Five interior angles of a 10-sided figure are each 185 degrees. If the remaining five interior angles are equal, then each of the five angles (in degrees) is equal to:
103
The key principle is the formula for the sum of the interior angles of any polygon with n sides:
Sum of interior angles = (n − 2) × 180°.
This comes from the fact that an n-sided polygon can be divided into (n − 2) triangles from a single vertex, and each triangle contributes 180°.
For a 10-sided figure (decagon), n = 10:
- Sum of all interior angles = (10 − 2) × 180° = 8 × 180° = 1440°.
- Five of the angles are each 185°, contributing 5 × 185° = 925°. (An interior angle above 180° means the decagon is non-convex/reflex at those vertices, which is allowed since the sum formula still holds.)
- The remaining five equal angles together account for 1440° − 925° = 515°.
- Each such angle = 515° ÷ 5 = 103°.
So the correct answer is 103. A common error is to forget that the total must be 1440° and instead guess values near 105° or 107°; only 103° makes the five equal angles sum exactly to the leftover 515°.
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Important Questions from Geometry
ABCDEF is a regular hexagon. Side of the hexagon is 36 cm. What is the area of the triangle AOB ?
If ∆ABC ~ ∆DEF, and BC = 4 cm, EF = 5 cm and the area of triangle ABC = 80 cm 2, then the area of the triangle DEF is:
If Δ ABC is right angled at B, AB = 12 cm and ∠CAB = 60°, determine the length of BC.
If ΔABC and ΔDEF are congruent triangles, then which of the following is FALSE?
D and E are points on the sides AB and AC, respectively, of ΔABC such that DE is parallel to BC and AD ∶ DB = 7 ∶ 9. If CD and BE intersect each other at F. then find the ratio of areas of ΔDEF and ΔCBF.
