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Lesson 8-4
Rectangles
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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