Rectangles

1. SECTION 6-4
Rectangles

2. ESSENTIAL QUESTIONS
How do you recognize and apply properties of rectangles?
How do you determine if parallelograms are rectangles?

3. RECTANGLE

4. RECTANGLE
A parallelogram with four right angles.

5. RECTANGLE
A parallelogram with four right angles.
Four right angles

6. RECTANGLE
A parallelogram with four right angles.
Four right angles
Opposite sides are parallel and congruent

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

10. THEOREMS
6.13 - Diagonals of a Rectangle:
6.14 - Diagonals of a Rectangle Converse:

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, find 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, find 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, find 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, 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.

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

18. EXAMPLE 2
Quadrilateral RSTU is a rectangle. If m∠RTU = (8x + 4)°
and m∠SUR = (3x − 2)°, find x.

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

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

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

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

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

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

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

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 find
the hypotenuse.

28. EXAMPLE 3

29. EXAMPLE 3
a2
+ b2
= c2

30. EXAMPLE 3
a2
+ b2
= c2
122
+ 352
= c2

31. EXAMPLE 3
a2
+ b2
= c2
122
+ 352
= c2
144 +1225 = c2

32. EXAMPLE 3
a2
+ b2
= c2
122
+ 352
= c2
144 +1225 = c2
1369 = c2

33. EXAMPLE 3
a2
+ b2
= c2
122
+ 352
= c2
144 +1225 = c2
1369 = c2
1369 = c2

34. EXAMPLE 3
a2
+ b2
= c2
122
+ 352
= c2
144 +1225 = c2
1369 = c2
1369 = c2
c = 37

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(−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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

56. 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
=
−6
2
= −3
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular

57. 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
=
−6
2
= −3
= −3
Diagonals must be congruent
Opposite sides parallel, consecutive sides perpendicular