By: Lisa Tauro and Lily Kosaka Unit 5 Review By: Lisa Tauro and Lily Kosaka

Concepts Triangle Sum Theorem -All the interior angle measures of a triangle always add up to 180. Exterior Angle Theorem -The measure of an exterior angle equals the sum of the 2 non-adjacent angles. Triangle Midsegment Theorem -If a segment joins two midpoints of a triangle, the newly formed small triangle and the original big triangle are similar. This means that the segment is parallel and 1/2 of the opposite side of the original triangle.

Concepts regular polygons -2D shapes with 3 or more sides in which all sides and angles are congruent convex polygons -In order to find the sum of all the interior angles of a convex polygon letting “n” be the number of sides/angles, use this formula: 180(n-2) exterior angles -All of the exterior angles in a convex polygon always add up to 360. diagonals in polygons -Letting “n” be the number of sides/angles, the formula for the number of diagonals “d” is: d=n(n-3)/2

Concepts ratios and proportions -A proportion is an equation that states that 2 ratios are equal Means -the numbers in the denominator of the first ratio and the numerator of the second Extremes -the numbers in the numerator of the first ratio and the denominator of the second cross multiplication product of the means = product of the extremes geometric mean -The geometric mean of 2 numbers is found by placing the 2 numbers into a proportion as the extremes, and setting the means as a variable arithmetic mean -the average of 2 numbers

Concepts similarity -In similar polygons, corresponding angles are congruent and corresponding sides are proportional proving triangles similar -AA~, SSS~, SAS~ Side Splitter Theorem -If a line is parallel to 1 side of a triangle and intersects the other 2 sides, then it divides those two sides proportionally Parallel and Transversals -If 3 or more parallel lines are intersected by 2 transversals, the parallel lines divide the transversals proportionally Angle Bisector Theorem -If a ray bisects an angle of a triangle, it divides the opposite side into segments that are proportional to the adjacent sides

Examples Find the measure of one interior angle of a regular heptagon: In a regular shape, all interior angles are congruent. A heptagon has seven interior angles. Using the theorem 180(n-2), the total sum of all seven interior angles becomes 900 degrees. So, one interior angle is about 128.57 (900/7). Find the geometric mean and the arithmetic mean of 12 and 3: First, set up a mean proportion with 12 and 3 as the extremes: 12/x=x/3. Cross multiply, x^2=36. Thus, the geometric mean (x) is 6. The arithmetic mean is simply the average which is 7.5. 2 Find the value of “x” (given that the segment is an angle bisector): 2/4=y/6 can be set up using the parallel and transversal lines theorem. By cross multiplying and solving, y=3. Then, the proportion 2/x=4/(3-x) can be established because of the angle bisector theorem. Thus, the value of “x” becomes 1. x 2 y 4 4 6

Common Mistakes/ Struggles If three or more parallel lines are intersected by transversals, parallel lines divide transversals proportionally Mistake: putting numbers in the wrong places in expression Decide what section or sections to use in the expression, and pay attention to where that piece of information should go into the expression used AB = DE AB = DE BC EF AC DF A D B E C F

Common Mistakes/Struggles n-gon (n= angles/sides) Sum of angle measures= (n-2)180 Mistake: using only this formula to determine one angle of a convex polygon Solution: ((n-2)180)/n

Common Mistakes/Struggles Always simplify Mistake: At the end of cross multiplying, not simplifying Solution: Recheck answers to confirm that all of them are completely simplified

Connections To Other Units Connection: Unit 7(1) Right Triangles and Trigonometry “Use Similar Triangles” Alt.- Leg Rules Both alt.- leg rules and Unit 5 require knowledge of proportions part of hyp = altitude altitude other part of hyp hypotenuse = leg leg projection of leg

Connections To Other Units Connection: Unit 3 Congruent Triangles Angle congruence Sides have to be proportional in similarity

Real Life Usage An engineer wants to confirm that a garden’s shape is similar to that of his original model. The garden has a ¾ circular water pool and a triangular flower bed. Are the two triangles similar? Yes, both triangles are similar by SAS~ and both circles are similar by the same scale factor, 5. 5 1 90 90 2 10 0.5 2.5

Thank you!!!