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In the parallelogram LOST shown below, $S N ⊥ O L$ and $S M ⊥ L T$ and . Find $∠ S T M, ∠ S O N$ . Find and $∠ N S M$ and , and choose the option which represents the same.

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50 °, 50 °, 50 °

60 °, 50 °, 70 °

60 °, 55 °, 75 °

50 °, 55 °, 65 °

answer is A.

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Detailed Solution

Here, we are given a parallelogram LOST with

S N ⊥ O L

and

S M ⊥ L T

.
We have, to calculate the angle measures of

∠ S T M, ∠ S O N

and

∠ N S M

.
Now, we know that the angle sum property of a triangle says that sum of all the interior angles of a triangle is always

180 ∘

.
Using this property in

△ M S T

, we get,

∠ T M S + ∠ M S T + ∠ S T M = 180 ° ∠ S T M = 180 ° − (90 ° + 40 °) ∠ S T M = 50 °

Next, it is the property of a parallelogram that its opposite angle pairs are equal.
So, using this property, we have,

∠ S O N = ∠ S T M = 50 ∘

Now, again using the angle sum property of a triangle in

△ S O N

, we get,

∠ O N S + ∠ O S N + ∠ S O N = 180 ° ∠ O S N = 180 ° − (90 ° + 50 °) ∠ O S N = 40 °

Again, it is the property of a parallelogram that the adjacent angle pairs are supplementary, i.e., they add up to

180 ∘

.
Applying this property on the parallelogram LOST, we get,

∠ S O N + ∠ T S O = 180 ° ∠ S O N + ∠ T S M + ∠ N S M + ∠ O S N = 180 ° 50 ° + 40 ° + ∠ N S M + 40 ° = 180 ° 130 ° + ∠ N S M = 180 ° ∠ N S M = 180 ° − 130 ° ∠ N S M = 50 °

So, the angle measures are found as,

∠ S T M = 50 °, ∠ S O N = 50 °

and

∠ N S M = 50 °

.
Hence, the correct option is (1).

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