Geometry

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# Geometry - PowerPoint PPT Presentation

Geometry . Midterm Review. Segment Addition Postulate. If 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

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### Geometry

Midterm Review

• If 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 KL38J 15 K L

JL = JK + KL

38 = 15 + KL

23 = KL

Substitute 38 and 15

Simple Algebra will give you a solution 23

Bisectors or Midpoints
• Midpoint
• 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 Lengths
• 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 Lengths
• 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 Algebra
• 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

5x

d

35

Laws of Logic
• Law 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 Detachment
• Mary 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:
Angles Formed by Transversals

t

1

m

5

n

• 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

t

m

3

6

n

• Alternate Interior Angles:
• Two angles that are on the opposite sides of the transversal and lie between the two accompanying lines
• 3& 6are on opposite or alternating sides of the transversal and lie on the inside of the two accompanying lines
Angles Formed by Transversals

t

2

m

n

7

• 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

t

m

4

6

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& 6are on the same side of the transversal and lie on the inside of the two accompanying lines
Properties of Slope
• Slope:
• Rise/Run
• (y2 – y1)/(x2 – x1)
• Negative Slope
• Moves down from left to right
• Positive Slope
• Moves up from left to right
• Undefined Slope
• Slope of Vertical Lines, y/0
• Zero Slope
• Slope of Horizontal Lines, 0/x
Identify the Parallel Lines

(2, 2)

(4, 2)

• Which 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

(-3, -1)

(3, -3)

(1, -4)

h

p

(-4, -6)

s

Slopes of Perpendicular Lines
• Two 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 reciprocals
• Example: (5/4)(-4/5) = -1
• Horizontal lines are perpendicular to vertical lines
Drawing a Perpendicular Line

w

(5, 6)

• Line 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)

(4, 4)

(1, -2)

Parts of a Right Triangle
• Hypotenuse
• 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: BC
• Legs: AB & AC

B

C

A

Using the Pythagorean Theorem to find…

The Hypotenuse

One of the legs

Hypotnuse2 = (leg1)2 + (leg2)2

102 = 62 + b2

100 = 36 + b2

b2 = 64

b = 8

• Hypotnuse2 = (leg1)2 + (leg2)2
• c2 = 32 + 42
• c2 = 9 + 16
• c2 = 25
• c = 5

c

10

3

6

4

b

Classifying Triangles using the Pythagorean Theorem

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

Obtuse

• 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

7

8

6

9

10

12

Classifying Triangles by their Sides
• Scalene Triangle
• Isosceles Triangle
• Equilateral Triangle

No Congruent Sides

3 Congruent Sides

At Least 2 Congruent Sides

Classifying Triangles by Angles
• Acute Triangle
• Right Triangle
• Obtuse Triangle
• Equiangular Triangle

3 Acute Angles

1 Obtuse Angle

3 Congruent Angles

1 Right Angle

Interior Angles of a Triangle

Triangle Sum Theorem

Corollary to the Triangle Sum Theorem

The Acute angles of a right triangle are complementary

mB+ mC = 90

• The sum of the measures of the angles of a triangle is 180°
• mA + mB + mC = 180

B

A

C

A

B

C

Exterior Angle Theorem
• The measure of the exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent or opposite angles
• m1 = mA + mB

A

B

1

Triangle Inequalities
• If 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

C

B

Midsegment
• Properties 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

C

D

B

A

E

Medians and Centroids
• A 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 Median
• AP = (2/3)(AX)
• If AX = 27, then AP = 18

X

P

A