Lesson 3-4: Polygons 1 Polygons

Lesson 3-4: Polygons 2 These figures are not polygonsThese figures are 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. 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 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. Classifications of a Polygon

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

Lesson 3-4: Polygons 6 Convex Polygon Formulas….. Diagonals of a Polygon: For a convex polygon with n sides: The sum of the interior angles is A segment connecting nonconsecutive vertices of a polygon 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: 1.Sum of the measures of the interior angles of a 11-gon is (n – 2)180°  (11 – 2)180 °  The measure of an exterior angle of a regular octagon is 3.The number of sides of regular polygon with exterior angle 72 ° is 4.The measure of an interior angle of a regular polygon with 30 sides

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

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

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°

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

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

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.