Quadrilaterals Diagonals and Angles of Polygons Quadrilaterals Diagonals

Quadrilaterals, Diagonals, and Angles of Polygons

Quadrilaterals, Diagonals, and Angles of Polygons • A Polygon is a simple closed plane figure, having three or more line segments as sides • A Quadrilateral is any four-sided closed plane figure • A Diagonal a line segment that connects one vertex to another (but not next to it) on a polygon

Classifying Polygons Number of Sides 3 Name of Polygon Triangle Number of Name of Sides Polygon 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon

Quadrilateral Angles • We know that the interior angles of a triangle add up to 180 degrees • How many degrees are in the interior angles of a quadrilateral?

Quadrilateral Angles • If we draw a diagonal from one vertex across to the opposite vertex, we see that we have formed two triangles • Therefore, the sum of two triangles will give you the measure of the interior angles of a quadrilateral • 180 + 180 = 360 degrees!

Quadrilateral Angles Checkpoint • Find the missing angle of a quadrilateral with the following measures: m 1 = 117 m 2 = 110 m 3 = 75 m 4 = 117 + 110 + 75 + x = 360 302 + x = 360 x = 58

Angles of Polygons Mini-Lab • Let’s explore this knowledge and how it relates to the angles of other polygons • Copy and complete the table below: Number of Sides Number of Triangles Sum of Angle Measurements 3 1 1(180) = 180 4 2 2(180) = 360 5 6 7 Sketch of Figure

Angles of Polygons Mini-Lab • Draw a pentagon with diagonals from one vertex to each opposing vertex

Angles of Polygons Mini-Lab • Let’s explore this knowledge and how it relates to the angles of other polygons • Copy and complete the table below: Number of Sides Number of Triangles Sum of Angle Measurements 3 1 1(180) = 180 4 2 2(180) = 360 5 3 3(180) = 540 6 7 Sketch of Figure

Angles of Polygons Mini-Lab • Draw a hexagon with diagonals from one vertex to each opposing vertex

Angles of Polygons Mini-Lab • Let’s explore this knowledge and how it relates to the angles of other polygons • Copy and complete the table below: Number of Sides Number of Triangles Sum of Angle Measurements 3 1 1(180) = 180 4 2 2(180) = 360 5 3 3(180) = 540 6 4 4(180) = 720 7 Sketch of Figure

Angles of Polygons Mini-Lab • Draw a heptagon with diagonals from one vertex to each opposing vertex

Angles of Polygons Mini-Lab • Let’s explore this knowledge in how it relates to the angles of other polygons • Copy and complete the table below: Number of Sides Sketch of Figure Number of Triangles Sum of Angle Measurements 3 1 1(180) = 180 4 2 2(180) = 360 5 3 3(180) = 540 6 4 4(180) = 720 7 5 5(180) = 900

Angles of Polygons Mini-Lab • What patterns do you see as a result of our experiment? • The number of triangles in any polygon is always two less than the number of sides. • Therefore, if n = the number of sides of the polygon; the sum of interior angles of any polygon can be expressed as (n – 2)180!

Angles of Polygons Checkpoint • Find the sum of the measures of the interior angles of each polygon: 15 -gon? 23 -gon? 30 -gon? (15 -sided figure) (23 -sided figure) (30 -sided figure) 13 x 180 = 2340

Angles of Polygons Checkpoint • Find the sum of the measures of the interior angles of each polygon: 15 -gon? 23 -gon? 30 -gon? (15 -sided figure) (23 -sided figure) (30 -sided figure) 13 x 180 = 2340 21 x 180 = 3780

Angles of Polygons Checkpoint • Find the sum of the measures of the interior angles of each polygon: 15 -gon? 23 -gon? 30 -gon? (15 -sided figure) (23 -sided figure) (30 -sided figure) 13 x 180 = 2340 21 x 180 = 3780 28 x 180 = 5040

Regular Polygons • A regular polygon is one that is equilateral (all sides congruent) and equiangular (all angles congruent) • Polygons that are not regular are said to be irregular • If the formula for finding the sum of measures of interior angles of a polygon is (n-2)180, how would you find the measure of each angle of a regular polygon? ( n – 2 )180 n

Regular Polygons Checkpoint • Find the sum of the measures of the interior angles of each regular polygon and the measure of each individual angle: 15 -gon? 23 -gon? 30 -gon? (15 -sided figure) (23 -sided figure) (30 -sided figure) 13 x 180 = 2340 / 15 = 156

Regular Polygons Checkpoint • Find the sum of the measures of the interior angles of each regular polygon and the measure of each individual angle: 15 -gon? 23 -gon? 30 -gon? (15 -sided figure) (23 -sided figure) (30 -sided figure) 13 x 180 = 2340 / 15 = 156 21 x 180 = 3780 / 23 = 164. 35

Regular Polygons Checkpoint • Find the sum of the measures of the interior angles of each regular polygon and the measure of each individual angle: 15 -gon? 23 -gon? 30 -gon? (15 -sided figure) (23 -sided figure) (30 -sided figure) 13 x 180 = 2340 / 15 = 156 21 x 180 = 3780 / 23 = 164. 35 28 x 180 = 5040 / 30 = 168

Homework • Skill 4: Polygons (both sides) • 6 -3 Skills Practice: Polygons and Angles • DUE TOMORROW!