4-1 Reasoning and Proof in Geometry

Transcription

1 4-1 Reasoning and Proof in Geometry Inductive reasoning Deductive reasoning onditional statement We can use symbols to represent our statements: if p, then p p implies p only if, if p hypothesis conclusion Example 1: Identify the hypothesis and conclusion in the following statements. If an angle measures 40, then the angle is an acute angle. If a triangle has two congruent sides, then the triangle is an isosceles triangle. You can write many sentences as a conditional. A rectangle has four right angles can be rewritten as If a figure is a rectangle, then it has four right angles. Example 2: Rewrite the following statements as conditionals. A suare has four congruent sides. An acute angle measures less than 90E. A conditional can have a truth value of either TRUE or FALSE. If we can find one counterexample, then the conditional is false. If x 2 0, then x 0. What values for x will make this conditional FALSE? A conditional can also be represented using a Venn Diagram. If a figure is a rectangle, then it has four right angles. Figures with 4 right angles Draw a Venn Diagram to represent If you live in San Diego, then you live in alifornia. rectangle Making onclusions: 1) Take a look at the following statements. a) If it is hot today, then I will get thirsty. b) It is hot today. or if p, then p p p LAW OF DETAHMENT

2 Using the Law of Detachment: Make a conclusion if possible. If not possible, then write not possible and explain. a) If the road is icy, then driving conditions are hazardous. The road is icy. onclusion: b) If you make a goal in basketball, then you score 2 points. Reed scored 2 points. onclusion: c) If points are on the same line, then they are collinear. Points A, and D lie on line p. onclusion: 2) Take a look at the following statements: a) If it is hot today, then I will get thirsty. b) If I get thirsty, then I will drink lots of water. or if p, then if, then r Therefore if p, then r p r p r LAW OF SYLLOGISM Using the Law of Syllogism Make a conclusion if possible. If not possible, then write not possible and explain. a) If a triangle has three congruent sides, then it is euilateral. If a triangle is euilateral, then it is also euianglular. onclusion: b) If Judy stays inside, then she will not get wet. If it rains, then Judy will stay inside. onclusion: Using the Law of Detachment and the Law of Syllogism Together: Look at the following statements and see if you can construct a conclusion. 1) If the circumference of a circle is 8π, then it s diameter is 8 inches. 2) If the diameter of a circle is 8 inches, then its radius is 4 inches. 3) If the radius is 4, then its area is 16π suare inches. 4) The circumference of circle O is 8π. Let s use symbols to help us figure it out! onclusion: Other Forms of the onditional: onverse of p Inverse of p ontrapositive of p Example 5: Find the converse, inverse, and contrapositive of the following statement If a triangle has at least two congruent sides, then it is an isosceles triangle. onverse: Inverse: ontrapositive:

3 You should examine the truth value of each statement. For example, given the statement, If an animal is a terrier, then it is a dog. The converse is. Notice that the converse is not true since there are many dogs that are not terriers. HOWEVER. When a statement and its converse are both true, they can be written as one statement called a biconditional. If p and p are both true, then we write p, which says p if and only if Example 6: Write the following two statements as a biconditional. 1) If two angles have the same measure, then they are congruent. 2) If two angles are congruent, then they have the same measure. iconditional: Important! Every definition can be written as a biconditional. We can tell if a definition is a good one by examining the conditional and the converse forms of the statement. Example 7: Is this a good definition? A segment is a part of a line.

4 4-1 Reasoning and Proof in Geometry Part 2 The following is important but is NOT in your textbooks. There are two postulates and a definition that need to be included as we begin our work with proofs: Segment Addition Postulate If three points, A,, and, are collinear and is between A and, then A + = A. A Angle Addition Postulate If point is in the interior of pao, or if pao is a straight angle, then mpao + mpo = mpao then mpao + mpo = 180 A O Definition of Midpoint A point is called a midpoint of a segment A if is between A and and A =. A O A Theorem 4.1 Vertical Angle Theorem If x and y are the measures of a pair of vertical angles, then x = y. (Vertical angles are congruent.) We re going to use deductive reasoning to prove this theorem in two formats: paragraph and statement-reason (which is also called a two-column proof). m n Given: Intersecting lines m and n form two vertical Prove: mp1 = mp3 Paragraph form: Given lines m and n intersect to form vertical Since p1 and p2 form a straight angle and p3 and p2 also form a straight angle, we have mp1 + mp2 = 180 and mp3 + mp2 = 180. y substitution (or transitivity) we have mp1 + mp2 = mp3 + mp2. y subtracting mp2 from both sides, we have mp1 = mp3. Two-olumn form: m n Given: Intersecting lines m and n form two vertical Prove: mp1 = mp3 Statements Reasons 1. Intersecting lines m and n form two vertical 1. Given

5 And now a review of some algebraic properties: Properties of Euality Addition Property If a = b, then a + c = b + c. Subtraction Property If a = b, then a - c = b - c. Multiplication Property If a = b, then a c = b c. Division Property If a = b and c 0, then a = b c c. Reflexive Property a = a Symmetric Property If a = b, then b = a. Transitive Property If a = b and b = c, then a = c. Substitution Property If a = b, then b can replace a in any expression. Fundamental Algebraic Properties Distributive Property a(b + c) = ab + ac ommutative Property of Addition a + b = b + a ommutative Property of Multiplication ab = ba Associative Property of Addition (a + b) + c = a + (b + c) Associative Property of Multiplicaton (ab)c = a(bc) Example 1: Prove the following conditional statement If two angles are supplementary and congruent, then they are right angles. Given: p1 is supplementary to p2. p1 p2 Prove: p1 and p2 are right angles 1 2 Paragraph form: Given p1 is supplementary to p2 and using the definition of supplementary angles, we know mp1 + mp2 = 180. Since p1 p2, we have mp1 = mp2 by the definition of congruent angles. Substituting, we have mp1 + mp1 = 180 or 2mp1 = 180. Dividing both sides by 2, we get mp1 = 90 and with substitution, mp2 = 90. Using the definition of right angles, we know that both p1 and p2 are right angles. Two-olumn form: Statements Reasons 1. p1 is supplementary to p2 1. Given p1 p2 3. Given