What is the proof for the converse of this statement: Given A exists on line l,...
What is the proof for the converse of this statement:
Given A exists on line l, and B does not exist on line l, then P exists on line AB(P does not equal A) --> P,B on the same side of line l.
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What is the proof for the converse of this statement:
Given A exists on line l,...
2) (i) State the converse of the Alternate Interior Angle Theorem in Neutral Geometry. (ii) Prove...
2) (i) State the converse of the Alternate Interior Angle Theorem in Neutral Geometry. (ii) Prove that if the converse of the Alternate Interior Angle Theorem is true, then all triangles have zero defect. [Hint: For an arbitrary triangle, ABC, draw a line through C parallel to side AB. Justify why you can do this.] 5) Consider the following statements: I: If two triangles are congruent, then they have equal defect. II: If two triangles are similar, then they have...
4. (a) Supply proof of the Menelaus Theorem concerning a transversal line L cutting the sides...
4. (a) Supply proof of the Menelaus Theorem concerning a transversal line L cutting the sides of ΔABC at points X,Y,Z respectively. (Hint) Drop perpendicular line segments from A, B, C to L and use similar triangles b)Centuries after Menelaus, Ceva discovered the Theorem that if P,Q, R are points on BC, CA and AB respectively so that AP, BQ, CR meet at a single point K, thern AR BP co RB PC QA Prove Ceva's theorem and its converse,...
7. (10) Find the flaw in the following attempted proof of the parallel postulate by Wolfgang Bolyai (Hungarian, 1775 - 1856) (see Fig. 3). Given any point P not on a line l, construct a line 1&#...
7. (10) Find the flaw in the following attempted proof of the parallel postulate by Wolfgang Bolyai (Hungarian, 1775 - 1856) (see Fig. 3). Given any point P not on a line l, construct a line 1' parallel to through P in the usual way: drop a perpendicular PQ to / and construct /" perpendicular to PQ. Let I" be any line through P distinct from l'. To see that /" intersects I, pick a point A on PQ between...
a. Define what it means for two logical statements to be equivalent b. If P and Q are two statements, show that the statement ( P) л (PvQ) is equivalent to the statement Q^ P c. Write the converse a...
a. Define what it means for two logical statements to be equivalent b. If P and Q are two statements, show that the statement ( P) л (PvQ) is equivalent to the statement Q^ P c. Write the converse and the contrapositive of the statement "If you earn an A in Math 52, then you understand modular arithmetic and you understand equivalence relations." Which of these d. Write the negation of the following statement in a way that changes the...
For the given direct statement, write the indicated related statement (converse, inverse, or contrapositive). q--> (p...
For the given direct statement, write the indicated related statement (converse, inverse, or contrapositive). q--> (p ∧ r) (contrapositive)
Validate each of the following proofs by evaluating each of the following. Foundation for the proof . a. Statement of what the author intends to show. b. Description, in your own words, of what t...
Validate each of the following proofs by evaluating each of the following. Foundation for the proof . a. Statement of what the author intends to show. b. Description, in your own words, of what the statement implies. c. Intuitive justification as to why this is likely to be true. Structure of the proof. . Identify what the author stated as a logical implication. What foundational assumptions will the author make? What will the author be required to demonstrate? Describe the...
Given: 21 and 23 are supplementary. Prove: а || b 3 alb 21 and 23 are...
Given: 21 and 23 are supplementary. Prove: а || b 3 alb 21 and 23 are supplementary. a. ? d. ? Supplements of the same are e._? b. ? Def. of linear pair 21 and 22 are supplementary. c. ? Which statement can be used in blank e? Converse of the same-side interior angles theorem O Converse of the corresponding angles postulate Converse of the same-side exterior angles theorem Given: 21 and Z3 are supplementary. Prove: a | b b...
For the first part it is just a reference. Validate each of the following proofs by evaluating each of the following. Foundation for the proof Statement of what the author intends to show. a. b. Des...
For the first part it is just a reference.
Validate each of the following proofs by evaluating each of the following. Foundation for the proof Statement of what the author intends to show. a. b. Description, in your own words, of what the statement implies c. Intuitive justification as to why this is likely to be true. a. Identify what the author stated as a logical implication. What foundational assumption:s b. .Structure of the proof. will the author make? What...
using these axioms prove proof number 5 1 - . Axiom 1: There exist at least...
using these axioms prove proof number 5
1 - . Axiom 1: There exist at least one point and at least one line Axiom 2: Given any two distinct points, there is exactly one line incident with both points Axiom 3: Not all points are on the same line. Axiom 4: Given a line and a point not on/ there exists exactly one linem containing Pouch that / is parallel tom Theorem 1: If two distinct lines are not parallet,...
Match the following: item 1. Indirect proof which assumes the opposite of a statement and shows...
Match the following: item 1. Indirect proof which assumes the opposite of a statement and shows this creates a logical inconsistency item 2, Indirect proof applied to a statement of the form P→Q which instead proves-Q→-P. item 3. Proves a "there exists" statement by finding a specific element for which the statement is true item 4. Disproves a "for all" statement by finding a specific element for which the statement is false. item 5. Proof where a statement is split...
4. (a) Supply proof of the Menelaus Theorem concerning a transversal line L cutting the sides of ΔABC at points X,Y,Z respectively. (Hint) Drop perpendicular line segments from A, B, C to L and use similar triangles b)Centuries after Menelaus, Ceva discovered the Theorem that if P,Q, R are points on BC, CA and AB respectively so that AP, BQ, CR meet at a single point K, thern AR BP co RB PC QA Prove Ceva's theorem and its converse,...
7. (10) Find the flaw in the following attempted proof of the parallel postulate by Wolfgang Bolyai (Hungarian, 1775 - 1856) (see Fig. 3). Given any point P not on a line l, construct a line 1' parallel to through P in the usual way: drop a perpendicular PQ to / and construct /" perpendicular to PQ. Let I" be any line through P distinct from l'. To see that /" intersects I, pick a point A on PQ between...
a. Define what it means for two logical statements to be equivalent b. If P and Q are two statements, show that the statement ( P) л (PvQ) is equivalent to the statement Q^ P c. Write the converse and the contrapositive of the statement "If you earn an A in Math 52, then you understand modular arithmetic and you understand equivalence relations." Which of these d. Write the negation of the following statement in a way that changes the...
Validate each of the following proofs by evaluating each of the following. Foundation for the proof . a. Statement of what the author intends to show. b. Description, in your own words, of what the statement implies. c. Intuitive justification as to why this is likely to be true. Structure of the proof. . Identify what the author stated as a logical implication. What foundational assumptions will the author make? What will the author be required to demonstrate? Describe the...
Given: 21 and 23 are supplementary. Prove: а || b 3 alb 21 and 23 are supplementary. a. ? d. ? Supplements of the same are e._? b. ? Def. of linear pair 21 and 22 are supplementary. c. ? Which statement can be used in blank e? Converse of the same-side interior angles theorem O Converse of the corresponding angles postulate Converse of the same-side exterior angles theorem Given: 21 and Z3 are supplementary. Prove: a | b b...
For the first part it is just a reference.
Validate each of the following proofs by evaluating each of the following. Foundation for the proof Statement of what the author intends to show. a. b. Description, in your own words, of what the statement implies c. Intuitive justification as to why this is likely to be true. a. Identify what the author stated as a logical implication. What foundational assumption:s b. .Structure of the proof. will the author make? What...
using these axioms prove proof number 5
1 - . Axiom 1: There exist at least one point and at least one line Axiom 2: Given any two distinct points, there is exactly one line incident with both points Axiom 3: Not all points are on the same line. Axiom 4: Given a line and a point not on/ there exists exactly one linem containing Pouch that / is parallel tom Theorem 1: If two distinct lines are not parallet,...
Match the following: item 1. Indirect proof which assumes the opposite of a statement and shows this creates a logical inconsistency item 2, Indirect proof applied to a statement of the form P→Q which instead proves-Q→-P. item 3. Proves a "there exists" statement by finding a specific element for which the statement is true item 4. Disproves a "for all" statement by finding a specific element for which the statement is false. item 5. Proof where a statement is split...
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