Geometry/Chapter 12

Interior angles are the angles inside a polygon. To find the sum of the interior angles, use the following expression: $(n-2)\cdot 180^{\circ }$ where $n$ is the number of sides of the polygon.

Example

What is the sum of all the degrees in a pentagon?

$(5-2)\cdot 180^{\circ }=3\cdot 180^{\circ }=540^{\circ }$ there are 540 degrees in a pentagon.

In order to find how many degrees are in each side of a regular pentagon (regular meaning same length and angle for each side), take the sum of all the interior angles and divide it by how many sides there are.

${\frac {540^{\circ }}{5}}=108^{\circ }$

In a regular pentagon, each angle is 108 degrees

Sum of the Interior Angles of a Triangle

The sum of the interior angles of a triangle is 180 degrees.

Example Problem:

What is the third angle of a triangle, given that the other two angles are 35 degrees and 75 degrees?

Answer: $35^{\circ }+75^{\circ }=110^{\circ }$ and $180^{\circ }-110^{\circ }=70^{\circ }$ so the third angle must be 70 degrees.

Triangle Exterior Angle Theorem

The exterior angle of a triangle is equal in measure to the sum of the two remote (not adjacent) interior angles of the triangle.

Example Problem

If the exterior angle of a triangle is 40 degrees and if one of the remote angles is 15 degrees, what is the measure of the other remote angle?

$40^{\circ }-15^{\circ }=25^{\circ }$ So the other remote angle is 25 degrees.

The Sum of Exterior Angles Theorem

The sum of exterior angles of a convex polygon taken one at each vertex is 360 degrees.

Exercises

Example Problem If a regular polygon has 15 sides, what is the measure of each exterior angle?

Answer: ${\frac {360^{\circ }}{15}}=24^{\circ }$ so each exterior angle is 24. The interior angles must add to 180 so $180^{\circ }-24^{\circ }=156^{\circ }$ so each interior angle is 156 degrees.

Navigation

• Geometry Main Page
• Motivation
• Introduction
• Geometry/Chapter 1 - HS Definitions and Reasoning (Introduction)
  • Geometry/Chapter 1/Lesson 1 Introduction
  • Geometry/Chapter 1/Lesson 2 Reasoning
  • Geometry/Chapter 1/Lesson 3 Undefined Terms
  • Geometry/Chapter 1/Lesson 4 Axioms/Postulates
  • Geometry/Chapter 1/Lesson 5 Theorems
  • Geometry/Chapter 1/Vocabulary Vocabulary
• Geometry/Chapter 2 Proofs
• Geometry/Chapter 3 Logical Arguments
• Geometry/Chapter 4 Congruence and Similarity
• Geometry/Chapter 5 Triangle: Congruence and Similiarity
• Geometry/Chapter 6 Triangle: Inequality Theorem
• Geometry/Chapter 7 Parallel Lines, Quadrilaterals, and Circles
• Geometry/Chapter 8 Perimeters, Areas, Volumes
• Geometry/Chapter 9 Prisms, Pyramids, Spheres
• Geometry/Chapter 10 Polygons
• Geometry/Chapter 11 $\mathbb {R} ,\mathbb {R} ^{2},\mathbb {R} ^{3}$
• Geometry/Chapter 12 Angles: Interior and Exterior
• Geometry/Chapter 13 Angles: Complementary, Supplementary, Vertical
• Geometry/Chapter 14 Pythagorean Theorem: Proof
• Geometry/Chapter 15 Pythagorean Theorem: Distance and Triangles
• Geometry/Chapter 16 Constructions
• Geometry/Chapter 17 Coordinate Geometry
• Geometry/Chapter 18 Trigonometry
• Geometry/Chapter 19 Trigonometry: Solving Triangles
• Geometry/Chapter 20 Special Right Triangles
• Geometry/Chapter 21 Chords, Secants, Tangents, Inscribed Angles, Circumscribed Angles
• Geometry/Chapter 22 Rigid Motion
• Geometry/Appendix A Formulae
• Geometry/Appendix B Answers to problems
• Appendix C. Geometry/Postulates & Definitions
• Appendix D. Geometry/The SMSG Postulates for Euclidean Geometry