Geometry Midterm Review
Segment Addition PostulateIf B is between A and C, then AB + BC = AC
(Converse): If AB + BC = AC, then B is between A and C.
A B C
AC
Application of Segment Addition Postulate: Use the Diagram to find KL
38
J 15 K L
JL = JK + KL
38 = 15 + KL
23 = KL
Segment Addition Postulate
Substitute 38 and 15
Simple Algebra will give you a solution 23
Bisectors or MidpointsMidpoint
A point that splits a segment into to equal halves
Bisectors Segment: A line or a Ray
that passes through the Midpoint of a segment
Angle: A line or a ray that cuts an angle in half
Find Segment Lengths1. M is the midpoint of AB, find AM and MB.Solution:M is the midpoint of AB, so AM is half of AB.
AM = ½ AB = ½ 26 = 13
MB = AM = 13
A M B
26
Find Segment Lengths1. P is the midpoint of RS, find PS and RS.Solution:P is the midpoint of RS, so PS = RP = 7.
RS = 2 RP = 2 7 = 14
PS = 7 and RS = 14
R P S
7
Using Algebra1. Line d is a segment bisector of
AB, find x.
Solution:
M is the midpoint, write an equation
Substitute values for AM and MB
Solve for x
AM = MB
5x = 35
X = 7
A M B
d5x 35
Laws of LogicLaw of Detachment
If the Hypothesis of a statement is true, then the conclusion is also true.
Law of Syllogism (aka The Chain Rule) If the hypothesis (p), then the conclusion (q) If the hypothesis (q), then the conclusion (r) If the hypothesis (p), then the conclusion (r)
The Law of DetachmentMary goes to the movies
every Friday and Saturday. Today is Friday 1st Identify the hypothesis and
conclusion of the statement
Hypothesis: “If it is Friday or
Saturday”
Conclusion: “Then Mary will go to the
movies.”
“Today is Friday” satisfies the hypothesis, so you can conclude that Mary will go to the movies.
The Law of Syllogism If Ron gets lunch today,
then he will get a sandwich.
If Ron gets a sandwich, then he will get a glass of milk.
If Ron gets lunch today, then he will get a glass of milk.
If p, then q
If q, then r
If p, then r
Types of Logical Statements If it is raining, then it is
cloudy. If it is cloudy, then it is
raining. If it is not raining, then it
is not cloudy. If it not cloudy, then it is
not raining.
Conditional Statement:
Converse:
Inverse:
Contrapositive:
Corresponding Angles:Two angles that are in
corresponding positions on both the transversal and accompanying lines
1 & 5 are to the left of the transversal and on the top of their accompanying lines
Angles Formed by Transversals
1
5
t
m
n
Alternate Interior Angles:Two angles that are on
the opposite sides of the transversal and lie between the two accompanying lines
3 & 6 are on opposite or alternating sides of the transversal and lie on the inside of the two accompanying lines
Angles Formed by Transversals
3
6
t
m
n
Alternate Exterior Angles:Two angles that are on
the opposite sides of the transversal and lie on the outside of accompanying lines
2 & 7 are on opposite or alternating sides of the transversal and lie on the outside of the two accompanying lines
Angles Formed by Transversals
2
7
t
m
n
Consecutive Interior Angles: (AKA Same Side Interior Angles)Two angles that are on
the same side of the transversal and lie between the two accompanying lines
4 & 6 are on the same side of the transversal and lie on the inside of the two accompanying lines
Angles Formed by Transversals
46
t
m
n
Properties of Slope Slope:
Rise/Run (y2 – y1)/(x2 – x1)
Negative SlopeMoves down from left to right
Positive SlopeMoves up from left to right
Undefined SlopeSlope of Vertical Lines, y/0
Zero SlopeSlope of Horizontal Lines, 0/x
Identify the Parallel LinesWhich of the lines if any are
parallel?Slope of p:
(-6 – (-1))/(-4 – (-3)) -5/-1 = 5
Slope of h: (2 – (-4))/(2 – 1) 6/1 = 6
Slope of s: (2 – (-3))/(4 – 3) 5/1 = 5
p s
(-4, -6)
(-3, -1)
(2, 2)
(1, -4)(3, -3)
(4, 2)
p h s
Slopes of Perpendicular LinesTwo nonvertical lines are perpendicular if and only
if the product of their slopes is -1 In other words the slopes of perpendicular lines are
opposite reciprocalsExample: (5/4)(-4/5) = -1
Horizontal lines are perpendicular to vertical lines
Drawing a Perpendicular LineLine w passes through (1, -2) and
(5, 6). Graph the line perpendicular to line w that passes through (2, 5)
Step 1: Find the slope of w (6 – (-2))/(5 – 1) = 8/4 = 2
Step 2: Determine the slope of the line perpendicular to w m = - ½
Step 3: Use rise and run to find a second point on the line
(2, 5)
(1, -2)
(5, 6)w
(4, 4)
Parts of a Right TriangleHypotenuse
Longest side of a right triangle
Side opposite the right angle
Legs of a Right Triangle Two shorter legs of a
right triangle The two legs that make
up the right angle
Label the Hypotenuse and the legs of the below Triangle
Hypotenuse: BCLegs: AB & AC
A
B
C
Using the Pythagorean Theorem to find… The Hypotenuse
Hypotnuse2 = (leg1)2 + (leg2)2
c2 = 32 + 42
c2 = 9 + 16c2 = 25c = 5
One of the legsHypotnuse2 = (leg1)2 + (leg2)2
102 = 62 + b2
100 = 36 + b2
b2 = 64b = 8
6
b
103
4
c
Classifying Triangles using the Pythagorean Theorem
Acute If the sum of the squares of the
two shorter sides is greater than the square of the largest side, then the triangle is acute
72 + 82 ? 102
49 + 64 ? 100 113 > 100 Therefore the Triangle is Acute
If the sum of the squares of the two shorter sides is less than the square of the largest side, then the triangle is obtuse
62 + 92 ? 122
36 + 81 ? 144 117 < 144 Therefore the Triangle is Obtuse
7 8
10
Obtuse
6 9
12
Classifying Triangles by their Sides
Scalene Triangle
Isosceles Triangle
Equilateral Triangle
No Congruent Sides
At Least 2 Congruent Sides
3 Congruent Sides
Classifying Triangles by Angles
Acute Triangle
Right Triangle
Obtuse Triangle
Equiangular Triangle
3 Acute Angles
1 Right Angle
1 Obtuse Angle
3 Congruent Angles
Interior Angles of a TriangleTriangle Sum Theorem
The sum of the measures of the angles of a triangle is 180°
mA + mB + mC = 180
Corollary to the Triangle Sum Theorem
The Acute angles of a right triangle are complementary
mB + mC = 90
A
B C A
B
C
Exterior Angle TheoremThe measure of the exterior angle of a triangle is
equal to the sum of the measures of the two nonadjacent or opposite angles
m1 = mA + mB
A
1B
Triangle InequalitiesIf one side of a triangle is longer than another,
then the angle opposite the longer side is larger than the angle opposite the shorter side.
If , then The converse is also true
A
B C
MidsegmentProperties of a Midsegment
Segment that connects the midpoints of two sides of a triangle
The Midsegment is half the length of the third side
The Midsgment is parallel to the third side
is a Midsegment
BD = ½ (AE) If AE = 12, then BD = 6
A
B
C
D
E
Medians and CentroidsA Median connects a vertex
of a triangle to a midpoint of the opposite side
The intersection of three Medians is a Centroid The distance from the vertex
to the Centroid is two-thirds the length of the Median
P is a Centroid
is a MedianAP = (2/3)(AX) If AX = 27, then AP = 18
P
A
X