University:
High School
Course:
High School Mathematics
Academic year:

2023
Views:

20

Pages:

5
Author:

anon

ANGLE RELATIONSHIPS IN POLYGONS Polygons - are 2dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up). Diagonal -A straight line inside a shape that goes from one corner to another (but not a n edge). Concave Polygon – a polygon with no part of any line segment joining two p oints on the polygon outside the polygon. Concave Polygon – a polygon with no part of any line segment joining two p oints on the polygon outside the polygon Regular Polygon – a polygon with all side equal and all interior angles. Complete the following table: Let n be the number of sides of a polygon. Polygon Tri an gle Qu ad rila ter al Pe nta go n He xa go n He pta go n Oct ago n Non ago n Dec ago n # of sides (n) 3 4 5 6 7 8 9 10 # of diagonals from one vertex 0 1 2 3 4 5 6 7 # of triangles in polygon 1 2 3 4 5 6 7 8 Interior angles 180 Exterior angles 360 ◦ ◦ ◦ 360 ◦ 360 ◦ 540 ◦ 360 ◦ 720 ◦ 360 ◦ 900 ◦ 360 ◦ 1080 ◦ 360 ◦ 1260 ◦ 360 3 ×180 = 540 First its “number of triangles in a pentagon” Second number (180) its “sum of interior angles in a triangle” What is the sum of the interior angles of a 20sided polygon? There are 18 triangles in a 20-sided polygon. ◦ ◦ 18 × 180 = 3240 How can we find the sum of the interior angles of an nsided polygon? ◦ 𝑆𝑢𝑚 = (𝑛 − 2) × 180 ◦ 1440 ◦ 360 Conclusions: ◦ Sum of Interior Angles in ANY Polygon is: (𝑛 − 2)× 180 ◦ Sum of Exterior Angles in ANY Polygon is: 360 Ex 1: Find the measures of a and b. Provide reasoning. Solution: 1) Find x. ◦ ◦ 𝑋 + 89 = 180 ◦ 𝑋 = 91 ◦ 2) The sum of the exterior angles in a quadrilateral is 360 , so ◦ ◦ ◦ 𝑋 + 55 + 𝑏 + 121 = 360 ◦ ◦ ◦ ◦ 91 + 55 + 𝑏 + 121 = 360 ◦ ◦ ◦ ◦ 𝑏 + 267 = 360 𝑏 = 360 − 267 ◦ 𝑏 = 93 3) a and b are supplementary angles. ◦ 𝑎 + 𝑏 = 180 ◦ 𝑎 + 93 = 180 ◦ ◦ 𝑎 = 180 − 93 ◦ 𝑎 = 87 ◦ 𝑎 = 87 ◦ 𝑏 = 93 Ex 2: Find the values of c and d. Provide reasoning. Solution: 1)d and 125^(\circ) are supp. Angles, so d+125=180 d=180-125 d=55 2) 7 sides, so n=7 ◦ Sum of interior angles = (𝑛 − 2) × 180 ◦ = (7 − 2) × 180 ◦ = 5 ×180 ◦ = 900 C+134+157+129+152+119+125=900 C+816=900 C=900-816 C=84 Ex 3: Find the values of x and y. Provide reasoning Solution: 1) We have an equilateral triangle (all side lengths are equal) So, 𝑥 = 2) 180 3 = 60 In a parallelogram, opposite sides are equal. Opposite angles are equal as well. Since x and y are supplementary angles, then ◦ 𝑥 + 𝑦 = 180 ◦ ◦ 60 + 𝑦 = 180 ◦ ◦ 𝑦 = 180 − 60 ◦ 𝑦 = 120 Ex 4: a) What is the measure of one interior angle in a regular hexagon? ◦ n=6, sum of int.angles = (𝑛 − 2) × 180 ◦ = (6 − 2) × 180 ◦ = 4 ×180 ◦ = 720 Interior angle = 720 6 ◦ = 120 b) What is the measure of one exterior angle in a regular hexagon? ◦ We know that the sum of the exterior angles in any polygon is 360 Exterior angle = = 60 360 6