1. Geometry
Midterm Review

2. 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

3. Application of Segment Addition Postulate: Use the Diagram to find KL
Application of Segment Addition Postulate: Use the Diagram to find KL 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

4. Bisectors or Midpoints
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

5. 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

6. 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

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

8. Laws of Logic Law of Detachment Law of Syllogism (aka The Chain Rule)
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)

9. 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.

10. 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

11. Types of Logical Statements
Conditional Statement:
Converse:
Inverse:
Contrapositive:
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.

12. Angles Formed by Transversals
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 t 1 m 5 n
Describe the position of 1 & 5

13. Angles Formed by Transversals
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 t m 3 6 n
Describe the position of 3 & 6

14. Angles Formed by Transversals
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 t 2 m n
Describe the position of 7 & 2 7

15. Angles Formed by Transversals
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 t m 4 6 n
Describe the position of 4 & 6

16. Properties of Slope Slope: Negative Slope Positive 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

17. Identify the Parallel Lines
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
(2, 2)
(4, 2)
(-3, -1)
(3, -3)
(1, -4) h p
(-4, -6) s

18. 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

19. Drawing a Perpendicular Line
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 w
(5, 6)
(2, 5)
(4, 4)
(1, -2)

20. Parts of a Right Triangle
Label the Hypotenuse and the legs of the below Triangle
Hypotenuse: BC
Legs: AB & AC
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 B A C

21. Using the Pythagorean Theorem to find…
The Hypotenuse
One of the legs
Hypotnuse2 = (leg1)2 + (leg2)2
c2 =
c2 =
c2 = 25
c = 5
Hypotnuse2 = (leg1)2 + (leg2)2
102 = 62 + b2
100 = 36 + b2
b2 = 64
b = 8 c 10 3 6 4 b

22. Classifying Triangles using the Pythagorean Theorem
Acute
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
? 102
? 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
? 122
? 144
117 < 144
Therefore the Triangle is Obtuse 7 8 6 9 10 12

23. Classifying Triangles by their Sides
Scalene Triangle
Isosceles Triangle
Equilateral Triangle
No Congruent Sides
3 Congruent Sides
At Least 2 Congruent Sides

24. Classifying Triangles by Angles
Acute Triangle
Right Triangle
Obtuse Triangle
Equiangular Triangle
3 Acute Angles
1 Obtuse Angle
1 Right Angle
3 Congruent Angles

25. Interior Angles of a Triangle
Triangle Sum Theorem
Corollary to the Triangle Sum Theorem
The sum of the measures of the angles of a triangle is 180°
mA + mB + mC = 180
The Acute angles of a right triangle are complementary
mB + mC = 90 B A C A B C

26. 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

27. 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 B C

28. Midsegment is a Midsegment Properties of a Midsegment BD = ½ (AE)
If AE = 12, then BD = 6
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 C B D A E

29. 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