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8-1: Find angle measures in polygons
Geometry Chapter 8 8-1: Find angle measures in polygons

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Warm-Up List all of the different polygons we have discussed in this class. Reminder, the names are determined by the number of sides the polygon has.

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Find Angle Measures in Polygons
Objective: Students will be able to find angle measures in various polygons using their structures. Agenda Definitions Theorems Practice

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Diagonals A diagonal of a polygon is a segment that joins two nonconsecutive vertices. Example: B C E A D Diagonals 𝑩𝑫 and 𝑩𝑬 are diagonals of vertex 𝑩

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Diagonals A diagonal of a polygon is a segment that joins two nonconsecutive vertices. Knowledge Connection: How many triangles are made by the diagonals? B C E A D

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Diagonals A diagonal of a polygon is a segment that joins two nonconsecutive vertices. Knowledge Connection: How many triangles are made by the diagonals? What is the combined measure of all the triangles? B C E A D

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Theorem 8.1 Theorem 8.1 – Polygon Interior Angles Theorem: The sum of the measures of the interior angles of a convex π‘›βˆ’π‘”π‘œπ‘› is (π’βˆ’πŸ)βˆ™πŸπŸ–πŸŽΒ°. Example: π‘š<1+…+π‘š<6=(6βˆ’2)βˆ™180 =4βˆ™180=πŸ•πŸπŸŽ 5 3 2 6 1 4 𝒏=πŸ”

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Example 1 Find the sum of the measures of the interior angles of the convex polygon given.

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Example 1 The given polygon has 8 sides (its an Octagon).
Find the sum of the measures of the interior angles of the convex polygon given. The given polygon has 8 sides (its an Octagon).

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Example 1 The given polygon has 8 sides (its an Octagon).
Find the sum of the measures of the interior angles of the convex polygon given. The given polygon has 8 sides (its an Octagon). Thus, its measure is (8βˆ’2)βˆ™180 6βˆ™180 πŸπŸŽπŸ–πŸŽΒ°

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Example 2 Find the number of sides of a convex polygon whose interior angles add to πŸ—πŸŽπŸŽΒ°. Give the name of this polygon.

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Example 2 Find the number of sides of a convex polygon whose interior angles add to πŸ—πŸŽπŸŽΒ°. Give the name of this polygon. π‘›βˆ’2 βˆ™180=900

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Example 2 Find the number of sides of a convex polygon whose interior angles add to πŸ—πŸŽπŸŽΒ°. Give the name of this polygon. π‘›βˆ’2 βˆ™180=900 π‘›βˆ’2=5 𝒏=πŸ•

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Example 2 Find the number of sides of a convex polygon whose interior angles add to πŸ—πŸŽπŸŽΒ°. Give the name of this polygon. The polygon has 𝑛=7 sides. Thus, this polygon is a Heptagon π‘›βˆ’2 βˆ™180=900 π‘›βˆ’2=5 𝒏=πŸ•

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Example 3 a.) A coin has the shape of a regular 11-gon. Find the sum of the measures of its interior angles. b.) The sum of the measures of the interior angles of a convex polygon is πŸπŸ’πŸ’πŸŽΒ°. Classify the polygon by the number of sides it has.

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Example 3 a.) A coin has the shape of a regular 11-gon. Find the sum of the measures of its interior angles. An 11-gon has 11 sides Thus, its measure is (11βˆ’2)βˆ™180 9βˆ™180 πŸπŸ”πŸπŸŽΒ°

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Example 3 b.) The sum of the measures of the interior angles of a convex polygon is πŸπŸ’πŸ’πŸŽΒ°. Classify the polygon by the number of sides it has. The polygon has 𝑛=10 sides. Thus, this polygon is a Decagon. π‘›βˆ’2 βˆ™180=1440 π‘›βˆ’2=8 𝒏=𝟏𝟎

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Corollary Corollary to Theorem 8.1 – Interior Angles of a Quadrilateral: The sum of the measures of the interior angles of a quadrilateral is πŸ‘πŸ”πŸŽΒ°. Example: In Quadrilateral ABCD, π’Ž<𝑨+π’Ž<𝑩+π’Ž<π‘ͺ+π’Ž<𝑫=πŸ‘πŸ”πŸŽΒ° B C A D

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Example 4 Find the value of x in the diagram shown. πŸπŸŽπŸ–Β° 𝟏𝟐𝟏° 𝒙° πŸ“πŸ—Β°

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Example 4 Find the value of x in the diagram shown. πŸπŸŽπŸ–Β° 𝟏𝟐𝟏° 𝒙° πŸ“πŸ—Β°
π‘₯ =360Β° π‘₯+288=360 𝒙=πŸ•πŸΒ°

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Example 5 Find the value of x in the diagram shown. 𝟏𝟏𝟎° πŸ—πŸΒ° 𝟏𝟎𝟎° 𝒙°
πŸ–πŸ’Β°

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Example 5 Find the value of x in the diagram shown. 𝟏𝟏𝟎° πŸ—πŸΒ°
𝟏𝟎𝟎° 𝟏𝟏𝟎° 𝒙° πŸ—πŸΒ° πŸ–πŸ’Β° The polygon is a Pentagon, with angle measure 5βˆ’2 βˆ™180=3βˆ™180=πŸ“πŸ’πŸŽΒ°

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Example 5 Find the value of x in the diagram shown. 𝟏𝟏𝟎° πŸ—πŸΒ°
𝟏𝟎𝟎° 𝟏𝟏𝟎° 𝒙° πŸ—πŸΒ° πŸ–πŸ’Β° π‘₯ =540 π‘₯+384=540 𝒙=πŸπŸ“πŸ”Β°

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Exterior Angles The sum of the measures of the Interior Angles depends on the number of sides the polygon has. Exterior Angles, however, always have the same angle measure, regardless of the number of sides the polygon has.

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Theorem 8.2 Theorem 8.2 – Polygon Exterior Angles Theorem: The sum of the measures of the exterior angles of a convex polygon, one angle at each vertex, is πŸ‘πŸ”πŸŽΒ°. 5 3 2 1 4 Example: π‘š<1+…+π‘š<5=πŸ‘πŸ”πŸŽΒ° 𝒏=πŸ“

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Example 6 Use the polygon exterior angles theorem to solve for the value of x. πŸπ’™Β° πŸ–πŸ—Β° 𝒙° πŸ”πŸ•Β°

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Example 6 Use the polygon exterior angles theorem to solve for the value of x. πŸπ’™Β° πŸ–πŸ—Β° 𝒙° πŸ”πŸ•Β° π‘₯+2π‘₯+89+67=360

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Example 6 Use the polygon exterior angles theorem to solve for the value of x. πŸπ’™Β° πŸ–πŸ—Β° 𝒙° πŸ”πŸ•Β° π‘₯+2π‘₯+89+67=360 3π‘₯+156=360 3π‘₯=204 𝒙=πŸ”πŸ–

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Example 7 Use the polygon exterior angles theorem to solve for the value of x. (𝒙+𝟐𝟎)Β° πŸ—πŸ”Β° 𝒙° 𝟏𝟏𝟐°

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Example 7 Use the polygon exterior angles theorem to solve for the value of x. (𝒙+𝟐𝟎)Β° πŸ—πŸ”Β° 𝒙° 𝟏𝟏𝟐° π‘₯+ π‘₯ =360

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Example 7 Use the polygon exterior angles theorem to solve for the value of x. (𝒙+𝟐𝟎)Β° πŸ—πŸ”Β° 𝒙° 𝟏𝟏𝟐° π‘₯+ π‘₯ =360 2π‘₯+228=360 2π‘₯=132 𝒙=πŸ”πŸ”

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Final Practice 1 The measures of three of the interior angles of a quadrilateral are πŸ–πŸ—Β°, 𝟏𝟏𝟎°, and πŸ’πŸ”Β°. Find the measure of the fourth angle.

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Final Practice 1 The measures of three of the interior angles of a quadrilateral are πŸ–πŸ—Β°, 𝟏𝟏𝟎°, and πŸ’πŸ”Β°. Find the measure of the fourth angle. π‘₯ =360Β° π‘₯+245=360 𝒙=πŸπŸπŸ“Β°

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Final Practice 2 The sum of the measures of a convex polygon is πŸπŸ‘πŸ’πŸŽΒ°. Classify the Polygon by the number of sides it has.

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Final Practice 2 The sum of the measures of a convex polygon is πŸπŸ‘πŸ’πŸŽΒ°. Classify the Polygon by the number of sides it has. The polygon has 𝑛=15 sides. Thus, this polygon is a 15-gon. π‘›βˆ’2 βˆ™180=2340 π‘›βˆ’2=13 𝒏=πŸπŸ“

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Final Practice 3 Find the sum of the measures of the interior angles of a convex nonagon.

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Final Practice 3 Find the sum of the measures of the interior angles of a convex nonagon. A nonagon has 9 sides Thus, its measure is (9βˆ’2)βˆ™180 7βˆ™180 πŸπŸπŸ”πŸŽΒ°

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Final Practice 4 A convex hexagon has exterior angles with measures πŸ‘πŸ’Β°, πŸ’πŸ—Β°, πŸ“πŸ–Β°, πŸ”πŸ•Β°, and πŸ•πŸ“Β°. Find the measure of exterior angle at the sixth vertex.

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Final Practice 4 A convex hexagon has exterior angles with measures πŸ‘πŸ’Β°, πŸ’πŸ—Β°, πŸ“πŸ–Β°, πŸ”πŸ•Β°, and πŸ•πŸ“Β°. Find the measure of exterior angle at the sixth vertex. π‘₯ =360 π‘₯+283=360 𝒙=πŸ•πŸ•Β°

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Extra Example Find measures of each interior and exterior angle of the regular polygon given.

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Extra Example Find measures of each interior and exterior angle of the regular polygon given. The given polygon has 12 sides (its a Dodecagon). Thus, its measure is (12βˆ’2)βˆ™180 10βˆ™180 πŸπŸ–πŸŽπŸŽΒ°

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Extra Example Find measures of each interior and exterior angle of the regular polygon given. For the Interior Angles: 1800Γ·12=πŸπŸ“πŸŽΒ° (Why?)

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Extra Example Find measures of each interior and exterior angle of the regular polygon given. For the Interior Angles: 1800Γ·12=πŸπŸ“πŸŽΒ° (Why?) For the Exterior Angles: 360Γ·12=πŸ‘πŸŽΒ° (Why?)

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Extra Example Find measures of each interior and exterior angle of the regular polygon given. Thus, each interior angle of a regular dodecagon has a measure of πŸπŸ“πŸŽΒ°, and each exterior angles has a measure of πŸ‘πŸŽΒ°.

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Extra Example Find measures of each interior and exterior angle of a regular 18-gon. An 18-gon has 18 sides Thus, its measure is (18βˆ’2)βˆ™180 16βˆ™180 πŸπŸ–πŸ–πŸŽΒ°

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Extra Example Find measures of each interior and exterior angle of a regular 18-gon. For the Interior Angles: 2880Γ·18=πŸπŸ”πŸŽΒ° For the Exterior Angles: 360Γ·18=𝟐𝟎° Thus, each interior angle of a regular 18-gon has a measure of πŸπŸ”πŸŽΒ°, and each exterior angle has a measure of 𝟐𝟎°.

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