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 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 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

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 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)

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 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.

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

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

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

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

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

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

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 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)

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

Using the Pythagorean Theorem to find… The Hypotenuse One of the legs Hypotnuse2 = (leg1)2 + (leg2)2 c2 = 32 + 42 c2 = 9 + 16 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

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 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 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 1 Right Angle 3 Congruent Angles

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

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 B C

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

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