Find the sum of the measures of the five acute angles of a star...?

Question

Please explain the answer clearly.....i figgured out one way to find it out but i need another way!!

Thanks a lot!!!

Answer 1

first draw a circle, and then put the star in it w/ the points touching the edge. that divides the circumference of the circle into 5 equal parts. so to find the degree of each one of those section divide 360 by 5 = 72. now look at the angle that is made from the lines from the ends of the individuals sections to the point of the star it goes to, which is the inscribed angle. now that angle which is one of the acute angles is half of its intercepted arc. so that makes 72 divided by 2= 36. so since you wanted to know the measure of all the acute angles and not just one multiply 36 times 5 = 180

Answer 2

Both of the answers so far have incorrectly assumed you were talking about a regular shape. However, that assumption is completely unnecessary - it works for any 5 pointed star at all.
When we connect up each of the vertices of the star, we get a (non-regular) pentagon in the middle. Lets call the 5 angles of that pentagon A,B,C,D,E, which we know add to (5-2)*180 = 540 degrees.
Now, look at one of the outside triangles - say we take the one which is next to the 'A' and 'B' angles. Then two angles of the triangle are 180-A and 180-B; thus to make the total 180, the last angle (an acute angle of the star) must be A+B-180.

Now we repeat that for the other 4 points. The total of all angles is:
A+B-180 + B+C-180 + C+D-180 + D+E-180 + E+A-180
= 2(A+B+C+D+E) - 5*180
= 2*540 - 5*180
= 180 degrees.

Answer 3

The centre of the star is a regular pentagon
Each interior angle = 108°
So the base angles of each of the isosceles triangles making the points of the star are 72° (supplement of 108°)
So the angle at each vertex of the star = 36° (180° - (72° + 72°))
Thus sum of 5 angles = 5 x 36°
= 180°

Alternately:
1. Draw the circumscribing circle

2. Join up all the vertices to form the regula pentagon around the circumscribing circle

3. Connect each vertex to the centre.

4. Each angle formed at the centre = 360°/5
= 72° (As equal chords subtend equal angles at the centre of a circle)

5. Therefore angle at the circumference = 36° (Angle at circumference is half the angle at the centre of a circle when subtended on the same arc)

6. So sum of angles = 5 x 36°
= 180°

Answer 4

My answer is 360; each angle measures 72.

Connect the tips of the star together to make it a 5 equal-sided polygon (pentagon). Draw straight lines from a midpoint of each sides of the polygon towards each of the opposite angles (dissecting an angle of the polygon into 2 equal angles). The lines passes through one common point of intersection (should be the midpoint of the polygon) and also creating 10 equal angles in reference with the point of intersection. Each of the angles has a measurement of 36 degrees, since the 10 angles make one circular rotation and 1 rotation is 360 degrees in measurement. Therefore 360/10 = 36 degrees.

Let as name the point of intersection as pt M, one of the tip of the star as pt A, and the end points of the line opposite to pt A are pts B and C.

Draw lines from pt M to pts B and C, creating a triangle BMC; and lines from pt A to points B and C, creating triangle BAC.

Triangles BMC and BAC are proportional with each other since they have the same base and both are isosceles triangels. And proportional triangles should have the same angles. Therefore angle M is equal to angle A.

Angle M has a measurement of 72 degrees since it is the sum of 2 angles from that 10 angles. And angle A is one of the acute angles of the star. Therefore 72 multiplied by 5 is 360.

Dindo