Sum of angles in circle (five-cornered star inscribed in circle; chords have equal lengths)

[shahar]

In the drawing in front of you there is a circle with the points A, B, C, D, and E as its circumference.
The cords shown in the drawing are of equal length.
What is the sum of the angles?

בשרטוט שלפניכם נתון מעגל שעל היקפו הנקודות A, B, C, D, ו-E.
המיתרים המוצגים בשרטוט הם בעלי אורך שווה.​

 

[Harry_the_cat]

Can you calculate each interior angle of the pentagon in the middle?
Can you then calculate the base angles of each isosceles triangle formed?
From there you should be able to find the size of the angles marked $\displaystyle \alpha$ to $\displaystyle \theta$ and therefore the sum.

 

[shahar]

Can you calculate each interior angle of the pentagon in the middle?
Can you then calculate the base angles of each isosceles triangle formed?
From there you should be able to find the size of the angles marked $\displaystyle \alpha$ to $\displaystyle \theta$ and therefore the sum.

I don't how to calculste the interal angle of the pentagon. Is it regual polygon? How can I show it?

 

[Harry_the_cat]

The chords are all the same length. There are lots of lines of symmetry.
Yes it is a regular pentagon.
The formula for calculating the sum of interior angles is ( n − 2 ) × 180 ∘ where n is the number of sides. All the interior angles in a regular polygon are equal.

 

[topsquark]

In the drawing in front of you there is a circle with the points A, B, C, D, and E as its circumference.
The cords shown in the drawing are of equal length.
What is the sum of the angles?
View attachment 36588

בשרטוט שלפניכם נתון מעגל שעל היקפו הנקודות A, B, C, D, ו-E.
המיתרים המוצגים בשרטוט הם בעלי אורך שווה.​

Or, how much of the circle of the circumference is, say, $\alpha$, intersecting? What is the relationship between the angle $\alpha$ and the size of that arc?

See here.

-Dan

 

[blamocur]

I believe that the sum in question does not depend on chords being the same length.

 

[pka]

In the drawing in front of you there is a circle with the points A, B, C, D, and E as its circumference.
The cords shown in the drawing are of equal length.
What is the sum of the angles?
View attachment 36588

בשרטוט שלפניכם נתון מעגל שעל היקפו הנקודות A, B, C, D, ו-E.
המיתרים המוצגים בשרטוט הם בעלי אורך שווה.​

This is a basic problem is angle-arc measures. I wish you had provided a translation!
Lacking a clear translation we must assume a regular star.
$m\left( {\widehat{CD}} \right) = 2m\left( {\sphericalangle \beta } \right)$
$\widehat{AB}~\cup~\widehat{BC}~\cup~\widehat{CD}~\cup~\widehat{DE}~\cup~\widehat{EA}$ is the whole circle,
So $m\left( {\sphericalangle \alpha } \right) + m\left( {\sphericalangle \beta } \right) + m\left( {\sphericalangle \gamma } \right) + m\left( {\sphericalangle \delta } \right) + m\left( {\sphericalangle \theta } \right)$= ???

$$$$

 

[Dr.Peterson]

I wish you had provided a translation!

Google translation:

In the drawing in front of you there is a circle with the points A, B, C, D, and E as its circumference.​

The strings [chords] shown in the drawing are of equal length​

This translation was provided. It does imply a regular pentagram. (And therefore I would tend to do it your way, though of course there are many approaches.)

I believe that the sum in question does not depend on chords being the same length.

True. In fact, it doesn't even depend on being inscribed in a circle!

But we may as well use the information given. (It's a more interesting problem in the general case, though!)

 

[shahar]

Google translation:

In the drawing in front of you there is a circle with the points A, B, C, D, and E as its circumference.​

The strings [chords] shown in the drawing are of equal length​

This translation was provided. It does imply a regular pentagram. (And therefore I would tend to do it your way, though of course there are many approaches.)


True. In fact, it doesn't even depend on being inscribed in a circle!

But we may as well use the information given. (It's a more interesting problem in the general case, though!)

Why does it not depend on being inscribed in a circle?

 

[khansaheb]

Why does it not depend on being inscribed in a circle?

Did you visit the website that was suggested in response #8 ?