Angles within Polygons

Unit Overview

In this unit, students will be able to:

· Identify regular polygons, as well as their individual interior angles.

· Determine missing angle values in polygons.

Key Concepts

· Regular polygons

· Interior angle sum of a polygon

· Remote Exterior Angle Theorem

Ohio’s Learning Standards

· 8.G.5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Calculators

· Here is a link to the Desmos scientific calculator that is provided for the Ohio State Test for 8^th Grade Mathematics in the spring.

· You are strongly encouraged to use the Texas Instruments TI-30XIIS handheld scientific calculator. It is extremely user friendly.

Polygons

· In a previous unit, you learned this information

· Example A Table and images:

· Notice that for the interior angle sum:

o It increases by 180° for ever side added, and the number of triangles that the polygon can be broken up into is 2 less than the number of sides

o If n = number of sides and S = sum of the interior angles of a polygon, then

§ S = 180(n – 2) → 180 multiplied by 2 less than the number of sides

· REGULAR POLYGON → see Example A set above

o Every side is equal (congruent) and every angle is equal

o Example B Set → irregular polygons

§ For the rhombus, all sides are congruent (symbolized with the same dash), but not all angles are congruent (the opposite angle pairs are congruent, marked by the same single or double arcs)

§ For the hexagon, some sides and angles are congruent, but not all.

o Determining individual interior angles of a regular polygon:

§ Sum of the interior angles ÷ number of angles

§ Example C Set:

o Determining missing angles polygons in polygons: set up algebraic equations and solve

§ Example D Set: solve for the variable.

5 sided figure: pentagon Right triangle Sum of angles = 540° 90+90+90+90+161+y=540                            y+431=540                               -431 -431                                   y=119	Right triangle Sum of angles = 180° 4x+2x+10+90=180             6x+100=180                  -100   -100                        6x=80                        ÷6 ÷6                          x = 13 2/3	Trapezoid Sum of angles = 360° 62+5x+12+122+58=360             5x+254=360                  -254   -254                        5x=106                        ÷5 ÷5                          x = 21.2

Further explanation and practice on polygons:

Let’s Practice:

---

Remote Exterior Angle Theorem:

∠A + ∠B = ∠D
Here is why this theorem works: ∠A + ∠B + ∠C = 180⁰ → sum of the angles in a triangle = 180°

∠D + ∠C = 180⁰ → ∠D and ∠C together make a straight line = 180°

Thus, ∠A + ∠B = ∠D

Example E set: Determine the value of the variable(s):

Long method: 1^st: Find the missing interior angle: ? = 180 – (66 + 44) = 70⁰ 2^nd: x + ? = 180 → they make a straight line x + 70 = 180       -70 -70          x = 110	y + 142 = 180 → they make a straight line      -142 -142             x = 38 sum of two remote interior ∠s = external ∠ 95⁰ + x = 142 -95 -95            x = 47

Short method → use the Exterior Angle Theorem OR use interior angles: x = 180 – (95+38) = 47

sum of two remote interior ∠s = external ∠

∠66 + ∠44 = ∠x

   110° = x

External Angle Sums for Polygons → always = 360°

Proofs: Triangle	Quadrilateral	Pentagon
all ∠s – interior∠s = exterior ∠s	all ∠s – interior∠s = exterior ∠s	all ∠s – interior∠s = exterior ∠s
540° – 180° = 360°	720° – 360° = 360°	900°– 540° = 360°

For further explanation and practice on remote exterior angles:

Let’s Practice: