The sum of the five point angles of a star (a five-pointed star) is a famous problem. Let us think about it using the rule for exterior angles.
Sum of the Point Angles of a Star
The sum of the five point angles of a five-pointed star is 180°.
Reason: Using the rule for the exterior angles of the triangles inside the star, the sum of the five point angles becomes exactly the sum of the interior angles of one triangle (180°).
The sum of the five point angles of a five-pointed star is 180°.
Reason: Using the rule for the exterior angles of the triangles inside the star, the sum of the five point angles becomes exactly the sum of the interior angles of one triangle (180°).
Explanation:
① Let the measures of the point angles A, B, C, D, and E be a, b, c, d, and e.
② Focusing on the triangles inside the star, each exterior angle of a triangle is the sum of two point angles of the star.
③ The sum of the interior angles of the original triangle is 180°.
④ Simplifying this gives a + b + c + d + e = 180°
① Let the measures of the point angles A, B, C, D, and E be a, b, c, d, and e.
② Focusing on the triangles inside the star, each exterior angle of a triangle is the sum of two point angles of the star.
③ The sum of the interior angles of the original triangle is 180°.
④ Simplifying this gives a + b + c + d + e = 180°
Example: Four point angles of a five-pointed star are 35°, 40°, 30°, and 25°. What is the remaining point angle?
Solution: 180 − (35 + 40 + 30 + 25) = 180 − 130 = 50°
Solution: 180 − (35 + 40 + 30 + 25) = 180 − 130 = 50°
For a six-pointed star (the Star of David), the sum of the angles has a different value. First, make sure you understand the shape of a five-pointed star.