2. ESSENTIAL QUESTIONS
How do you recognize and apply properties of rectangles?
How do you determine if parallelograms are rectangles?
Monday, April 22, 2013
6. RECTANGLE
A parallelogram with four right angles.
Four right angles
Opposite sides are parallel and congruent
Monday, April 22, 2013
7. RECTANGLE
A parallelogram with four right angles.
Four right angles
Opposite sides are parallel and congruent
Opposite angles are congruent
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
10. THEOREMS
6.13 - Diagonals of a Rectangle:
6.14 - Diagonals of a Rectangle Converse:
Monday, April 22, 2013
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:
Monday, April 22, 2013
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
Monday, April 22, 2013
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.
Monday, April 22, 2013
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.
Monday, April 22, 2013
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.
Monday, April 22, 2013
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.
Monday, April 22, 2013
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
Monday, April 22, 2013
18. EXAMPLE 2
Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)°
and m∠SUR = (3x − 2)°, find x.
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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?
Monday, April 22, 2013
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.
Monday, April 22, 2013
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
Monday, April 22, 2013
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.
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
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
Monday, April 22, 2013
