Rectangles

1
Lesson 8-4

• Rectangles

2
Transparency 8-4
5-Minute Check on Lesson 8-3

• Determine whether each quadrilateral is a
  parallelogram. Justify your answer.
• 2.
• Determine whether the quadrilateral with the
  given vertices is a parallelogram using the
  method indicated.
• 3. A(,), B(,), C(,), D(,) Distance formula
• 4. R(,), S(,), T(,), U(,) Slope formula
• 5.
  Which set of statements will prove LMNO a
  parallelogram?

L
M
Standardized Test Practice
N
O
LM // NO and LO ? MN
LO // MN and LO ? MN
A
B
LM ? LO and ON ? MN
LO ? MN and LO ? ON
C
D
Click the mouse button or press the Space Bar to
display the answers.
3
Transparency 8-4
5-Minute Check on Lesson 8-3

• Determine whether each quadrilateral is a
  parallelogram. Justify your answer.
• 2.
• Determine whether the quadrilateral with the
  given vertices is a parallelogram using the
  method indicated.
• 3. A(,), B(,), C(,), D(,) Distance formula
• 4. R(,), S(,), T(,), U(,) Slope formula
• 5.
  Which set of statements will prove LMNO a
  parallelogram?

Yes, diagonal bisect each other
Yes, opposite angles congruent
Yes, opposite sides equal
No, RS not // UT
L
M
Standardized Test Practice
N
O
LM // NO and LO ? MN
LO // MN and LO ? MN
A
B
LM ? LO and ON ? MN
LO ? MN and LO ? ON
C
D
Click the mouse button or press the Space Bar to
display the answers.
4
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

5
Vocabulary

• Rectangle quadrilateral with four right angles.

6
Polygon Hierarchy
Polygons
Quadrilaterals
Parallelograms
Kites
Trapezoids
IsoscelesTrapezoids
Rhombi
Rectangles
Squares
7
Example 4-1a
Quadrilateral RSTU is a rectangle. If RT 6x 4
and SU 7x - 4 find x.
Definition of congruent segments
Substitution
Subtract 6x from each side.
Add 4 to each side.
Answer 8
8
Example 4-1c
Quadrilateral EFGH is a rectangle. If FH 5x 4
and GE 7x 6, find x.
Answer 5
9
Solve for x and y in the following rectangles
A
B
Hint Special Right Triangles
x
60
8
30
D
C
y
A
B
x
2y 8
4y -12
3x - 8
D
C
Hint p is perimeter
2x
A
B
P 36 feet
x
x
D
C
2x
x
Hint 2 Equations, 2 Variables ? Substitution
A
B
3y
3x -9
2y
D
C
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Example 4-2a
Quadrilateral LMNP is a rectangle. Find x.
?MLP is a right angle, so m?MLP 90
Angle Addition Theorem
Substitution
Simplify.
Subtract 10 from each side.
Divide each side by 8.
Answer 10
11
Quadrilateral LMNP is a rectangle. Find y.
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Example 4-2d
Since a rectangle is a parallelogram, opposite
sides are parallel. So, alternate interior angles
are congruent.
Alternate Interior Angles Theorem
Substitution
Simplify.
Subtract 2 from each side.
Divide each side by 6.
Answer 5
13
Example 4-2e
Quadrilateral EFGH is a rectangle.
a. Find x.
b. Find y.
Answer 11
Answer 7
14
Example 4-3a
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Quadrilateral Characteristics Summary
Convex Quadrilaterals
4 sided polygon 4 interior angles sum to 360 4
exterior angles sum to 360
Parallelograms
Trapezoids
Bases Parallel Legs are not Parallel Leg angles
are supplementary Median is parallel to
basesMedian ½ (base base)
Opposite sides parallel and congruent Opposite
angles congruent Consecutive angles
supplementary Diagonals bisect each other
Rhombi
Rectangles
IsoscelesTrapezoids
All sides congruent Diagonals perpendicular Diagon
als bisect opposite angles
Angles all 90 Diagonals congruent
Legs are congruent Base angle pairs congruent
Diagonals are congruent
Squares
Diagonals divide into 4 congruent triangles
16
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