Geometry Mini Quiz 12/10/15 1) 3) Fill in the chart. Write the name of the point of concurrency (where they meet). 2) AltitudesAngle Bisector MediansPerpendicular Bisector WRITE DOWN! Use ½ sheet of paper 1) ASSUME There are not at least 7 dogs OR There are 7 or more dogs 2) Assume N is not odd OR N is even Orthocenter Incenter Centroid Circumcenter

5-5 Review Format for Indirect Proof Step 1 Assume temporarily that ________________________________. Step 2 _____________________________________________________ Step 3 Then this contradicts the given statement that _________. Step 4 The temporary assumption that ____________ must be false. Step 5 So we can conclude that_____________________.

Ex. 1) Step 1 Assume temporarily that ________________________________. Step 2 Step 3 Then this contradicts the given statement that _________. Step 4 The temporary assumption that ____________ must be false. Step 5 So we can conclude that_____________________. X = y = 48 Plug in x = 4 into 8(x + y) = 48 8(4 + y) = y = 16 y = 2 X = 4

5-6 INEQUALITIES IN ONE TRIANGLE WRITE DOWN!

Triangle Inequality Theorem A C B  Smallest Side Opposite Smallest Angle Side AB C Watch!

Triangle Inequality Theorem A B C  Biggest Side Opposite Biggest Angle Side AC B Watch!

Ex 2) Triangle Inequality Theorem  Smallest Side =  Smallest Angle =  Medium Side =  Medium Angle =  Biggest Side =  Biggest Angle =  Make a note: Biggest Side Opposite Biggest Angle A B C AC B CB A AB C LOOK AT Opposite Side & Angle

Ex 3) 60 ANGLES: Small = 58 = U Medium = 60 = T Large = 62 = V SIDES: Small = opposite of U = TV Medium = opposite of T = UV Large = opposite of V = UT Find the measure of angle T

Ex 4 : List the measures of the sides of the triangle, in order of least to greatest. 10x - 10 = 180 Solving for x: Therefore, AB < AC < BC least greatest 2x + 1 4x 4x -11 2x x + 4x - 11 =180 10x = 190 X = 19 A = 76 o ; B = 65 o ; C = 39 o A = 4(19) = 76 o B = 4(19) – 11 = 65 o C = 2(19) + 1 = 39 o Smallest = Angle C Medium = Angle B Largest = Angle A Smallest = Side AB Medium = Side AC Largest = Side BC Plug back in to angles:

Triangle Inequality Theorem The sum of the 2 SMALLEST sides must be greater than the 3 rd side.

Ex 5) Determine if the following lengths are legs of triangles a) Good Exampleb) Bad Example Small sides: = 6 3 rd side: 5 3+3> 5 YES! Sum of small sides is greater than 3 rd side. Yes, it is a triangle. Small sides: = 4 3 rd side: 5 2+2>5 NO! Sum of small sides is NOT greater than 3 rd side No, it is not a triangle.

Ex 6) Determine if the following lengths are sides of triangles a)4, 9, ? 9 9 > 9 We choose the smallest two of the three sides and add them together. Comparing the sum to the third side: b) 9, 5, 5 Since the sum is not greater than the third side, this is not a triangle ? 9 10 > 9 Since the sum is greater than the third side, this is a triangle

Ex 7) YES! No!

Ex 8) Angles: S = 24 smallest O = 130 largest X = 26 medium Sides (opposite from angle): OX = smallest XS = largest OS = medium Shortest side = OX because angle S (opposite from side) is smallest. Find measure of angle X

Ex 9) Two sides of a triangle are 5 ft and 8 ft long. What is the range of possible lengths for the third side? Do ALL Combinations. Add 2 sides. Must be greater than 3 rd side.

Review