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Lesson 7-1 Angles of Polygons

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Click the mouse button or press the Space Bar to display the answers.

5-Minute Check on Chapter 5 State whether this sentence is always, sometimes, or never true. The three altitudes of a triangle intersect at a point inside the triangle. Find n and list the sides of ΔPQR in order from shortest to longest if mP = 12n – 15, mQ = 7n + 26, and mR = 8n – 47. State the assumption you would make to start an indirect proof of the statement If –2x ≥ 18, then x ≤ –9. Find the range for the measure of the third side of a triangle given that the measures of two sides are 43 and 29. Write an inequality relating mABD and mCBD. Write an equation that you can use to find the measures of the angles of the triangle. Sometimes – when all angles are acute all angle add to 180 so n = 8; PQ < QR < PR x > -9 43 – 29 = 14 < n < 72 = Since 11 > 10, then mABD < mCBD 111 = 3x + (x – 5) Click the mouse button or press the Space Bar to display the answers.

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Objectives Use the interior angle measures of polygons

Sum of Interior angles = (n-2) • 180 One Interior angle = (n-2) • 180 / n Use the exterior angle measures of polygons Sum of Exterior angles = 360 One Exterior angle = 360/n Exterior angle + Interior angle = 180 (linear pair)

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Vocabulary Convex – no line that contains a side of the polygon goes into the interior of the polygon Diagonal – a segment of a polygon that joins two nonconsecutive vertices Equilateral polygon – all sides of the polygon are congruent Equiangular polygon – all interior angles of the polygon are congruent Exterior angles – angle outside the polygon formed by an extended side Interior angles – an angle inside the polygon Regular polygon – convex polygon that is both equilateral and equiangular

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Angles in a Polygon 1 2 3 4 5 6 7 8 Octagon n = 8

8 180° – 360° (center angles) = (8-2) • 180 = 1080 Sum of Interior angles = (n-2) • 180

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Sum of Interior Angles: so each interior angle is (n – 2) * 180

Angles in a Polygon Sum of Interior Angles: (n – 2) * where n is number of sides so each interior angle is (n – 2) * 180 n Octagon n = 8 Interior Angle Sum of Exterior Angles: so each exterior angle is n Interior Angle + Exterior Angle = 180 Exterior Angle Octagon Sum of Exterior Angles: 360 Sum of Interior Angles: 1080 One Interior Angle: 135 One Exterior Angle: 45

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Polygons Sides Name Sum of Interior Angles One Interior Angle

Sum Of Exterior Angles One Exterior Angles 3 Triangle 180 60 360 120 4 Quadrilateral 90 5 Pentagon 540 108 72 6 Hexagon 720 7 Heptagon 900 129 51 8 Octagon 1080 135 45 9 Nonagon 1260 140 40 10 Decagon 1440 144 36 12 Dodecagon 1800 150 30 n N – gon (n-2) * 180 180 – Ext 360 ∕ n =

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Angle Theorems Note: Sum of interior angles in a polygon is found by S = (n – 2) × 180°

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Angle Theorems Note: Sum of exterior angles in any polygon is 360°

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Example 1 Find the sum of the measures of the interior angles of the figure. Since the figure is a convex decagon, we can use the Polygon Interior Angles Theorem. 𝑺 = 𝒏 – 𝟐 ×𝟏𝟖𝟎= 𝟏𝟎−𝟐 ×𝟏𝟖𝟎=𝟏𝟒𝟒𝟎 Answer: The sum of the measures of the angles is 1440.

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Example 2 The sum of the measures of the interior angles of a convex polygon is 1800°. Classify the polygon by the number of sides. Since the figure is a convex decagon, we can use the Polygon Interior Angles Theorem. 𝑺 = 𝒏 – 𝟐 ×𝟏𝟖𝟎 𝟏𝟖𝟎𝟎=(𝒏−𝟐)×𝟏𝟖𝟎 𝟏𝟎=(𝒏−𝟐) 𝟏𝟐 = 𝒏 Answer: The figure is a dodecagon, 12-sided figure.

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Example 3 Find the value of x in the diagram.

Since the figure is a convex decagon, we can use the Polygon Interior Angles Theorem. 𝑺 = 𝒏 – 𝟐 ×𝟏𝟖𝟎= 𝟔−𝟐 ×𝟏𝟖𝟎=𝟕𝟐𝟎 𝟕𝟐𝟎=𝒙+𝟏𝟏𝟖+𝟏𝟏𝟏+𝟏𝟐𝟒+𝟏𝟏𝟓+𝟏𝟒𝟓 𝟕𝟐𝟎=𝒙+𝟔𝟏𝟑 𝟏𝟎𝟕=𝒙 Answer: x = 107.

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Example 4 Is the polygon regular? Explain your reasoning

Find the measures of B, D, E, and G. Answer: no – not all angles = 𝑺 = 𝒏 – 𝟐 ×𝟏𝟖𝟎= 𝟕−𝟐 ×𝟏𝟖𝟎=𝟗𝟎𝟎 𝟗𝟎𝟎=𝒙+𝟏𝟎𝟎+𝒙+𝟏𝟒𝟎+𝒙+𝒙+𝟏𝟔𝟎 𝟗𝟎𝟎=𝟒𝒙+𝟒𝟎𝟎 𝟓𝟎𝟎=𝟒𝒙 𝟏𝟐𝟓=𝒙=∡𝑩=∡𝑫=∡𝑬=∡𝑮

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Example 5 Find the value of x in the diagram. 𝑺𝒖𝒎 𝑬𝒙𝒕 ∡=𝟑𝟔𝟎

𝟑𝟔𝟎=𝟓𝒙+𝟔𝟐+𝟑𝟓+𝟔𝟗+𝟒𝟏+𝟓𝟖 𝟑𝟔𝟎=𝟓𝒙+𝟐𝟔𝟓 𝟗𝟓=𝟓𝒙 𝟏𝟗=𝒙 Answer: x = 19

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Example 6 Each face of the dodecahedron is shaped like a regular pentagon. Find the measure of each interior angle of a regular pentagon. Find the measure of each exterior angle of a regular pentagon. 𝑰𝒏𝒕 ∡+𝑬𝒙𝒕 ∡=𝟏𝟖𝟎 𝑰𝒏𝒕 ∡+𝟕𝟐=𝟏𝟖𝟎 𝑰𝒏𝒕 ∡=𝟏𝟎𝟖 𝑺𝒖𝒎 𝑬𝒙𝒕 ∡=𝟑𝟔𝟎 𝑬𝒙𝒕∡= 𝟑𝟔𝟎 𝟓 =𝟕𝟐

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Polygon Hierarchy Polygons Quadrilaterals Parallelograms Kites

Trapezoids Isosceles Trapezoids Rectangles Rhombi Squares

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Quadrilaterals Venn Diagram

Trapezoids Parallelograms Isosceles Trapezoids Rhombi Kites Squares Rectangles

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Quadrilateral Characteristics Summary

Convex Quadrilaterals 4 sided polygon 4 interior angles sum to 360 4 exterior angles sum to 360 Parallelograms Trapezoids Bases Parallel Legs are not Parallel Leg angles are supplementary Median is parallel to bases Median = ½ (base + base) Opposite sides parallel and congruent Opposite angles congruent Consecutive angles supplementary Diagonals bisect each other Rectangles Rhombi Isosceles Trapezoids All sides congruent Diagonals perpendicular Diagonals bisect opposite angles Angles all 90° Diagonals congruent Legs are congruent Base angle pairs congruent Diagonals are congruent Squares Diagonals divide into 4 congruent triangles

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Summary & Homework Summary: Homework:

If a convex polygon has n sides and sum of the measures of its interior angles is S, then S = 180(n-2)° The sum of the measures of the exterior angles of a convex polygon is 360° Interior angle + Exterior angle = 180 (linear pair) Homework: SOL Polygon’s Angles WS

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