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A playground is in the form of a rectangle ATEF as shown below. Two players are standing at the points F and B where EF = EB. Find the values of x and y.

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x = 135 °

and

y = 45 °

x = 45 °

and

y = 135 °

x = 155 °

and

y = 65 °

x = 65 °

and

y = 155 °

answer is A.

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Detailed Solution

Here, we have to calculate the angle measures x and y, given that ATEF is a rectangular playground, with EF = EB.
So, in

△ F E B

, by the angle sum property, we have,

∠ E F B + ∠ E B F + ∠ F E B = 180 °

………(i)
Now, we know that sides FE and EB are equal and the triangle is isosceles.
So, the angles opposite to the equal sides, i.e.,

∠ E F B

and

∠ E B F

must be equal.
Also, as the playground is a rectangle, thus by its property,

∠ F E B

is a right angle.
Using these values, we can rewrite the equation (i) as follows,

∠ E F B + ∠ E F B + 90 ∘ = 180 ∘ 2 ∠ E F B = 180 ∘ − 90 ° 2 ∠ E F B = 90 ° ∠ E F B = 90 ∘ 2 ∠ E F B = 45 ∘

Again, by the property of rectangle, we have,

∠ E F A = 90 ∘

, or

∠ E F B + ∠ y = 90 ∘

Now, substituting the value of

∠ E F B

as found above, we have, the value of

∠ y

as follows,

45 ∘ + ∠ y = 90 ∘ ∠ y = 90 ∘ − 45 ∘ ∠ y = 45 ∘

Next, as

∠ E F B

and

∠ E B F

are equal, so

∠ E B F = 45 ∘

. Also, as EBT is a straight line, so the angles

∠ E B F

and

∠ T B F

form a linear pair as they lie on the same line. Thus, we have,

∠ E B F + ∠ T B F = 180 ∘ 45 ∘ + ∠ x = 180 ∘ ∠ x = 180 ∘ − 45 ∘ ∠ x = 135 ∘

So, the values of x and y are found to be

135 ∘

and

45 ∘

respectively.
Hence, the correct option is (1).

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