1 Geometry Unit 1 Basics of Geometry
2 Using inductive reasoning - Looking for patterns and making conjectures is part of a process called inductive reasoning Conjecture- an unproven statement that is based on observations. (in other words it is a guess or prediction made from observations)
3 Section Patterns and Inductive Reasoning 1. Look for a Pattern Diagram or table 2. Make a conjecture A guess or prediction based on observations 3. Verify the conjecture Use logical reasoning to show the prediction is true.
4 Describing a visual pattern Example 1- Draw the next figure
5 Example 2 Draw the next figure
6 Describing a number pattern Look for a relationship between consecutive numbers In example 3, we multiply each number by 4 1 x 4 = 4, 4 x 4 = 16, 16 x 4 = 64, 64 x 4 = 256 In example 4, we add consecutive multiples of = -2, = 4, = 13, = 25
7 Example 5: 27, 9, 3, 1, What is the relationship between consecutive numbers? To get the next number you divide by 3 So the next number would be 1 divided by 3 or 1/3
8 Proving conjectures are 1. To prove that a conjecture is true, you need to prove it is true for all cases. (a proof). 2. To prove that a conjecture is false, find a counterexample. Counterexample an example that shows it does not work.
9 Example 6 The difference between two positive numbers is always positive Find a counterexample A counterexample, in this case, would be an example that showed the difference between two positive numbers was not positive (meaning it was negative) 5 4 = -1, 5 and 4 are positive, yet the difference is negative. Name another counterexample.
10 Example 7 For all real numbers x, the expression x 2 is greater than or equal to x Find a counterexample We need to show an example where x 2 is less than x (meaning not greater than and not equal to x) If x = ½, then (½) 2 = x 2 = ¼ and ¼ < ½. So when x = ½, x 2 is not greater than or equal to x.
11 Start-up for Predict the next number: 11, 23, 47, 95, 2. Predict the next number: 3, -11, -53, -179, 3. Find a counterexample to the conjecture: No two prime numbers are consecutive.
12 Section 1.2 Points, Lines, and Planes Definition- using known words to describe new words. Undefined term- a word that is not formally defined, although there is a general agreement on what the word means. Ex. Point, line, and plane
13 Point- has no dimension, represented by a small dot. Line- one dimension, represented by a line with arrows. *** Plane- two dimensions, represented by a flat shape. ***It extends forever even though you can see edges
14 collinear Points- Points that lie on the same line. Coplanar Points- Points that lie on the same plane.
15 Example 1 Name 3 points that are collinear Name 4 points that are coplanar Name 3 points that are noncollinear.
16 Important Terms and Symbols 1. Line Line AB or A B AB 2. Line Segment Segment AB or A AB B
17 Ray (Initial point A) Ray AB or Opposite Rays A B AB A C B If C is between A and B, then CB are opposite rays CA and
18 Example 2 Draw Three non-collinear points and label them A, B, and C. Draw line AB Draw segment BC Draw ray CA
19 A B C
20 Start-up for 1.2 True or False 1. A, B, and E are collinear. E D C 2. A, B, and D are coplanar A 3. D and F lie on a ray. j F 4. B lies on the intersection of two lines. 5. D lies on the intersection of a segment and a line. B
21 Section 1.3 Segments and Their Measures Postulate (axioms)- Rules that are not proven, but we accept. Theorem- Rules that are proven
22 Ruler Postulate: Each coordinate on a number line can be mapped to a real number The distance between two points on a number line is x1 x2 (absolute value) Notation- AB (distance or length of AB)
23 Segment Addition Postulate If B is between A and C, then AB + BC = AC A B C If AB + BC = AC, then B is between A and C.
24 Congruent Segments Congruent Segments - segments that have the same length
25 Example 1 Suppose M is between L and N. Use the Segment Addition Postulate to solve. Then find the lengths of LM, MN, and LN. L 1. LM = 3x + 8, MN = 2x-5, and LN = 23. M N
26 Example 2 LM = ½ z + 2, MN = 3z + 1.5, LN = 5z + 2
27 Distance Formula If A(x 1, y 1 ) and B(x 2, y 2 ) are points in a coordinate plane, then the distance between points A and B is ( ) 2 x ( ) 2 x1 + y2 y1 2
28 Example 3- Let A= (-1, 1) B= (-4, 3) C=(3, 2) and D= (2, -1) 1. Find AB 2. Find AC 3. Find BD
29 Start-up 1.3 Points A, D, F, and X are on a segment in this order. AD = 15, AF = 22, and AX = 30. Find each length. 1. DF 2. FX Use the distance formula to decide whether PQ = QR. 3. P(2,4), Q(4,10), R(0,6) 4. P(-1,-2), Q(2,0), R(4,3)
30 SECTION 1.4 Angles and Their Measures
31 Section 1.4 Angles and Their Measures Angle- Consists of two different rays that have the same initial point. A Sides Rays ( BA and ) Vertex Initial point ( B ) Notation ABC Always put the vertex in the middle B C
32 Congruent Angles Angles that have the same measure. B D E A Interior- the inside of an angle (Points E and F) Exterior- outside of an angle (point D) C F
33 Angle Addition Postulate: (Remember the Segment Addition Postulate) RST If P is the interior of, then m RSP + m PST = m RST R P S T
34 Classifying Angles Acute Obtuse- Right- Straight- Zero- = 0 0 < x < < x < 180 x = 90 x = 180 x
35 Adjacent Angles- 2 angles are adjacent if they share a common vertex and side, but no common interior points.
36 Start-up #1.4 In a coordinate plane, plot the points A(-6,2), B(0,0), and C(1,3) and sketch angle ABC. Then answer the questions below. 1. Name the vertex 2. Write two names for the angle. 3. Find the measure of angle ABC. 4. Write the coordinates of a point that lies in the interior of the angle.
37 Section 1.5 Segment and Angle Bisectors Midpoint A point that divides a segment into two congruent segments Segment Bisector A segment, ray, line, or plane that intersects a segment at its midpoint.
38 Midpoint Formula If A(x 1, y 1 ) and B(x 2, y 2 ) are points in a coordinate plane, then the midpoint of segment AB has coordinates x1 + x2 y1 + y ' Note that this is just the average of the x s and the average of the y s
39 Example 2- Find the midpoint a. (-2, 3) and (5, -2) (x 1, y 1 ) (x 2, y 2 )
40 b. (3, 5) and (-4, 0) (x 1, y 1 ) (x 2, y 2 ) So the midpoint is + + = 2 0 5, = 2 5, 2 1
41 Example 3 Finding coordinates of endpoints *The midpoint of segment RP is M(2, 4). One endpoint is R(-1, 7). Find the coordinate of P.
42 Angle Bisector A ray that divides an angle into 2 adjacent angles that are congruent. A D C B
43 Example 5 Example 5 Find the measure of the angles. *We know that the rays are angle bisectors. A D x x - 20 C B
44 b. B A x + 4 D 3x - 30 C
45 Section 1.6 Angle Pair Relationships
46 Vertical Angles- 2 angles are vertical if their sides form 2 pairs of opposite rays and opposite angles are equal. Linear Pair 2 adjacent angles whose sum is 180 0
47 In the diagram the vertical angles are 1 and 3 2 and 4 The linear pairs are 1 and 2 1 and 4 2 and
48 Example 1 Find the angle measures.? s to consider. Can I solve for both variables at the same time? What is the relationship between the angles with the same variable? y x+5 4y-15 x+15
49 The angles with the x variable are a linear pair, so they add to 180 Same with the ones with y Then 3x+5+x+15=180 4x+20=180 4x=160 x=40 Solve for y y=33 Could these be the right angle measures for this figure??? y x+5 4y-15 x+15
50 Example 2-Find the angle measures x+15=3x+5 2x=10 x=5 2y+100 = y +130 y = 30 2y+100 x+15 3x+5 y +130 Are these valid angle measures??
51 Complementary Angles- Two angles that add up to 90 0 Supplementary Angles- Two angles that add to What s the difference between supplementary angles and a linear pair???
53 Section 1.7- Perimeter, Area, and Circumference.
54 Square All sides are equal Four right angles Perimeter = 4s Area = s 2 s s s s
55 Rectangle Opposite sides are equal Four right angles Perimeter = l+l+w+w=2(l)+2(w) Area= l * w l w w l
56 Triangle Perimeter= a + b + c Area = ½ * b * h a b c
57 Circle Perimeter = Circumference Circumference = = Area = πr 2 πd d 2πr r
58 Section 2.1 Conditional Statements
59 Conditional Statement A logical statement containing 2 parts, a hypothesis and a conclusion. If Then Form If = The hypothesis Then = The conclusion
60 Example 1 Write the following as a conditional statement in if then form. a. A number divisible by 9 is also divisible by 3 If a number is divisible by 9, then it is also divisible by 3.
61 b. All sharks have a boneless skeleton. If it is a shark, then it has a boneless skeleton. c. Two points are collinear if they lie on the same line If two points lie on the same line, then they are collinear
62 Recall: Counterexample- an example showing a statement is false Converse- Switching the hypothesis and conclusion
63 Example 2- Find a counterexample and then write the converse. a. If x 2 = 16, then x = 4 Counterexample is x = -4 (-4) 2 = 16 Converse If x = 4, then x 2 = 16
64 b. If a number is odd, then it is divisible by 3. Counterexample is 7 7 is odd, but not divisible by three Converse If a number is divisible by 3, then it is odd
65 Negation Negation- Saying the opposite Inverse- Negating the hypothesis and conclusion Contrapositive- Negating the converse of a conditional statement
66 Example 3 Write the inverse, converse, and contrapositive. If an animal is a fish, then it can swim. Inverse- If an animal is not a fish, then it cannot swim Converse- If an animal can swim, then it is a fish Contrapositive- If an animal cannot swim, then it s not a fish
67 Equivalent Statements- Equivalent Statements- When two statements are both true or both false. Original and Contrapositive are equivalent. Converse and Inverse are equivalent.
68 Point-Line-Plane Postulates Through any 2 points there is exactly one line A line consists of at least 2 points A plane consists of at least 3 noncollinear points. Through any 3 noncollinear points, there is a plane
69 P-L-P Postulates Cont. If 2 points are in a plane, then the line containing them are in the same plane If 2 planes intersect, their intersection is a line If 2 lines intersect, their intersection is a point
71 Section 2.2 Definitions & Biconditional Statements Perpendicular lines- 2 lines that form a right angle *Every definition can be written as 2 different conditional statements, If Then The original statement and Its converse.
72 Biconditional Statement a conditional statement that contains if and only if (Can be denoted as iff).
73 Example 1 Rewrite the biconditional statement as a conditional statement and then write its converse. Then decide if it is a true biconditional statement, meaning both are true.
74 Two lines intersect if and only if their intersection is exactly one point. Conditional: If two lines intersect, then their intersection is at one point. Converse: If two lines contain exactly one point, then the two lines intersect. TRUE
75 x = 3 if and only if x 2 = 9 Conditional: If x = 3, then x 2 = 9 Converse: If x 2 = 9, then x = 3 FALSE, x could equal -3
76 Example 2 The following statements are true. Write the converse and decide whether it is true or false. If the converse is true, combine it with the original to form a biconditional.
77 If x 2 = 4, then x = 2 or 2 Converse: If x = 2 or 2, then x 2 = 4 TRUE Biconditional: x 2 = 4 iff x = 2 or -2
78 If a number ends in 0, then the number is divisible by 5. Converse: If a number is divisible by 5, then the number ends in 0. FALSE Counterexample: 25 is divisible by 5 but does not end in 0
80 Chapter 1 Three Strikes! 1. Give a counterexample to the conjecture: All mammals live on land.
81 Chapter 1 Three Strikes! 2. Show that Goldbach s conjecture (all even # s greater than 2 can be written as the sum of two primes) is true for 84.
82 Chapter 1 Three Strikes! 3. What is the distance from A(6, -1) to B(-2, 5)?
83 Chapter 1 Three Strikes! 4. Name all of the congruent segments that can be formed using the endpoints: A(-3, 8), B(6, 5), C(0, 2), and D(2, -4).
84 Chapter 1 Three Strikes! 5. Name a pair of adjacent angles in the figure below: C B D E A
85 Chapter 1 Three Strikes! 6. Find B given endpoint A(4, 1), and M(-2, 4) the midpoint of. AB
86 Chapter 1 Three Strikes! uuur 7. Given BD bisects ABC, find the value of x: A (5x + 5) B D (6x - 2) C
87 Chapter 1 Three Strikes! 8. Given A and B are supplementary, find the measure of each angle if: m A = 4x 6 m B = 2x + 12
88 Chapter 1 Three Strikes! 9. Find the perimeter and area of the rectangle: A B 12 8 D C