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