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If PQRSTU is a regular hexagon, then determine each angle of $Δ P Q T .$

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60 °, 45 °, 30 °

90 °, 60 °, 30 °

70 °, 60 °, 45 °

90 °, 70 °, 60 °

answer is B.

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Detailed Solution

It is given that PQRSTU is a regular hexagon.
Question Image

As we know that the sum of all interior angles of a hexagon is

720 o

.
As there are six angles, so the measure of each angle is

\frac{720 °}{6} = 120 ° .

That is,

∠ U P Q = ∠ P Q R = ∠ U T S = 120 °

Here,

∠ T

is opposite of

∠ Q

, i.e, the line TQ is the bisector of the angle

∠ P Q R

,
So,

∠ T Q P = 12 ∠ P Q R = 12 × 120 ° = 60 °

.
From the figure,

∠ P T Q = 14 ∠ U T S = 14 × 120 ° = 30 °

.
Let,

∠ T P Q = x °

.
Now, in

Δ T P Q

, by applying angle sum property, we get,

∠ TPQ + ∠ TQP + ∠ PTQ = 180 °

⇒ x ° + 60 ° + 30 ° = 180 ° ⇒ x ° + 90 ° = 180 ° ⇒ x ° = 90 °

So, the measures of all angles of the

Δ T P Q

are

90 °, 60 °, 30 °

.
Hence, the correct option is 2.

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