simplify.

Answer: 5

Quadrilateral EFGH is a rectangle.

a. Find x. b. Find y.

Answer: 11 Answer: 7

Kyle is building a barn for his horse. He measures the diagonals of the door opening to make sure that they bisect each other and they are congruent. How does he know that the corners are angles?

We know that A parallelogram with congruent diagonals is a rectangle. Therefore, the corners are angles.

Answer:

Quadrilateral Characteristics Summary

Convex Quadrilaterals

Squares

RhombiRectangles

Parallelograms Trapezoids

IsoscelesTrapezoids

Opposite sides parallel and congruentOpposite angles congruentConsecutive angles supplementaryDiagonals bisect each other

Bases ParallelLegs are not ParallelLeg angles are supplementary Median is parallel to basesMedian = ½ (base + base)

Angles all 90°Diagonals congruent

Diagonals divide into 4 congruent triangles

All sides congruentDiagonals perpendicularDiagonals bisect opposite angles

Legs are congruent Base angle pairs congruent Diagonals are congruent

4 sided polygon4 interior angles sum to 3604 exterior angles sum to 360

Summary & Homework

• Summary:– A rectangle is a quadrilateral with four right angles

and congruent diagonals– If the diagonals of a parallelogram are congruent,

then the parallelogram is a rectangle

• Homework: – pg 428-429; 10-13, 16-20, 42