Question

$<b><i>$ [1o0 points]

find the sum of All the colored angles in the star ?

▶easy trick......required :)◀​

Answer 1

Mark every angle as a, b, c, d and e

Start from top and continue clockwise which means top angle will be angle a then going clockwise the angles will be b, c, d and e

See the triangle with angle a

In this triangle see the left angle which is the exterior angle of triangle comprising angle b and angle d

According to remote interior angle property the sum of thise two angles will be equal to its exterior angle

Again,

In the same triangle apply the same property

Now,

Apply sumof measures of all angles of a triangle property

We get

<a+<b+<c+<d+<e = 180

Which means the sum of all coloured angles is 180


Another method :-

Draw a circle around the star such that all the vertex is touching the circle

Name the angles as it was named previously

Use the property of Inscribed angle theorem in all the angles

The measures of arcs will be 2a, 2b, 2c, 2d, 2e.

Those 5 arcs will constitute the whole circle which means that the sym of the arcs will be 360

2a + 2b + 2c + 2d + 2e = 360

If we divide it by 2 we get

a + b + c + d + e = 180

Answer 2

In the center is a 5-sided regular polygon (a regular

pentagon). The sum of the interior angles of a polygon

is gotten by the formula:

SUM OF INTERIOR ANGLES = (NUMBER OF SIDES - 2) * 180�

For a 5-sided polygon (pentagon) this is (5-2)*180� = 3*180�=540�

Since all 5 angles of a regular pentagon are equal, each

interior angle of the regular pentagon is 540%2F5� = 108�

So I'll mark one of the 108� interior angles of the pentagon:

Its suppplement is found by subtracting 180�-108�=72�.

We'll mark it 72�:

That 72� angle is one of the base angles of an isosceles

triangle. So we'll mark the other base angle 72� also.

Now we can find the angle at the top point of the star by

adding the two equal base angles and subtracting from 180�.

72� + 72� = 144�

180� - 144� = 36�

So each point of the star is 36�.

You wanted the sum of the points interior angles of

the points. There are 5 of them, so 5 times 36� is

180�.

See the attached file to know more and clearly