Question

RS and ST are 2 sides of a regular 12-sided polygon.
RT is a diagonal of the polygon.
Work out the size of angle STR.
You must show your working.

Answer 1

Answer:

  15°

Step-by-step explanation:

The exterior angle at vertex S is 360°/12 = 30°. That angle has a measure that is equal to the sum of the congruent angles at R and T of ΔRST. In other words, ...

  ∠T = 30°/2 = 15°

The size of angle STR is 15°.

Answer 2

The sides of a regular polygon are congruent.

The size of STR is 15 degrees

The polygon is 12-sided.

This means that:

$\mathbf{n =12}$

The sum of angles in a regular hexagon is 360.

So, the angle at vertex S is:

$\mathbf{\theta = \frac{360}{n}}$

This gives

$\mathbf{\theta = \frac{360}{12}}$

$\mathbf{\theta = 30^o}$

The external angle of a triangle equals the sum of the opposite internal angles.

This means that:

$\mathbf{\theta = \angle STR + \angle SRT}$

Where:

$\mathbf{ \angle STR = \angle SRT}$

So, we have:

$\mathbf{\theta = \angle STR + \angle STR}$

$\mathbf{\theta = 2\angle STR}$

Substitute $\mathbf{\theta = 30^o}$

$\mathbf{30^o = 2\angle STR}$

Divide both sides by 2

$\mathbf{15^o = \angle STR}$

Rewrite as:

$\mathbf{\angle STR = 15^o }$

Hence, the size of STR is 15 degrees

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