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Five line segments of equal lengths, $\text{PR, PS, QS, QT}$ and $\text{RT}$ are used to form a star as shown in the figure above.

The value of $\theta$, in degrees, is _______________

1. $36$
2. $45$
3. $72$
4. $108$
Migrated from GO Civil 1 year ago by Arjun

### 1 comment

Your answer is correct but approach is wrong. Angle between S& R never be 2 theta. If it so then theta would be 45 degree. Consider triangle TQX then 4theta=180 & theta will be 45 degree.

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$\textbf{Central Angle Theorem:}$  The Central Angle Theorem states that the central angle from two chosen points $A$ and $B$ on the circle is always twice the inscribed angle from those two points. The inscribed angle can be defined by any point along the outer arc $AB$ and the two points $A$ and $B.$

Now, we can use the above theorem, and get the below diagram,

Now, $10\theta = 360^{\circ}$

$\implies \theta = 36^{\circ}.$

So, the correct answer is $(A).$

### 1 comment

Provided link isn’t working.. Can you elaborate how you assumed intersection point of SQ and RT as origin to apply theorem.

Instead one can observe that Pentagon is regular due to symmetry,as all line segments are equal; Hence each of its interior angles is 108deg, Using this we can arrive at Ѳ=36deg.

Here if we get that its a regular pentagon then it becomes easy

by

I have proved 2 triangles like that all 5 can be proved congruent and because of that their bases are same and hence proves that its a regular pentagon

Sir if u say i will edit and add this image in main ans
Now, how did you prove that their bases are the same? :)
Parts of 2 congruent triangles are equal so by congruency i m able to say their bases must be equal