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From symmetry, there is a circle that circumscribes the pentagram. The 5 alpha angles divide the circle (360 degrees) into 5 equal parts (alpha angles are angles with vertices on the circle - angles inscribed in the circle). So alpha = (360/5)/2=36 (/2 because the angles have vertices on the circle)

Interior angles for a regular polygon are (n-2) x 180º/n, for a pentagon is 3 x 180º/5 = 108.  So the two base angles of the arm of the star are 72º, making the final angle 36º.

I swear you are better than a therapist for making people feel better about themselves!!! Love your videos breath of fresh air!!!

I already knew the answer when I saw the thumbnail - the golden triangle is one of those gems of geometry.

The exterior angles of a polygon add up to 360. So 360/5=72. 180-72*2=36.

The sum of angles of a five-pointed star is also 180°
180 / 5 equals 36 👍

I used the inscribed angle theorem.  The angle between two chords that meet on a circle is half the arc length of the distance between where those chords intersect the circle.  "The inscribed angle theorem states that the measure of an inscribed angle is always exactly half the measure of its intercepted arc"

Since a Pentagram divides the circle into five arc segments.  I.e trace from alpha to the adjacent alpha five times and you get back to the original alpha.  The Arc Length is 360 / 5 or 72 degrees.  Half of 72 is 36.

Alpha is 36

Pentagrams have 5 points.
All the angles in the points add up to 180 degrees.
180/5 = 36
Each angle is 36 degrees

36.   Centre angle (360 / 5) equals twice the Inscribed angle (vertex ON the circle).

Greetings from Odesa, Ukraine. Thanks for the fun math.

Solution:
A standard regular pentagon consists of 5 isosceles triangles if one connects the vertices to the center. The central angle of these triangles is 360°/5 = 72°. The two base angles each measure (180° – 72°)/2 = 54°. Consequently, the angle at each vertex of the pentagon is 2 × 54° = 108°. The two base angles of the outer triangle each measure 180° – 108° = 72°. Finally, the angle α—the one we are looking for—measures 180° – 2 × 72° = 36°.

I quickly did it in my head.

I will stipulate that the solution given is appropriate for a general audience, and I will confess that I would not have thought to do it that way. Instead I observed that each vertex of the pentagram subtends an arc equal to one fifth of a circle. With 360 degrees in circle each arc has 72 degrees. The angles are therefore one half of that value or 36 degrees. Since the subtended angle conclusion rests on a result less well known than the sum of the angles in a triangle or the observation that the angles in the pentagon are congruent, I guess I wouldn't offer this solution to an audience that wasn't 100 percent math people.

I always love your patient, clear, and accessible explanations. Hope they continue to get people inspired by the topics.

Just draw a circle in the center of the figure and radius to the points of the star. These points divide the circle in five equal arcs. From the center of the circle the angle of each arc will be 360÷5=72 degrees. From the points of the stars the angle will be half,  36 degrees.

I forgot the equation for calculating the sum of the interior angles ( (n-2)*180 ) and discovered that if you divide 360 by the number of sides, you get the complement angle...

sides   |                interior              |   complement
    3     |      (3-2)*180/3 = 60       |   360/3 = 120
    4     |      (4-2)*180/4 = 90       |   360/4 = 90
    5     |      (5-2)*180/5 = 108     |   360/5 = 72
    ...
    9     |      (9-2)*180/9 = 140     |   360/9 = 40
    ...
   15    |   (15-2)*180/15 = 156   |   360/15 = 24

Math solutions made delightful 😁

Hi, Suzanne

Since this star is made from a regular pentagon, each arc connecting the star's tips is 360° / 5 = 72°. Therefore, each tip of the star represents an inscribed angle that is half the arc. That means the angle can only be 36°.

It's faster this way and can be done in your head.

Hi Suzzana.  Love your enthusiasm and smile. I am retired but while browsing YouTube I see your math challenges, and get tempted to solve them to see if I still remember my basics from school and college days.
Some times I am able to solve them, other times I have to watch you solve them. Thank you very much.

The method I used is really basic, using minimal knowledge (because my maths lessons ended, I suspect, before our host was born).

I started with the interior pentagon, and divided it into five equal triangles joined at the centre. Now, given that a full circle is 360 degrees, the central angle of each triangle must be 360/5, or 72 degrees. Each of the other angles of these triangles must be (180-72)/2, or 54 degrees.

As a result, each interior angle of the pentagon must be 2x54, or 108 degrees. Angle Beta must be 180-108, or 72 degrees. Angle Alpha is therefore 180-(2x72), or 36 degrees.