in exercises 49–54, the diagonals of square lmnp intersect at k. given that lk = 1, find the indicated measure.
49. ( mangle mkn )
50. ( mangle lmk )
51. ( mangle lpk )
52. ( kn )
53. ( ln )
54. ( mp )

Question

Accepted Answer

Let's solve each problem one by one:

### Problem 49: $ m\angle MKN $
# Explanation:
## Step1: Recall square diagonal properties
In a square, the diagonals are perpendicular to each other. So the angle between the diagonals ( $ \angle MKN $) is $ 90^\circ $.
# Answer:
$ 90^\circ $

### Problem 50: $ m\angle LMK $
# Explanation:
## Step1: Diagonals in square bisect angles
In a square, the diagonals bisect the interior angles. Each interior angle of a square is $ 90^\circ $, so the diagonal bisects it into two angles of $ 45^\circ $. So $ \angle LMK = 45^\circ $.
# Answer:
$ 45^\circ $

### Problem 51: $ m\angle LPK $
# Explanation:
## Step1: Diagonals in square bisect angles
In a square, the diagonals bisect the interior angles. Each interior angle of a square is $ 90^\circ $, so the diagonal bisects it into two angles of $ 45^\circ $. Also, triangle $ LPK $ is an isosceles right triangle? Wait, no. Wait, in square $ LMNP $, $ LP $ and $ LM $ are sides, diagonals bisect angles. So $ \angle LPK $ is $ 45^\circ $ because diagonal bisects the $ 90^\circ $ angle at $ P $.
# Answer:
$ 45^\circ $

### Problem 52: $ KN $
# Explanation:
## Step1: Diagonals in square bisect each other
In a square, the diagonals bisect each other. So $ LK = KN $ (since diagonals bisect each other). Given $ LK = 1 $, so $ KN = 1 $.
# Answer:
$ 1 $

### Problem 53: $ LN $
# Explanation:
## Step1: Diagonals in square bisect each other
In a square, the diagonals bisect each other. So $ LN = LK + KN $. We know $ LK = 1 $ and $ KN = 1 $ (from problem 52). So $ LN = 1 + 1 = 2 $.
# Answer:
$ 2 $

### Problem 54: $ MP $
# Explanation:
## Step1: Diagonals in square are equal
In a square, the diagonals are equal in length. We found $ LN = 2 $ (from problem 53), so $ MP = LN = 2 $.
# Answer:
$ 2 $

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QUESTION IMAGE

Question

Explanation:

Problem 49: \( m\angle MKN \)

Step1: Recall square diagonal properties

Step1: Diagonals in square bisect angles

Step1: Diagonals in square bisect angles

Answer:

Problem 50: \( m\angle LMK \)