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Postulates and Paragraph Proofs. Notes. Postulate: a statement that is accepted to be true and describes a fundamental relationship between the basic terms of geometry. P2.1 Through any 2 points, there is exactly one line.
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Notes • Postulate: a statement that is accepted to be true and describes a fundamental relationship between the basic terms of geometry. • P2.1 Through any 2 points, there is exactly one line. • P2.2 Through any 3 points not on the same line, there is exactly one plane. • P2.3 A line contains at least 2 points • P2.4 A plane contains at least 3 points NOT on the same line • P2.5 If two points lie in a plane, then the entire line containing those points lies in that plane. • P2.6 If two lines intersect, then their intersection is exactly one point • P2.7 If two planes intersect, then their intersection is a line.
Always, Sometimes, Never Practice! • If points A, B, & C lie in plane M, then they are collinear. • ____________ • There is exactly one plane that contains noncollinear points P,Q, and R. • ____________ • There are at least two lines through points M and N. • ____________
Notes • Theorem: a statement or conjecture that has been proven true using logic. • Theorems can be used just like postulates and definitions to prove other statements. • Proof: a logical argument in which each step towards the conclusion is supported by a definition, postulate, or theorem i.e. supporting statements. • Paragraph Proof: a type of proof that gives the steps and supporting statements in paragraph form • Informal Proof: paragraph proofs are a type of informal proof
Example: • Recall: • The definition of a midpoint is the point half way between the endpoints of a segment. • AB means the measure or length of the segment AB • while refers to the segment with endpoints A and B • Given: M is the mid point of • Prove: • Proof: From the definition of midpoint of a segment, PM = MQ. i.e. the measures of the segments are equal. This means that and have the same measure. By definition of congruence, if two segments have the same measure, then they are congruent. Thus, .
Notes • The proof we just did was the proof of the Midpoint Theorem. We can now use this theorem in other proofs. • i.e. If M is the midpoint of segment PQ, then segment PM is congruent to segment MQ • Steps to writing a good proof: • List the given information. • State what is to be proven. • Draw a diagram of the given information. • Develop an argument that properly uses deductive reasoning. • Each step follows from the previous • Each step is supported by accepted facts.
On Your Own: • In the figure at the right, P is the midpoint of and , and . • Write a paragraph proof to show that PQ = PT • Given: • P is the midpoint of and • and • Prove: PQ = PT • From the definition of midpoint of a segment, QP = PR. From the segment addition postulate, QR = QP + PR. Substituting QP for PR, QR = QP + QP. Simplifying gives QR = 2QP. Likewise, ST = 2PT. From the definition of congruent segments, QR = ST from the given information. Substituting 2QP for QR and 2PT for ST gives 2QP = 2PT. Dividing both sides by 2 gives QP = PT. Since QP is another way of naming the segment PQ, we can rewrite the last equation as PQ = PT and reach our desired conclusion.
