2.
ESSENTIAL QUESTIONS
How do you recognize and apply properties of rectangles?
How do you determine if parallelograms are rectangles?
Tuesday, April 29, 14
4.
RECTANGLE
A parallelogram with four right angles.
Tuesday, April 29, 14
5.
RECTANGLE
A parallelogram with four right angles.
Four right angles
Tuesday, April 29, 14
6.
RECTANGLE
A parallelogram with four right angles.
Four right angles
Opposite sides are parallel and congruent
Tuesday, April 29, 14
7.
RECTANGLE
A parallelogram with four right angles.
Four right angles
Opposite sides are parallel and congruent
Opposite angles are congruent
Tuesday, April 29, 14
8.
RECTANGLE
A parallelogram with four right angles.
Four right angles
Opposite sides are parallel and congruent
Opposite angles are congruent
Consecutive angles are supplementary
Tuesday, April 29, 14
9.
RECTANGLE
A parallelogram with four right angles.
Four right angles
Opposite sides are parallel and congruent
Opposite angles are congruent
Consecutive angles are supplementary
Diagonals bisect each other
Tuesday, April 29, 14
10.
THEOREMS
6.13 - Diagonals of a Rectangle:
6.14 - Diagonals of a Rectangle Converse:
Tuesday, April 29, 14
11.
THEOREMS
6.13 - Diagonals of a Rectangle: If a parallelogram is a
rectangle, then its diagonals are congruent
6.14 - Diagonals of a Rectangle Converse:
Tuesday, April 29, 14
12.
THEOREMS
6.13 - Diagonals of a Rectangle: If a parallelogram is a
rectangle, then its diagonals are congruent
6.14 - Diagonals of a Rectangle Converse: If diagonals of a
parallelogram are congruent, then the parallelogram is a
rectangle
Tuesday, April 29, 14
13.
EXAMPLE 1
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.
Tuesday, April 29, 14
14.
EXAMPLE 1
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.
Since we have a rectangle, the
diagonals are congruent.
Tuesday, April 29, 14
15.
EXAMPLE 1
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.
Since we have a rectangle, the
diagonals are congruent.
The diagonals also bisect each other,
so JN = LN and KN = MN.
Tuesday, April 29, 14
16.
EXAMPLE 1
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.
Since we have a rectangle, the
diagonals are congruent.
The diagonals also bisect each other,
so JN = LN and KN = MN.
So JN = LN = KN = MN = 6.5 feet and KM = KN + MN.
Tuesday, April 29, 14
17.
EXAMPLE 1
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.
Since we have a rectangle, the
diagonals are congruent.
The diagonals also bisect each other,
so JN = LN and KN = MN.
So JN = LN = KN = MN = 6.5 feet and KM = KN + MN.
KM = 13 feet
Tuesday, April 29, 14
18.
EXAMPLE 2
Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)°
and m∠SUR = (3x − 2)°, ﬁnd x.
Tuesday, April 29, 14
19.
EXAMPLE 2
Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)°
and m∠SUR = (3x − 2)°, ﬁnd x.
m∠RTU + m∠SUR = 90
Tuesday, April 29, 14
20.
EXAMPLE 2
Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)°
and m∠SUR = (3x − 2)°, ﬁnd x.
m∠RTU + m∠SUR = 90
8x + 4 + 3x − 2 = 90
Tuesday, April 29, 14
21.
EXAMPLE 2
Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)°
and m∠SUR = (3x − 2)°, ﬁnd x.
m∠RTU + m∠SUR = 90
8x + 4 + 3x − 2 = 90
11x + 2 = 90
Tuesday, April 29, 14
22.
EXAMPLE 2
Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)°
and m∠SUR = (3x − 2)°, ﬁnd x.
m∠RTU + m∠SUR = 90
8x + 4 + 3x − 2 = 90
11x + 2 = 90
−2 −2
Tuesday, April 29, 14
23.
EXAMPLE 2
Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)°
and m∠SUR = (3x − 2)°, ﬁnd x.
m∠RTU + m∠SUR = 90
8x + 4 + 3x − 2 = 90
11x + 2 = 90
−2 −2
11x = 88
Tuesday, April 29, 14
24.
EXAMPLE 2
Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)°
and m∠SUR = (3x − 2)°, ﬁnd x.
m∠RTU + m∠SUR = 90
8x + 4 + 3x − 2 = 90
11x + 2 = 90
−2 −2
11x = 88
11 11
Tuesday, April 29, 14
25.
EXAMPLE 2
Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)°
and m∠SUR = (3x − 2)°, ﬁnd x.
m∠RTU + m∠SUR = 90
8x + 4 + 3x − 2 = 90
11x + 2 = 90
−2 −2
11x = 88
11 11
x = 8
Tuesday, April 29, 14
26.
EXAMPLE 3
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
of the diagonals of the frame. If AB = 12 inches, BC = 35
inches, CD = 12 inches, and DA = 35 inches, how long do
the lengths of the diagonals need to be?
Tuesday, April 29, 14
27.
EXAMPLE 3
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
of the diagonals of the frame. If AB = 12 inches, BC = 35
inches, CD = 12 inches, and DA = 35 inches, how long do
the lengths of the diagonals need to be?
The diagonal forms a right triangle
with legs of 12 and 35. We need to ﬁnd
the hypotenuse.
Tuesday, April 29, 14
35.
EXAMPLE 3
a2
+ b2
= c2
122
+ 352
= c2
144 + 1225 = c2
1369 = c2
1369 = c2
c = 37
The diagonals must both be 37 inches
Tuesday, April 29, 14
36.
EXAMPLE 4
Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2),
and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
Tuesday, April 29, 14
37.
EXAMPLE 4
Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2),
and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
Diagonals must be congruent
Tuesday, April 29, 14
38.
EXAMPLE 4
Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2),
and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2
+ (3 + 2)2
Diagonals must be congruent
Tuesday, April 29, 14
39.
EXAMPLE 4
Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2),
and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2
+ (3 + 2)2
= (−5)2
+ 52
Diagonals must be congruent
Tuesday, April 29, 14
40.
EXAMPLE 4
Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2),
and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2
+ (3 + 2)2
= (−5)2
+ 52 = 25 + 25
Diagonals must be congruent
Tuesday, April 29, 14
41.
EXAMPLE 4
Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2),
and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2
+ (3 + 2)2
= (−5)2
+ 52 = 25 + 25 = 50
Diagonals must be congruent
Tuesday, April 29, 14
42.
EXAMPLE 4
Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2),
and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2
+ (3 + 2)2
= (−5)2
+ 52 = 25 + 25 = 50
KM = (1 − 0)2
+ (4 + 3)2
Diagonals must be congruent
Tuesday, April 29, 14
43.
EXAMPLE 4
Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2),
and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2
+ (3 + 2)2
= (−5)2
+ 52 = 25 + 25 = 50
KM = (1 − 0)2
+ (4 + 3)2
= 12
+ 72
Diagonals must be congruent
Tuesday, April 29, 14
44.
EXAMPLE 4
Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2),
and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2
+ (3 + 2)2
= (−5)2
+ 52 = 25 + 25 = 50
KM = (1 − 0)2
+ (4 + 3)2
= 12
+ 72 = 1 + 49
Diagonals must be congruent
Tuesday, April 29, 14
45.
EXAMPLE 4
Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2),
and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2
+ (3 + 2)2
= (−5)2
+ 52 = 25 + 25 = 50
KM = (1 − 0)2
+ (4 + 3)2
= 12
+ 72 = 1 + 49 = 50
Diagonals must be congruent
Tuesday, April 29, 14
46.
EXAMPLE 4
Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2),
and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2
+ (3 + 2)2
= (−5)2
+ 52 = 25 + 25 = 50
KM = (1 − 0)2
+ (4 + 3)2
= 12
+ 72 = 1 + 49 = 50
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
Tuesday, April 29, 14
47.
EXAMPLE 4
Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2),
and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2
+ (3 + 2)2
= (−5)2
+ 52 = 25 + 25 = 50
KM = (1 − 0)2
+ (4 + 3)2
= 12
+ 72 = 1 + 49 = 50
m( JK) =
4 − 3
1 + 2
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
Tuesday, April 29, 14
48.
EXAMPLE 4
Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2),
and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2
+ (3 + 2)2
= (−5)2
+ 52 = 25 + 25 = 50
KM = (1 − 0)2
+ (4 + 3)2
= 12
+ 72 = 1 + 49 = 50
m( JK) =
4 − 3
1 + 2
=
1
3
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
Tuesday, April 29, 14
49.
EXAMPLE 4
Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2),
and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2
+ (3 + 2)2
= (−5)2
+ 52 = 25 + 25 = 50
KM = (1 − 0)2
+ (4 + 3)2
= 12
+ 72 = 1 + 49 = 50
m( JK) =
4 − 3
1 + 2
=
1
3
m(LM) =
−3 + 2
0 − 3
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
Tuesday, April 29, 14
50.
EXAMPLE 4
Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2),
and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2
+ (3 + 2)2
= (−5)2
+ 52 = 25 + 25 = 50
KM = (1 − 0)2
+ (4 + 3)2
= 12
+ 72 = 1 + 49 = 50
m( JK) =
4 − 3
1 + 2
=
1
3
m(LM) =
−3 + 2
0 − 3
=
−1
−3
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
Tuesday, April 29, 14
51.
EXAMPLE 4
Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2),
and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2
+ (3 + 2)2
= (−5)2
+ 52 = 25 + 25 = 50
KM = (1 − 0)2
+ (4 + 3)2
= 12
+ 72 = 1 + 49 = 50
m( JK) =
4 − 3
1 + 2
=
1
3
m(LM) =
−3 + 2
0 − 3
=
−1
−3
=
1
3
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
Tuesday, April 29, 14
52.
EXAMPLE 4
Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2),
and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2
+ (3 + 2)2
= (−5)2
+ 52 = 25 + 25 = 50
KM = (1 − 0)2
+ (4 + 3)2
= 12
+ 72 = 1 + 49 = 50
m( JK) =
4 − 3
1 + 2
=
1
3
m(LM) =
−3 + 2
0 − 3
=
−1
−3
=
1
3
m(KL) =
−2 − 4
3 − 1
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
Tuesday, April 29, 14
53.
EXAMPLE 4
Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2),
and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2
+ (3 + 2)2
= (−5)2
+ 52 = 25 + 25 = 50
KM = (1 − 0)2
+ (4 + 3)2
= 12
+ 72 = 1 + 49 = 50
m( JK) =
4 − 3
1 + 2
=
1
3
m(LM) =
−3 + 2
0 − 3
=
−1
−3
=
1
3
m(KL) =
−2 − 4
3 − 1
=
−6
2
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
Tuesday, April 29, 14
54.
EXAMPLE 4
Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2),
and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2
+ (3 + 2)2
= (−5)2
+ 52 = 25 + 25 = 50
KM = (1 − 0)2
+ (4 + 3)2
= 12
+ 72 = 1 + 49 = 50
m( JK) =
4 − 3
1 + 2
=
1
3
m(LM) =
−3 + 2
0 − 3
=
−1
−3
=
1
3
m(KL) =
−2 − 4
3 − 1
=
−6
2
= −3
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
Tuesday, April 29, 14
55.
EXAMPLE 4
Quadrilateral JKLM has vertices J(−2, 3), K(1, 4), L(3, −2),
and M(0, −3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (−2 − 3)2
+ (3 + 2)2
= (−5)2
+ 52 = 25 + 25 = 50
KM = (1 − 0)2
+ (4 + 3)2
= 12
+ 72 = 1 + 49 = 50
m( JK) =
4 − 3
1 + 2
=
1
3
m(LM) =
−3 + 2
0 − 3
=
−1
−3
=
1
3
m(KL) =
−2 − 4
3 − 1
=
−6
2
m( JM) =
−3 − 3
0 + 2
= −3
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
Tuesday, April 29, 14
59.
PROBLEM SET
p. 422 #1-31 odd, 41, 49, 55, 59, 61
“Character - the willingness to accept responsibility for
one's own life - is the source from which self respect
springs.” - Joan Didion
Tuesday, April 29, 14
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