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Polygons

Polygons. Polygons. Definition:. A closed figure formed by a finite number of coplanar segments so that each segment intersects exactly two others, but only at their endpoints. These figures are not polygons. These figures are polygons. Classifications of a Polygon. Convex:.

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Polygons

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  1. Polygons Lesson 3-4: Polygons

  2. Polygons Definition: A closed figure formed by a finite number of coplanar segments so that each segment intersects exactly two others, but only at their endpoints. These figures arenot polygons These figures are polygons Lesson 3-4: Polygons

  3. Classifications of a Polygon Convex: No line containing a side of the polygon contains a point in its interior Concave: A polygon for which there is a line containing a side of the polygon and a point in the interior of the polygon. Lesson 3-4: Polygons

  4. Classifications of a Polygon Regular: A convex polygon in which all interior angles have the same measure and all sides are the same length Irregular: Two sides (or two interior angles) are not congruent. Lesson 3-4: Polygons

  5. Polygon Names Triangle 3 sides 4 sides Quadrilateral 5 sides Pentagon 6 sides Hexagon 7 sides Heptagon 8 sides Octagon 9 sides Nonagon 10 sides Decagon Dodecagon 12 sides n sides n-gon Lesson 3-4: Polygons

  6. Convex Polygon Formulas….. Diagonals of a Polygon: A segment connecting nonconsecutive vertices of a polygon For a convex polygon with n sides: The sum of the interior angles is The measure of one interior angle is The sum of the exterior angles is The measure of one exterior angle is Lesson 3-4: Polygons

  7. Examples: • Sum of the measures of the interior angles of a 11-gon is • (n – 2)180°  (11 – 2)180 °  1620 • The measure of an exterior angle of a regular octagon is • The number of sides of regular polygon with exterior angle 72 ° is • The measure of an interior angle of a regular polygon with 30 sides Lesson 3-4: Polygons

  8. TRIANGLE FUNDAMENTALS Triangle Sum Theorem: The sum of the interior angles in a triangle is 180˚

  9. Example: 5. In Δ ABC, m<A = 45°, m<B = 90°, find m<C. m<C= 180°- (45° + 90°) m<C= 45°

  10. Example: 6.If 2x, x and 3x are the measures of the angles of a triangle, find the angles. 2x+x+3x=180° 6x= 180° x=30° Angles are 90°, 30° and 60°

  11. Exterior Angle Theorem The measure of the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. Remote Interior Angles Exterior Angle

  12. Example 7: Find the mA. 3x - 22 = x + 80 3x – x = 80 + 22 2x = 102 X=51 m<A=51°

  13. Corollaries: 1.If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent. 2. Each angle in an equiangular triangle is 60˚ 3. Acute angles in a right triangle are complementary. 4. There can be at most one right or obtuse angle in a triangle.

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