7. RECTANGLE
A parallelogram with four right angles.
Four right angles
Opposite sides are parallel and congruent
Opposite angles are congruent
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
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
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:
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
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, ο¬nd KM.
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, ο¬nd KM.
Since we have a rectangle, the
diagonals are congruent.
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, ο¬nd KM.
Since we have a rectangle, the
diagonals are congruent.
The diagonals also bisect each other,
so JN = LN and KN = MN.
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, ο¬nd 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.
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, ο¬nd 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
19. EXAMPLE 2
Quadrilateral RSTU is a rectangle. If mβ RTU = (8x + 4)Β°
and mβ SUR = (3x β 2)Β°, ο¬nd x.
mβ RTU + mβ SUR = 90
20. EXAMPLE 2
Quadrilateral RSTU is a rectangle. If mβ RTU = (8x + 4)Β°
and mβ SUR = (3x β 2)Β°, ο¬nd x.
mβ RTU + mβ SUR = 90
8x + 4 + 3x β 2 = 90
21. EXAMPLE 2
Quadrilateral RSTU is a rectangle. If mβ RTU = (8x + 4)Β°
and mβ SUR = (3x β 2)Β°, ο¬nd x.
mβ RTU + mβ SUR = 90
8x + 4 + 3x β 2 = 90
11x + 2 = 90
22. EXAMPLE 2
Quadrilateral RSTU is a rectangle. If mβ RTU = (8x + 4)Β°
and mβ SUR = (3x β 2)Β°, ο¬nd x.
mβ RTU + mβ SUR = 90
8x + 4 + 3x β 2 = 90
11x + 2 = 90
β2 β2
23. EXAMPLE 2
Quadrilateral RSTU is a rectangle. If mβ RTU = (8x + 4)Β°
and mβ SUR = (3x β 2)Β°, ο¬nd x.
mβ RTU + mβ SUR = 90
8x + 4 + 3x β 2 = 90
11x + 2 = 90
β2 β2
11x = 88
24. EXAMPLE 2
Quadrilateral RSTU is a rectangle. If mβ RTU = (8x + 4)Β°
and mβ SUR = (3x β 2)Β°, ο¬nd x.
mβ RTU + mβ SUR = 90
8x + 4 + 3x β 2 = 90
11x + 2 = 90
β2 β2
11x = 88
11 11
25. EXAMPLE 2
Quadrilateral RSTU is a rectangle. If mβ RTU = (8x + 4)Β°
and mβ SUR = (3x β 2)Β°, ο¬nd x.
mβ RTU + mβ SUR = 90
8x + 4 + 3x β 2 = 90
11x + 2 = 90
β2 β2
11x = 88
11 11
x = 8
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?
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 ο¬nd
the hypotenuse.
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
36. EXAMPLE 4
Quadrilateral JKLM has vertices J(β2, 3), K(1, 4), L(3, β2),
and M(0, β3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
37. EXAMPLE 4
Quadrilateral JKLM has vertices J(β2, 3), K(1, 4), L(3, β2),
and M(0, β3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
Diagonals must be congruent
38. EXAMPLE 4
Quadrilateral JKLM has vertices J(β2, 3), K(1, 4), L(3, β2),
and M(0, β3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (β2 β 3)2
+ (3+ 2)2
Diagonals must be congruent
39. EXAMPLE 4
Quadrilateral JKLM has vertices J(β2, 3), K(1, 4), L(3, β2),
and M(0, β3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (β2 β 3)2
+ (3+ 2)2
= (β5)2
+ 52
Diagonals must be congruent
40. EXAMPLE 4
Quadrilateral JKLM has vertices J(β2, 3), K(1, 4), L(3, β2),
and M(0, β3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (β2 β 3)2
+ (3+ 2)2
= (β5)2
+ 52
= 25 + 25
Diagonals must be congruent
41. EXAMPLE 4
Quadrilateral JKLM has vertices J(β2, 3), K(1, 4), L(3, β2),
and M(0, β3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (β2 β 3)2
+ (3+ 2)2
= (β5)2
+ 52
= 25 + 25 = 50
Diagonals must be congruent
42. EXAMPLE 4
Quadrilateral JKLM has vertices J(β2, 3), K(1, 4), L(3, β2),
and M(0, β3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (β2 β 3)2
+ (3+ 2)2
= (β5)2
+ 52
= 25 + 25 = 50
KM = (1β0)2
+ (4 + 3)2
Diagonals must be congruent
43. EXAMPLE 4
Quadrilateral JKLM has vertices J(β2, 3), K(1, 4), L(3, β2),
and M(0, β3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (β2 β 3)2
+ (3+ 2)2
= (β5)2
+ 52
= 25 + 25 = 50
KM = (1β0)2
+ (4 + 3)2
= 12
+ 72
Diagonals must be congruent
44. EXAMPLE 4
Quadrilateral JKLM has vertices J(β2, 3), K(1, 4), L(3, β2),
and M(0, β3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (β2 β 3)2
+ (3+ 2)2
= (β5)2
+ 52
= 25 + 25 = 50
KM = (1β0)2
+ (4 + 3)2
= 12
+ 72
= 1+ 49
Diagonals must be congruent
45. EXAMPLE 4
Quadrilateral JKLM has vertices J(β2, 3), K(1, 4), L(3, β2),
and M(0, β3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (β2 β 3)2
+ (3+ 2)2
= (β5)2
+ 52
= 25 + 25 = 50
KM = (1β0)2
+ (4 + 3)2
= 12
+ 72
= 1+ 49 = 50
Diagonals must be congruent
46. EXAMPLE 4
Quadrilateral JKLM has vertices J(β2, 3), K(1, 4), L(3, β2),
and M(0, β3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (β2 β 3)2
+ (3+ 2)2
= (β5)2
+ 52
= 25 + 25 = 50
KM = (1β0)2
+ (4 + 3)2
= 12
+ 72
= 1+ 49 = 50
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
47. EXAMPLE 4
Quadrilateral JKLM has vertices J(β2, 3), K(1, 4), L(3, β2),
and M(0, β3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (β2 β 3)2
+ (3+ 2)2
= (β5)2
+ 52
= 25 + 25 = 50
KM = (1β0)2
+ (4 + 3)2
= 12
+ 72
= 1+ 49 = 50
m( JK) =
4 β 3
1+ 2
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
48. EXAMPLE 4
Quadrilateral JKLM has vertices J(β2, 3), K(1, 4), L(3, β2),
and M(0, β3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (β2 β 3)2
+ (3+ 2)2
= (β5)2
+ 52
= 25 + 25 = 50
KM = (1β0)2
+ (4 + 3)2
= 12
+ 72
= 1+ 49 = 50
m( JK) =
4 β 3
1+ 2
=
1
3
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
49. EXAMPLE 4
Quadrilateral JKLM has vertices J(β2, 3), K(1, 4), L(3, β2),
and M(0, β3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (β2 β 3)2
+ (3+ 2)2
= (β5)2
+ 52
= 25 + 25 = 50
KM = (1β0)2
+ (4 + 3)2
= 12
+ 72
= 1+ 49 = 50
m( JK) =
4 β 3
1+ 2
=
1
3
m(LM) =
β3+ 2
0β 3
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
50. EXAMPLE 4
Quadrilateral JKLM has vertices J(β2, 3), K(1, 4), L(3, β2),
and M(0, β3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (β2 β 3)2
+ (3+ 2)2
= (β5)2
+ 52
= 25 + 25 = 50
KM = (1β0)2
+ (4 + 3)2
= 12
+ 72
= 1+ 49 = 50
m( JK) =
4 β 3
1+ 2
=
1
3
m(LM) =
β3+ 2
0β 3
=
β1
β3
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
51. EXAMPLE 4
Quadrilateral JKLM has vertices J(β2, 3), K(1, 4), L(3, β2),
and M(0, β3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (β2 β 3)2
+ (3+ 2)2
= (β5)2
+ 52
= 25 + 25 = 50
KM = (1β0)2
+ (4 + 3)2
= 12
+ 72
= 1+ 49 = 50
m( JK) =
4 β 3
1+ 2
=
1
3
m(LM) =
β3+ 2
0β 3
=
β1
β3
=
1
3
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
52. EXAMPLE 4
Quadrilateral JKLM has vertices J(β2, 3), K(1, 4), L(3, β2),
and M(0, β3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (β2 β 3)2
+ (3+ 2)2
= (β5)2
+ 52
= 25 + 25 = 50
KM = (1β0)2
+ (4 + 3)2
= 12
+ 72
= 1+ 49 = 50
m( JK) =
4 β 3
1+ 2
=
1
3
m(LM) =
β3+ 2
0β 3
=
β1
β3
=
1
3
m(KL) =
β2 β 4
3β1
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
53. EXAMPLE 4
Quadrilateral JKLM has vertices J(β2, 3), K(1, 4), L(3, β2),
and M(0, β3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (β2 β 3)2
+ (3+ 2)2
= (β5)2
+ 52
= 25 + 25 = 50
KM = (1β0)2
+ (4 + 3)2
= 12
+ 72
= 1+ 49 = 50
m( JK) =
4 β 3
1+ 2
=
1
3
m(LM) =
β3+ 2
0β 3
=
β1
β3
=
1
3
m(KL) =
β2 β 4
3β1
=
β6
2
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
54. EXAMPLE 4
Quadrilateral JKLM has vertices J(β2, 3), K(1, 4), L(3, β2),
and M(0, β3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (β2 β 3)2
+ (3+ 2)2
= (β5)2
+ 52
= 25 + 25 = 50
KM = (1β0)2
+ (4 + 3)2
= 12
+ 72
= 1+ 49 = 50
m( JK) =
4 β 3
1+ 2
=
1
3
m(LM) =
β3+ 2
0β 3
=
β1
β3
=
1
3
m(KL) =
β2 β 4
3β1
=
β6
2
= β3
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
55. EXAMPLE 4
Quadrilateral JKLM has vertices J(β2, 3), K(1, 4), L(3, β2),
and M(0, β3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (β2 β 3)2
+ (3+ 2)2
= (β5)2
+ 52
= 25 + 25 = 50
KM = (1β0)2
+ (4 + 3)2
= 12
+ 72
= 1+ 49 = 50
m( JK) =
4 β 3
1+ 2
=
1
3
m(LM) =
β3+ 2
0β 3
=
β1
β3
=
1
3
m(KL) =
β2 β 4
3β1
=
β6
2
m( JM) =
β3β 3
0+ 2
= β3
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
56. EXAMPLE 4
Quadrilateral JKLM has vertices J(β2, 3), K(1, 4), L(3, β2),
and M(0, β3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (β2 β 3)2
+ (3+ 2)2
= (β5)2
+ 52
= 25 + 25 = 50
KM = (1β0)2
+ (4 + 3)2
= 12
+ 72
= 1+ 49 = 50
m( JK) =
4 β 3
1+ 2
=
1
3
m(LM) =
β3+ 2
0β 3
=
β1
β3
=
1
3
m(KL) =
β2 β 4
3β1
=
β6
2
m( JM) =
β3β 3
0+ 2
=
β6
2
= β3
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular
57. EXAMPLE 4
Quadrilateral JKLM has vertices J(β2, 3), K(1, 4), L(3, β2),
and M(0, β3). Determine whether JKLM is a rectangle by
using the distance formula, then slope.
JL = (β2 β 3)2
+ (3+ 2)2
= (β5)2
+ 52
= 25 + 25 = 50
KM = (1β0)2
+ (4 + 3)2
= 12
+ 72
= 1+ 49 = 50
m( JK) =
4 β 3
1+ 2
=
1
3
m(LM) =
β3+ 2
0β 3
=
β1
β3
=
1
3
m(KL) =
β2 β 4
3β1
=
β6
2
m( JM) =
β3β 3
0+ 2
=
β6
2
= β3
= β3
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular