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(a) Statement is true. Its converse, a polygon is a quadrilateral, is not necessarily true; it might be a triangle. (b) Statement is true. Its converse, an angle with greater measure than a right angle is an obtuse angle, is not necessarily true; it might be a straight angle. (c) Statement is true. Its converse, a state of the United States is Florida, is not necessarily true; it might be any one of the other 49 states. (d) Statement is true. Its converse, if I am your teacher, then you are my pupil, is also true. (e) The statement, a definition, is true. Its converse, a triangle that has all congruent sides is an equilateral triangle, is also true.
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CHAPTER 2 Methods of Proof
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2.5 Proving a Theorem
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Theorems should be proved using the following step-by-step procedure. The form of the proof is shown in the example that follows the procedure. Note that accepted symbols and abbreviations may be used. 1. Divide the theorem into its hypothesis (what is given) and its conclusion (what is to be proved). Underline the hypothesis with a single line, and the conclusion with a double line. 2. On one side, make a marked diagram. Markings on the diagram should include such helpful symbols as square corners for right angles, cross marks for equal parts, and question marks for parts to be proved equal. 3. On the other side, next to the diagram, state what is given and what is to be proved. The Given and To Prove must refer to the parts of the diagram. 4. Present a plan. Although not essential, a plan is very advisable. It should state the major methods of proof to be used. 5. On the left, present statements in successively numbered steps. The last statement must be the one to be proved. All the statements must refer to parts of the diagram. 6. On the right, next to the statements, provide a reason for each statement. Acceptable reasons in the proof of a theorem are given facts, definitions, postulates, assumed theorems, and previously proven theorems.
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Step 1: Prove: All right angles are equal in measure. s Steps 2 Given: j A and j B are rt. j and 3: To Prove: mj A mj B Step 4: Plan: Since each angle equals 90 , the angles are equal in measure, using Post. 1: Things equal to the same thing are equal to each other. Steps 5 and 6:
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Statements s 1. j A and j B are rt. j. 2. mj A and mj B each 3. mj A mj B
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90 .
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Reasons 1. Given 2. m(rt. j ) 90 3. Things to same thing
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SOLVED PROBLEM
2.14 Proving a theorem Use the proof procedure to prove that supplements of angles of equal measure have equal measure.
Step 1: Prove: Supplements of angles of equal measure have equal measure. Steps 2 Given: ja sup. j1, jb sup. j2 and 3: mj1 mj2 To Prove: mja mjb Step 4: Plan: Using the subtraction postulate, the equal angle measures may be subtracted from the equal sums of measures of pairs of supplementary angles. The equal remainders are the measures of the supplements.
CHAPTER 2 Methods of Proof
Steps 5 and 6:
Statements
1. 2. 3. 4. 5. j a sup. j 1, j b sup. j 2 mj a mj 1 180 mj b mj 2 180 mj a mj 1 mj b mj 2 mj 1 mj 2 mj a mj b 1. 2. 3. 4. 5.
Reasons
Given s s Sup. j are j the sum of whose measures 180 . Things to the same thing each other. Given If s are subtracted from s, the differences are .
SUPPLEMENTARY PROBLEMS
2.1. Complete each statement. In (a) to (e), each letter, such as C, D, or R, represents a set or group. (a) If A is B and B is H, then _____ . (b) If C is D and P is C, then _____ . (c) If _____ and B is R, then B is S. (d) If E is F, F is G, and G is K, then _____ . (e) If G is H, H is R, and _____ , then A is R. (f) If triangles are polygons and polygons are geometric figures, then _____ . (g) If a rectangle is a parallelogram and a parallelogram is a quadrilateral, then _____ . 2.2. State the conclusions which follow when Postulate 1 is applied to the given data, which refer to Fig. 2-29. (2.2)
(2.3)
Fig. 2-29