Unit 5 ReviewBy: 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

4 6

2

4

xFind 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.

y

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 = DEBC EF AC DF

A D

B E

FC

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 = legleg 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?

90 901

5

2 10

Yes, both triangles are similar by SAS~ and both circles are similar by the same scale factor, 5.

0.5 2.5

Thank you!!!