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CC Geometry H
Aim #30: How do we prove a parallelogram is a rectangle?
f
Do Now: 1) A rectangle is a parallelogram: Always/Sometimes/Never
2) Which is not true about a rectangle?
a) Diagonals bisect each other.
b) Opposite angles are congruent.
c) Diagonals bisect the angles.
d) Diagonals are congruent.
3) In rectangle ABCD, AE = 3x + y, EC = 2x + y + 7 and DE = 2y + 3x - 1. Find the
values of x and y.
D
C
E
A
B
Proving a property of a rectangle:
If a parallelogram is a rectangle, then its diagonals are congruent.
Given: Rectangle GHIJ
Prove: GI ≅ HJ
Statements
G
H
J
I
Reasons
Given: Rect. RSTU, M is midpoint of RS
Prove: ΔUMT is isosceles
Statements
1a) In
Reasons
B
ABCD, AE = 7x - 1, and EC = 5x + 5. Find AC.
C
E
A
D
b) If DB = 10x + 10, find DB.
c) What kind of parallelogram is ABCD and justify your response.
2) The length of a rectangle is seven more than the width. A diagonal is one more
than twice the width. Find the width, length and the length of the diagonal using
an algebraic solution.
To prove a parallelogram is a rectangle, prove one of the following:
1. ________________________________________________
2. _______________________________________________
3. ***An equiangular quadrilateral is a rectangle.***
B
C
Given:
Prove: ECBF is a rectangle
E
Statements
D
Reasons
F
A
Given: Rect. ABCD
Prove: ≮CAD ≅ ≮BDA
E
A
Statements
Given: Rect. PQRS
Prove: ≮1 ≅ ≮2
Reasons
R
Q
1
M
2
P
Statements
C
B
S
Reasons
D
Name ____________________________
CC Geometry H
Date ____________
HW #30
1) Rectangle ABCD is shown below. Find x:
C
B
2y E
­
4x
20
3x
+ y
D
A
2) The length of two adjacent sides of a rectangle differ by 17. If the perimeter
of the rectangle is 146, compute a diagonal and the area of the rectangle. Solve
algebraically.
3) Given: Rect. ABCD,
Prove: a) ΔABP ≅ ΔDCN
B
A
E
b)
N
P
C
D
Statements
Reasons
OVER
4) In rectangle ABCD shown below, AC and BD are diagonals. If m≮1 = 49, find
m≮ADB.
C
B
1
D
A
For #s 5 and 6, refer to rectangle ABCD shown below, with diagonals AC and BD
C
B
intersecting at R.
5) If DR = 4(3x - 10) and CR = 3(x - 2) + 12,
R
find x, AR, AC, and BD.
A
6) If AC = 3(2x + 5) -
(4x + 4) and BD = (12x - 3) + 5x, find x, AC, and DR.
Mixed review:
1) Construct the following using a compass and straightedge:
a. Median from vertex A
b. Altitude from vertex A.
A
A
2) In the figure below AB || CD. What is the measure of ≮CAD?
C
A
D
108o
127
o
38o
B
D