Geometry Mini Quiz 12/10/15
1)
1) ASSUME There are not at least 7 dogs
OR There are 7 or more dogs
2)
2) Assume N is not odd OR
N is even
3) Fill in the chart. Write the name of the point of concurrency (where they meet).
Altitudes
Orthocenter
Angle
Bisector
Medians Perpendicular
Bisector
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
X=4
• Assume temporarily that ________________________________.
• Step 2
Plug in x = 4 into 8(x + y) = 48
8(4 + y) = 48
32 + 8y = 48
- 32
-32
8y = 16
• Step 3
y=2
• Then this contradicts the given statement that _________.
• Step 4
X=4
• The temporary assumption that ____________
must be false.
• Step 5
• So we can conclude that_____________________.
5-6 INEQUALITIES IN
ONE TRIANGLE
Triangle Inequality Theorem
Smallest Side Opposite
Side AB
A
Smallest Angle C
3
7
B
C
5
Triangle Inequality Theorem
Biggest Side Opposite
Side AC
A
Biggest Angle B
7
3
B
C
5
Ex 2) Triangle Inequality Theorem
LOOK AT Opposite Side & Angle
Smallest Side =
AB
Smallest Angle =
Medium Side =
A
C
7
CB
Medium Angle =
3
A
B
Biggest Side =
Biggest Angle =
AC
B
C
5
Ex 3)
60
Find the measure of angle T
ANGLES:
SIDES:
Small = 58 = U
Small = opposite of U = TV
Medium = 60 = T
Medium = opposite of T = UV
Large = 62 = V
Large = opposite of V = UT
Ex 4: List the measures of the sides of the
triangle, in order of least to greatest.
B
Solving for x:
2x +1 + 4x + 4x - 11 =180
4x -11
10x - 10 = 180
10x = 190
A
4x
X = 19
2x + 1
Plug back in to angles:
A = 4(19) = 76o
B = 4(19) – 11 = 65o
C
Smallest = Angle C
Medium = Angle B
Largest = Angle A
Smallest = Side AB
Medium = Side AC
Largest = Side BC
C = 2(19) + 1 = 39o
Therefore, AB < AC < BC
A = 76o; B = 65o; C = 39o
least
greatest
Triangle Inequality Theorem
The sum of the 2 SMALLEST sides must be greater than the 3rd side.
Ex 5) Determine if the following lengths are legs of triangles
a) Good Example
b) Bad Example
Small sides: 3 + 3 = 6
3rd side: 5
Small sides: 2 + 2 = 4
3rd side: 5
3+3> 5 YES! Sum of small
sides is greater than 3rd side.
2+2>5 NO! Sum of small sides
is NOT greater than 3rd side
Yes, it is a triangle.
No, it is not a triangle.
Ex 6) Determine if the following lengths are sides of triangles
a)
4, 9, 5
b)
9, 5, 5
We choose the smallest two of the three sides and add
them together. Comparing the sum to the third side:
4+5 ? 9
5+5 ? 9
9>9
10 > 9
Since the sum is
not greater than
the third side,
this is not a
triangle
Since the sum is
greater than the
third side, this is
a triangle
Ex 7)
YES!
No!
Ex 8)
Find measure of angle X
Angles:
Sides (opposite from
Shortest side =
angle):
S = 24 smallest
O = 130 largest OX = smallest
X = 26 medium XS = largest
OS = medium
OX
because angle S
(opposite from
side) is smallest.
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 3rd side.
Review