Twinkle Twinkle Angled Star

Geometry Level 2

For a five pointed star as shown, compute the value of ∠ A + ∠ B + ∠ C + ∠ D + ∠ E . \large \angle A + \angle B + \angle C + \angle D + \angle E .

15 0 ∘ 150^\circ 12 0 ∘ 120^\circ 30 0 ∘ 300^\circ 18 0 ∘ 180^\circ 36 0 ∘ 360^\circ 54 0 ∘ 540^\circ

Samer Ayoub by samer Ayoub , 46, Canada

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

8 solutions

Nihar Mahajan
Feb 21, 2016

First step: a+d

Second step: a+d+c

Third step: e+b+(a+d+c)=180

The reasons for the steps are stated in the pic

30 Helpful 0 Interesting 0 Brilliant 1 Confused

Simpler solution, great!

Sandeep Bhardwaj - 5 years, 3 months ago

Log in to reply

Thank you!

Nihar Mahajan - 5 years, 3 months ago

One more solution- in the Pentagon taking the exterior angles clock wise sum will be 360 again taking the exterior angles counter clockwise sum will be 360 (as in every polygon sum of exterior angles is 360) so the total sum is 720 and this will form the sum of the base angles of the 5 triangles and we know that sum of angles of a triangle is 180 so for 5 triangles sum will be 900 now subtracting

Ankur Verma - 4 years, 4 months ago

Log in to reply

900-720=180

Ankur Verma - 4 years, 4 months ago

By the way I am new to brilliant so where can we write solution

Ankur Verma - 4 years, 4 months ago

Log in to reply

When you answer a problem correctly, at the top of solution discussions, you will be allowed to enter your solution

Calvin Lin Staff - 4 years, 4 months ago

I could not type a solution in the proper place, so with apologies, I'll put it here. Connect D to E;Then DE is parallel to BC. By alternate interior angles, <CDE=<DCB and <BED=<CBE. Then we have formed the triangle ADE which includes all 5 angles, so the answer is 180. Ed Gray

Edwin Gray - 3 years, 9 months ago

What's the first step?

Calvin Lin Staff - 5 years, 3 months ago

Log in to reply

By first step , you mean e+b+(a+d+c)=180?

Nihar Mahajan - 5 years, 3 months ago

Log in to reply

No, i meant the first step of the solution (after drawing the diagram).

For example: - First step: Show that the angle is a+d
- Second step: Show that the angle is a+d+c
- Third step: etc

It's not clear how to approach your solution, because you just draw a bunch of stuff in there and magically came up with a+b+c+d+e = 180.

Calvin Lin Staff - 5 years, 3 months ago

Log in to reply

@Calvin Lin – First step: a+d

Second step: a+d+c

Third step: e+b+(a+d+c)=180

The reasons for the steps are stated in the pic.

Nihar Mahajan - 5 years, 3 months ago

Log in to reply

@Nihar Mahajan – Great. Can you make that obvious in your solution? Not everyone is a mind reader. You should list out the steps that you took explicitly.

Calvin Lin Staff - 5 years, 3 months ago
Elbert Rasay
Feb 26, 2016

I KNEW IT. READING THOSE CHINESE COMICS HELP IN MY MATH HOMEWORK

14 Helpful 0 Interesting 0 Brilliant 0 Confused
Samer Ayoub
Feb 20, 2016

7 Helpful 0 Interesting 0 Brilliant 0 Confused
Adam Hamada
Aug 19, 2016

In a regular pentagram (5-pointed star), the angle in each point is 36. 36 x 5 = 180.

1 Helpful 0 Interesting 0 Brilliant 0 Confused
Kaustubh Khulbe
Jan 15, 2018

You would connect a line from angle D to angle E. Then, the bottom left and right angle fo the triangle we made would be 2x each because it is twice as much as angle A. Then you would simply solve the equation 5x = 180 and get x = 36. Since each triangle on the outer edges of the pentagon in the center of the star is similar, we would do 36 * 5 and get 180 as the answer.

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Jack Lam
Feb 28, 2016

For an n n -pointed star, that goes around it's inside twice, the angle sum in a polygon formula applies for n 2 \frac{n}{2} .

S = 360 ( n − 2 ) = 360 ( 2.5 − 2 ) = 180 S = 360(n-2) =360 (2.5-2) = 180

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Tom Engelsman
Feb 27, 2016

Imagine a circle that is circumscribed about the 5-pointed star. Using the fact that any inscribed angle inside of a circle measures 1/2 of the intercepted circular arc:

A = arc DE/2; B = arc CE/2, C = arc BD/2, D = arc AC/2, E = arc AB/2.

Arcs AC, CE, ED, DB, and BA must sum to 360 degrees. Hence, the sum of the angles A, B, C, D, & E equals 180 degrees.

0 Helpful 0 Interesting 0 Brilliant 0 Confused
Prasit Sarapee
Feb 26, 2016

Angle corner of pentagon = 540/5=108,
and each corner of star =108/3=36.
Total angles corner of star =36*5=180.

0 Helpful 0 Interesting 0 Brilliant 0 Confused

You are assuming that the star is regular. What happens if the star is not regular?

Calvin Lin Staff - 5 years, 3 months ago

Log in to reply

I think total interior angles of "the star" is the same. (...Interior angles of any pentagon is 540)

Prasit Sarapee - 5 years, 3 months ago

If the star is not regular, we merely add another structure in relation to a given that makes the star regular, produce the given calculation, and then subtract the added structure.

Refath Bari - 4 years ago

regular and not regular is the same total sum of inner angle.

Prasit Sarapee - 3 years, 11 months ago

Log in to reply

I agree that the "total sum is the same". Can you make that more explicit in your solution, so that we don't rely on the assumption that it is regular?

Calvin Lin Staff - 3 years, 11 months ago

The star is irregular (angles are not equal) so you can't apply regular pentagon rules

samer Ayoub - 5 years, 3 months ago

Log in to reply

um...i think so. ?#!@.

Prasit Sarapee - 3 years, 11 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...