Lab: Pythagorean Theorem Converse

Transcription

1 Lab: Pythagorean Theorem Converse Name Pd ***Your group should have a bag that contains 9 squares and a protractor. Directions: 1. Put together the squares listed below to form a triangle. a. Set the hypotenuse aside. b. Connect the other two sides to make a right angle. c. Insert the hypotenuse; Does it fit? If not, do you have to make the angle bigger or smaller?. Explain how the length of c effects what happens to the angle. Use the side lengths of each square to create a triangle (if a triangle can be created). The triangle should be similar to the triangle created above. Determine if the triangle formed is acute, right, or obtuse by measuring the angles. Triangle side lengths in increasing order a, b, c Area of the largest square c Area of one of the smaller squares a Area of the remaining square b a + b Compare c to a + b (<, >, =) Is the triangle formed acute, right or obtuse? 3, 4, 5 5, 1, 13 5, 6, 10 6, 10, 1 9, 11, 13 4, 5, 6 Looking at the completed table, determine the following answers to the questions below. 1. For what type triangle is c = a + b?. If c a + b, for what kind of triangle is c < a + b? 3. If c a + b, for what kind of triangle is c > a +b? This relationship between c and a + b is sometimes called the UN Pythagorean Theorem. In your own words write what you discovered about the relationship between c, a + b, and the type triangle that is formed.

2 10 th Grade Pythagorean Theorem Questions: 1. The sides of squares can be used to form triangles. The areas of the squares that form right triangles have a special relationship. Using the dimensions of the squares below, determine which set of squares will form a right triangle? A. 1 B C. 13 D The drawing below shows how 3 squares can be joined at their vertices to form a right triangle. Which is closest to the area in square inches of the largest square? in F. 638 in G. 130 in H. 36 in J. 78 in 14.5 in

3 3. Look at the triangle shown below. Which of the following could be the triangle s dimensions? A. 7, 13.4, 14. B. 1.5,,.5 C. 7, 4, 6 D. 8, 10, The drawing below shows 3 square parking lots that enclose a grassy area shaped like a right triangle. Lot C Lot A Lot B If Lot A s perimeter is 300 yards and Lot B s perimeter is 400 yards, what is perimeter of Lot C? F. 500 yd G. 700 yd H yd J. 000 yd

4 Name: Period: Date: Conditional Statements and Venn Diagrams A conditional sentence can be translated into if then form: If, then. The first blank holds the condition or hypothesis, and the second blank holds the conclusion. If hypothesis, t hen conclusion. In the following examples, the hypothesis is underlined once, and the c onclusion is underlined twice. Examples: Original Sentences If then form All birds lay eggs. If it is a bird, then it lays eggs. A square is a quadrilater al. If it is a square, then it is a quadrilateral. An insect is not an anim al. If it is an insect, then it is not an animal. Triangl es are polygons. If it is a triangle, then it is a polygon. Key Point: The hypothesis does not include the word if, and the conclusion does not include the word then. Traditio nally, the small letters, p, q, r, s, and t represent sentences. We can form two conditionals from the two sentences, p and q: (1) If p, then q. ( p q) () If q, then p. (q p) Let each circle in a Venn diagram represent a sentence. p means being inside the circle representing p, and q means beings inside the circle representing q. There are four possible Venn diagrams to consider for the circle representing p and the circle representing q. q p q p p q p q Figure 1 Figure Figure 3 Figure 4 Consider the case (1): If p, then q. (p q) If it is inside the circle representing p, then it is inside the circle representing q. W hich circle represents q p (if q, then p)?

5 Name: Period: Date: Exercises 1. Draw a Venn diagram to represent the following statement: All pelicans eat fish.. Sever al if then statements are listed below. Which of them seem to be true if the diagram you have drawn (in the last problem) represents a true statement? a. If a bird is a pelican, then it eats fish. b. If a creature eats fish, then it is a pelican. c. If a bird is not a pelican, then it doesn t eat fish. d. If a creature doesn t eat fish, then it is not a pelican. 3. Draw a Venn diagram to represent the following statement: Professional basketball players are not midgets. 4. Whi ch of the statements below are true if your diagram represents a true statement? a. If a fellow is a professional basketball player, then he is not a midget. b. If a fellow is not a professional basketball player, then he is not a midget. c. If a fellow is not a midget, then he is a professional basketball player. d. If a fellow is a midget, then he is not a professional basketball player. 5. Draw a Venn diagram for the general conditional s tatement: p q. What other if then statement does it represent? 6. Given the following Venn diagram, write the if then statements it represents. w t 7. Given this Venn diagram, write the if then statements that it represents. r s

6 Kinds of Reasoning Name Period Date Kinds of Reasoning Some are better than others! Reasoning patterns are part of everyday life. Sometimes you just know something. At other times you may have observed patterns and based a conclusion on that. Then, there are the times when you are absolutely certain that an event will occur. Intuitive Reasoning Reasoning by using beliefs and hunches. Intuitive thinking involves sensing that something is true and just feeling sure that you are correct. It is jumping to a conclusion without any real evidence. (Example: Mrs. Jones doesn t want her son to go to the movies because she just has a feeling he will get himself into trouble.) Inductive Reasoning Reasoning by finding a general principle based upon the evidence of specific cases. It is an educated guess based on data or observations. Cases include making decisions on the basis of polls, drawing a conclusion from a computer lab investigation, and making decisions based on observations in science labs. Since it is not possible to examine every situation, there is always the possibility that a contradiction will be found. Example: After asking the ages of 5 freshmen, Judy reasoned that all freshmen are at least 13 years old. Deductive Reasoning Reasoning based on some statements that have been accepted to be true (rules to reason by) reasoning without any guessing. The conclusion is absolutely certain there is no room for doubt. New facts are deduced from accepted facts. Example: Today is Tuesday, so tomorrow must be Wednesday. Mathematicians use both inductive and deductive reasoning. Inductive reasoning leads to conjectures or educated guesses. Then, deductive reasoning is applied to determine if the conjecture is true. It takes only one false example to show that a conjecture is false. The false example is called a counterexample. The next example clarifies the connection to if-then statements: Tom makes a promise to his friend by saying, If I find $5, then I will take my friend to the movies. What are the possibilities? 1. Tom finds $5 and takes his friend to the movies.. Tom finds $5 and doesn t take his friend to the movies. 3. Tom doesn t find $5 and takes his friend to the movies. 4. Tom doesn t find $5 and doesn t take his friend to the movies. p = Tom finds $5 Case 1 Tom kept his promise. Case Tom broke his promise. Case 3 Promise doesn t apply. Case 4 Promise doesn t apply. q = Tom takes his friend to the movies p q p q T T T T F F F T T F F T

7 Name Date Class LESSON -1 Challenge Discovery Through Patterns The pattern shown is known as Pascal s Triangle. 1. If the pattern is extended, find the terms in row 7. 1, 6, 15, 0, 15, 6, 1. Make a conjecture for the pattern. Each row has 1 as the first and last number. Each of the other numbers is found by adding the two numbers that appear just above it. 3. Make a conjecture about the sum of the terms in each row. The sum of each row of terms after the first is twice the sum of the terms in the previous row. Refer to the pattern of figures for Exercises 4 and 5. Figure 1 Figure Figure 3 4. If the pattern continues, how many black triangles will there be in Figure 4? in Figure 5? 7; Write an algebraic expression for the number of black triangles in figure n. 3 n 1 Find a counterexample for each statement. 6. For every integer x, x x 1 is divisible by. Sample answer: x 7. For every integer n, n n is prime. Sample answer: n 8. Make a table of values for the expression 4 a 1, where a is a positive integer. Make a conjecture about the type of number that is generated by the rule. a 4 a Sample answer: All values of 4 a 1 are divisible by 3. Copyright by Holt, Rinehart and Winston. 8 Holt Geometry All rights reserved.

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9 PLAR WS #4 (Pre-AP) Preparing for Angle Relationships and Parallel Lines Name: Date: Period: Find the measures of each of the indicated angles. Also find x and y if necessary. (Fraction and decimal answers are ok.) Show your work or write your reasoning. (Linear pair are supplementary, vertical angles are, etc) x x-3 7 3y+3 x 18 x y y y x-5 x y x+3 x y y x+0 x y

10 PLAR WS #7 (Pre-AP) Two-Column Proofs with Parallel Lines Name: Date: Period: 1) p131 exercise 10 Statements: Reasons: Given: a b, c d Prove: 1 3 1) a b ) 3) c d 4) 5) 1 3 1) ) Alternate Interior Angles Theorem 3) 4) Corresponding Angles Postulate 5) ) p13 exercise 8 Statements: Reasons: Given: a b, 1 4 Prove: 3 1) ) 3) 3 and 4 are supplementary 4) 5) 3 1) Given ) Same-side Interior Angles Theorem 3) 4) Given 5) 3) p13 exercise 9 Statements: Reasons: Given: a b Prove: 1 and are supplementary 1) ) 3) m 3 + m = 180 4) 5) 1) ) Corresponding Angles Postulate 3) 4) Substitution Property of Equality 5) Definition of supplementary s 4) p139 exercise 38 Statements: Reasons: Given: l n, 1 8 Prove: j k 1), ) 8 6 3) 6 4 4) 5) j k 1) Given ) 3) 4) Transitive Property of Congruence 5) 5) p143 exercise Statements: Reasons: Given: In a plane, a b, b c, and c d. Prove: a d. 1) a b, b c ) 3) 4) a d. 1) Given ) 3) Given 4) Theorems in textbook lesson 3-3: Thm 3-9: If two lines are parallel to the same line, then they are to each other. Thm 3-10: In a plane, if two lines are perpendicular to the same line, then they are to each other. Thm 3-11: In a plane, if a line is perpendicular to one of two parallel lines, then it is to the other.

11 Multi-step problem-solving with Parallel Lines PLAR WS #6 Name: Date: Period: First, look for an angle relationship that will help you solve for the unknown. Write a sentence explaining the relationship between the angles. (Example: If parallel lines, then corresp s are congruent. If parallel lines, then s-s int s are suppl. Vertical s are congruent. Linear pair adds to 180. Etc.) Next, set up an algebraic equation for this angle relationship. Finally, solve for the unknown variables, and answer the question asked! 1. Find the measure of each angle. w = v = x = y =. On the map below, First Avenue and Second Avenue are parallel. A city planner proposes to locate a small garden and park on the triangular island formed by the intersections of four streets shown below. What are the measures of the three angles of the garden? 3. In the diagram of the gate, the horizontal bars are parallel and the vertical bars are parallel. Find x and y. (13b 4) (7a + 1) d + 4. The figure below shows Aaron s recent hiking course, which started at point L, went to point M and then point P, and then returned to point L. What is the measure of LMP formed by Aaron s hiking course? Hint: Extend the parallel lines!

12 Parallel Lines Angle Puzzle PLAR WS #5 Name: Date: Period: Use the Patty Paper Conjectures to calculate the measure of the lettered angles in the diagram below. Fill in the diagram with what you know: Solve vertical angles first, because they are the easiest! And linear pairs next. Then use the Parallel Lines Conjectures to solve for angles. Highlight the two parallel lines; the Parallel Lines Conjectures only apply to angle pairs formed by the parallel lines. If necessary, use the Triangle-Angle Sum Theorem: The sum of the measures of the angles of a triangle is. And also the sum of the measures of the angles of a quadrilateral is. As you find the measure of each angle, write it on the diagram, because you need to see them on the diagram to figure out the rest. Don t write your answers at the bottom until you re completely done. 110 v w 115 f g u a h d b 135 c j k x 68 r 4 m n p q s 14 t 19 a b c d f g h j k m n p q r s t u v w x Pg 153 # 10: Find the measures of the angles of each triangle. Classify each triangle by its angles.

13 PLAR WS #9 (Pre-AP) Name: 3- & 3-3 Skills Practice and Word Problem Practice (from Glencoe) Date: Period: 3. Line a is parallel to line b, and line a is parallel to line c. Find the values of x and y.

14 4. Line a is parallel to line b. Find the value of x. Word Problem Practice