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Name: ____________________________________ Date: __________________________ Geometry Unit 3 Day 4 Introduction to Proofs What is a proof? A proof is a convincing argument that something is true. In mathematics, a proof starts with things that are agreed upon, called postulates or axioms, and then uses logic to reach a conclusion. Conclusions are often reached in geometry by observing data and looking for patterns. This type of reasoning is called inductive reasoning. The conclusions reached by inductive reasoning is called a conjecture. A proof in geometry consists of a sequence of statements, each supported by a reason, that starts with a given se of premises and leads to a valid conclusion. This type of reasoning is called deductive reasoning. Each statement in a proof follows from one or more of the previous statements. A reason for a statement can come from the set of given premises or from one of the four types of other premises: definitions; postulates; properties of algebra, equality, or congruence; or previously proven theorems. Once a conjecture is proved, it is called a theorem. As a theorem, it becomes a premise for geometric arguments you can use to prove other conjectures. The four common methods of geometric proofs are: 1) Paragraph proofs – a proof in which the statements and their justifications are written together in logical order in complete sentences. There is always a diagram and a statement of the given and prove sections before the paragraph. 2) Two-column proofs - a proof in which the statements are listed in logical order in one column, and the reason each statement is true is in another column. The last statement is always what is proven. 3) Flow chart proofs - a concept map that shows the statements and reasons needed for a proof in a structure that helps indicate the logical order. Statements, written in the logical order, are placed in the boxes. The reason for each statement is placed under that box. 4) Coordinate proofs - a proof involving midpoints, slopes, and distances on the coordinate plane Think of proofs like a game. The object of the proof game is to have all the statements in your chain linked so that one fact leads to another until you reach the prove statement. However, before you start playing the proof game, you should survey the playing field (your figure), look over the given and the prove parts, and develop a plan on how to win the game. Here are some important theorems we will prove before using them in formal proofs. Vertical Angle Theorem: If two angles are vertical angles, then they have equal measures. Linear Pair Theorem: If two angles form a linear pair, then they are supplementary. Complements of the Same Angele Theorem: If two angles are complements of the same angle, then they have equal measures. Angles 1 and 3 are both complements to angle 2, therefore, they have the same measure. Supplements of the Same Angle Theorem: If two angles are supplements of the same angle, then they have equal measures. Angles 1 and 3 are both supplements to angle 2, therefore, they have the same measure. Alternate Interior Angles Theorem: If two parallel lines are cut by a transversal, then the alternate interior angles have equal measures. Converse of Alternate Interior Angles Theorem: If two coplanar lines are cut by a transversal so that two alternate interior angles have the same measure, then the lines are parallel. Corresponding Angles Theorem: If two parallel lines are cut by a transversal, then the corresponding angles have equal measures. Converse of Corresponding Angles Theorem: If two coplanar lines are cut by a transversal so that two corresponding angles have the same measure, then the lines are parallel. What does the word converse mean in the above context? Vertical Angle Theorem If two angles are vertical angles, then they have equal measures. Given: Lines AB and CD intersect at E Prove: AEC BED & AED BEC For this proof, we will use transformations, specifically, a rotation. Using patty paper, trace the intersecting lines and points. Then rotate AEC to map onto BED. On the diagram, mark and label C’ and A’. Then do the same for AED. Paragraph Proof A rotation of ______about point ______ will map point A onto EB such that A will lie on EB since we are dealing with straight segments. It will also map point C onto _______such that C will lie on ________. The rotation will create A’EC’, which will be congruent to ________ since they are the same angles with the same sides (rays) and the same vertex. Since A’EC’ is a 180o rotation of _______ about point _______, A’EC’ AEC since rotations are rigid transformations which preserve ____________________________ and congruent angles are equal in measure. AEC BED by the __________________ property of congruence (or substitution). The same argument will apply to proving AED BEC. Two Column Proof Statements 1) Lines AB and CD intersect at E Reasons 1) 2) 1, 2 & 3, 4 & 1, 4 form linear pairs 2) A linear pair is 3) 1, 2 & 3, 4 & 1, 4 are supplementary 3) 4) m 1 + m 2 = 180 m 3 + m 4 = 180 m 1 + m 4 = 180 5) m 1 + m 2 = m 1 + m 4 m1 + m4 = m 3 + m 4 4) Supplementary angles are 6) m 1 = m 1; m 4 = m 4 6) 7) m 2 = m 4; m 1 = m 3 7) 8) 2 4; 1 3 or 8) 5) AEC BED & AED BEC Complements of the Same Angele Theorem If two angles are complements of the same angle, then they have equal measures. Given: 1 and 2 are complementary 2 and 3 are complementary Prove: 1 3 Flow Chart Proof 1 and 2 are complementary 2 and 3 are complementary m m 1+m 2+m 2 =90 3 = 90 m 1+m 2= m 2+m 3 m 2=m 2 m 1=m 3 1 3 Linear Pair Theorem If two angles form a linear pair, then they are supplementary Given: 1 and 2 form a linear pair Prove: 1 and 2 are supplementary Two Column Proof Statements 1) 1 and 2 form a linear pair 2) BA and BC are opposite rays Reasons 1) 2) 3) ABC is a straight angle 3) 4) m ABC = 180o 4) 5) m 1 + m 2 = m ABC 5) 6) m 1 + m 2 = 180o 6) 7) 1 and 2 are supplementary 7) Corresponding Angles Theorem If two parallel lines are cut by a transversal, then the corresponding angles have equal measures Given: AD ║ BC Prove: ABC EAD Paragraph Proof After a translation of ABC by vector map onto BA , B’ will map onto A, A’ will map onto AE , and C’ will AD . A’BC’ ABC since a translation is a rigid transformation that preserves ________________________________ and angles are congruent if they have equal measure. A’BC’ EAD since they are the same angles with the same sides (rays) and the same vertex. ABC EAD by the _____________________ property of congruence (or substitution). Teacher’s Copy Vertical Angle Theorem If two angles are vertical angles, then they have equal measures. Given: Lines AB and CD intersect at E Prove: AEC BED & AED BEC For this proof, we will use transformations, specifically, a rotation. Using patty paper, trace the intersecting lines and points. Then rotate AEC to map onto BED. On the diagram, mark and label C’ and A’. Then do the same for AED. Paragraph Proof A rotation of 180o about point E will map point A onto EB such that A will lie on EB since we are dealing with straight segments. It will also map point C onto ED such that C will lie on ED . The rotation will create A’EC’, which will be congruent to BED since they are the same angles with the same sides (rays) and the same vertex. Since A’EC’ is a 180o rotation of AEC about point E A’EC’ AEC since rotations are rigid transformations which preserve angle measure and congruent angles are equal in measure. AEC BED by the transitive property of congruence (or substitution). The same argument will apply to proving AED BEC. Two Column Proof Statements 1) Lines AB and CD intersect at E 1) Given Reasons 2) 1, 2 & 3, 4 & 1, 4 form linear pairs 2) A linear pair is a pair of adjacent angles that form a straight line. 3) 1, 2 & 3, 4 & 1, 4 are supplementary 3) Angles that form a linear pair are supplementary 4) m 1 + m 2 = 180 m 3 + m 4 = 180 m 1 + m 4 = 180 5) m 1 + m 2 = m 1 + m 4 m1 + m4 = m 3 + m 4 4) Supplementary angles are two angles the sum of whose measures is 180o 6) m 1 = m 1; m 4 = m 4 5) The transitivity of congruence. (Quantities equal to the same quantity are equal to each other) 6) Reflexive Property (Quantity is = to itself) 7) m 2 = m 4; m 1 = m 3 7) Subtraction Property of Equality 8) 2 4; 1 3 or AEC BED & AED BEC 8) Congruent angles are angles equal in measure Complements of the Same Angele Theorem If two angles are complements of the same angle, then they have equal measures. Given: 1 and 2 are complementary 2 and 3 are complementary Prove: 1 3 Flow Chart Proof 1 and 2 are complementary GIVEN 2 and 3 are complementary GIVEN m 1+m 2 =90 Complementary angles are two adjacent angles whose sum is 90o m 2+m 3 = 90 Complementary angles are two adjacent angles whose sum is 90o m 1+m 2= m 2+m 3 Transitive Property of Equality m 2=m Reflexive Property 2 m 1=m Subraction Property of Equality 3 1 3 Congruent angles have equal measures Linear Pair Theorem If two angles form a linear pair, then they are supplementary Given: 1 and 2 form a linear pair Prove: 1 and 2 are supplementary Two Column Proof Statements 1) 1 and 2 form a linear pair 2) BA and BC are opposite rays Reasons 1) Given 2) Linear pairs are formed by two adjacent angles whose noncommon sides are opposite rays 3) ABC is a straight angle 3) Opposite rays form a straight angle 4) m ABC = 180o 4) A straight angle measure 180o 5) m 1 + m 2 = m ABC 5) Angle Addition Postulate 6) m 1 + m 2 = 180o 6) Transitive Property of Equality (Substitution) 7) 1 and 2 are supplementary 7) Supplementary angles are two angles the sum of whose measures is 180o Corresponding Angles Theorem If two parallel lines are cut by a transversal, then the corresponding angles have equal measures Given: AD ║ BC Prove: ABC EAD Paragraph Proof After a translation of ABC by vector map onto BA , B’ will map onto A, A’ will map onto AE , and C’ will AD . A’BC’ ABC since a translation is a rigid transformation that preserves angle measure and angles are congruent if they have equal measure. A’BC’ EAD since they are the same angles with the same sides (rays) and the same vertex. ABC EAD by the transitive property of congruence (or substitution).