1. By: Madi and Hrishika

2. Interior Angles: angles inside a polygon Exterior Angles: angles outside of a polygon that are formed by extending the sides Regular Polygons: polygons with all angles and sides congruent Helpful Definitions

3. Triangle Theorems Triangle Sum Theorem: the sum of the interior angles of a triangle is always 180 ゜ Exterior Angle Theorem: the measure of an exterior angle of a triangle is equal to the sum of the two opposite angles 1 2 ∠ 1+ ∠ 2 = ∠ 3 3

4. Triangle Theorems Cont. T hird Angle Theorem or No Choice Theorem: If two angle pairs of a triangle are congruent to two angles of another triangle, then the third pair of angles are also congruent. These theorems are used to find measurements in different problems.

5. 13 2 A: ∠ 3= ∠ 1+ ∠ 2, 180= ∠ 4+ ∠ 3 6x+5y+38=2x+5y+10+3x+7y-14 180=8x-10y-14+6x+5y+38 14(42)=(-x+7y)14 156=14x-5y 588=-14x+98y 156=14x-5(8) 156=14x-5y 196=14x 744=93y 14=x 8=y ∠ 1=2x+5y+10=2(14)+5(8)+10=28+40+10= 4 + 78° Q: Find the measure of ∠ 1 If ∠ 1=2x+5y+10, ∠ 2=3x+7y-14 ∠ 3=6x+5y+38, ∠ 4=8x-10y-14

6. Formulas n-gon n=number of sides sum of the external angles of any polygon adds up to 360° number of non-overlapping triangles=(n-2) sum of angle measures=(n-2)180 number of diagonals=(n-3)n 2

7. A: First you find how many sides the polygon has 360÷15=24=n Then you use the formula to find the number of diagonals (n-3)n=(24-3)24=21×24=504= 2 2 2 2 Q: How many diagonals does a regular polygon have if its exterior angles are 15°? 252 diagonals

8. Ratios can be written in many different forms: a:b, a, a to b b Proportions compare two ratios: a=c b d Geometric mean: √ab Arithmetic mean is the average of two numbers. Mean proportion: a=x x r Ratios, Proportions, and Geometric Mean

9. Q: What is the geometric mean and the arithmetic mean of 4 and 16. Q: The shadow of a 36 foot tall tree is 12 feet. Paul is 6 feet tall. How tall is Paul’s shadow? A: GM: √4×16=√64= AM: (4+16)/2=20/2= ±8 10 A: First you set up the proportion, then you solve for x. 36=6 36x=72 12 x x= 2 feet

10. Similar Polygons Similar polygons have proportional corresponding sides and congruent angles A scale factor is the ratio of lengths of similar shapes

11. Ways to Prove Similarity AA ∼ : two pairs of corresponding angles are congruent SSS ∼ : all three sides of the triangles are proportional to each other SAS ∼ : two pairs of proportional sides, and the angles in between those two sides are congruent

12. Proportionality Theorems If a line is parallel to the base of a triangle, then its sides are proportionate. AE=AD=ED AB AC BC Angle Bisector Theorem: AB=AC BD CD E A CB D B A C D

13. Proportionality Theorems Cont. If three or more lines are intersected by two transversals, then the transversals divide the parallel lines proportionally These theorems are used to set up proportions when finding a segment length

14. A: First you set up the proportion. AB=BC 8=20 DE EF 10 x Then you cross multiply and solve for x. 8x=200 x= A B C DEF Q: AD ∥ BE ∥ CF and AB=8, BC=20, DE=10. Find the length of EF 25

15. Common Mistakes ● Mistake: people get confused on which are the corresponding sides of a triangle Solution: You can find the side lengths by using the name of the triangle and corr. sides are on the same spot in both of the triangles ● Mistake: people incorrectly label similar triangles Solution: Check to make sure that corresponding sides are listed in the same order ● Mistake: write the wrong proportion between sides Solution: Be careful and double check your work

16. Using These Skills in Life ● Finding the mean proportion, the geometric mean, and the arithmetic mean can be used with data ● Setting up and solving proportions can be used for finding measurements of various objects ● You can convert different measurements when you are baking ● The skills in this unit would also be useful in careers such as an architect or an engineer

17. Connections to Other Units ● Unit 7; Altitude and Leg Rules: had to set up proportions and solve for unknown segment lengths ● Unit 9; Dilations: had to find and use scale factors between two similar figures