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Proving Statements in Geometry Eleanor Roosevelt High School Geometry Mr. Chin-Sung Lin ERHS Math Geometry Inductive Reasoning Mr. Chin-Sung Lin ERHS Math Geometry Visual Pattern Describe and sketch the fourth figure in the pattern: ? Fig. 1 Fig. 2 Fig. 3 Fig. 4 Mr. Chin-Sung Lin ERHS Math Geometry Visual Pattern Describe and sketch the fourth figure in the pattern: Fig. 1 Fig. 2 Fig. 3 Fig. 4 Mr. Chin-Sung Lin ERHS Math Geometry Visual Pattern Describe and sketch the fourth figure in the pattern: ? Fig. 1 Fig. 2 Fig. 3 Fig. 4 Mr. Chin-Sung Lin ERHS Math Geometry Visual Pattern Describe and sketch the fourth figure in the pattern: Fig. 1 Fig. 2 Fig. 3 Fig. 4 Mr. Chin-Sung Lin ERHS Math Geometry Visual Pattern Describe and sketch the fourth figure in the pattern: ? Fig. 1 Fig. 2 Fig. 3 Fig. 4 Mr. Chin-Sung Lin ERHS Math Geometry Visual Pattern Describe and sketch the fourth figure in the pattern: Fig. 1 Fig. 2 Fig. 3 Fig. 4 Mr. Chin-Sung Lin ERHS Math Geometry Visual Pattern Describe and sketch the fourth figure in the pattern: ? Fig. 1 Fig. 2 Fig. 3 Fig. 4 Mr. Chin-Sung Lin ERHS Math Geometry Visual Pattern Describe and sketch the fourth figure in the pattern: Fig. 1 Fig. 2 Fig. 3 Fig. 4 Mr. Chin-Sung Lin ERHS Math Geometry Number Pattern Describe the pattern in the numbers and write the next three numbers: 1 4 7 10 ? ? ? Mr. Chin-Sung Lin ERHS Math Geometry Number Pattern Describe the pattern in the numbers and write the next three numbers: 1 4 3 7 3 10 13 16 3 3 3 19 3 Mr. Chin-Sung Lin ERHS Math Geometry Number Pattern Describe the pattern in the numbers and write the next three numbers: 1 4 9 16 ? ? ? Mr. Chin-Sung Lin ERHS Math Geometry Number Pattern Describe the pattern in the numbers and write the next three numbers: 1 4 3 9 7 5 2 16 25 36 2 11 9 2 2 49 13 2 Mr. Chin-Sung Lin ERHS Math Geometry Conjecture An unproven statement that is based on observation Mr. Chin-Sung Lin ERHS Math Geometry Inductive Reasoning Inductive reasoning, or induction, is reasoning from a specific case or cases and deriving a general rule You use inductive reasoning when you find a pattern in specific cases and then write a conjecture for the general case Mr. Chin-Sung Lin ERHS Math Geometry Weakness of Inductive Reasoning Direct measurement results can be only approximate We arrive at a generalization before we have examined every possible example When we conduct an experiment we do not give explanations for why things are true Mr. Chin-Sung Lin ERHS Math Geometry Strength of Inductive Reasoning A powerful tool in discovering new mathematical facts (making conjectures) Inductive reasoning does not prove or explain conjectures Mr. Chin-Sung Lin ERHS Math Geometry Make a Conjecture Give five colinear points, make a conjecture about the number of ways to connect different pairs of the points Mr. Chin-Sung Lin ERHS Math Geometry Make a Conjecture Give five colinear points, make a conjecture about the number of ways to connect different pairs of the points No of Points 1 2 3 4 5 0 1 3 6 ? Picture No of Connections Mr. Chin-Sung Lin ERHS Math Geometry Make a Conjecture No of Points 1 2 3 4 5 0 1 3 6 ? Picture No of Connections 1 2 3 ? Mr. Chin-Sung Lin ERHS Math Geometry Make a Conjecture No of Points 1 2 3 4 5 0 1 3 6 10 Picture No of Connections 1 2 3 4 Conjecture: You can connect five colinear points 6 + 4 = 10 different ways Mr. Chin-Sung Lin ERHS Math Geometry Prove a Conjecture No of Points 1 2 3 4 5 0 1 3 6 10 Picture No of Connections 1 2 3 4 Conjecture: You can connect five colinear points 6 + 4 = 10 different ways Mr. Chin-Sung Lin ERHS Math Geometry Prove a Conjecture To show that a conjecture is true, you must show that it is true for all cases Mr. Chin-Sung Lin ERHS Math Geometry Disprove a Conjecture To show that a conjecture is false, you just need to find one counterexample A counterexample is a specific case for which the conjecture is false Mr. Chin-Sung Lin ERHS Math Geometry Exercise: Disprove a Conjecture Conjecture: the sum of two number is always greater than the larger number Mr. Chin-Sung Lin ERHS Math Geometry Exercise: Disprove a Conjecture Conjecture: the value of x2 always greater than the value of x Mr. Chin-Sung Lin ERHS Math Geometry Exercise: Disprove a Conjecture Conjecture: the product of two numbers is even, then the two numbers must both be even Mr. Chin-Sung Lin ERHS Math Geometry Analyzing Reasoning Mr. Chin-Sung Lin ERHS Math Geometry Analyzing Reasoning Use inductive reasoning to make conjectures Use deductive reasoning to show that conjectures are true or false Mr. Chin-Sung Lin ERHS Math Geometry Analyzing Reasoning Example What conclusion can you make about the product of an even integer and any other integer 2 * 5 = 10 (-4) * (-7) = 28 2 * 6 = 12 6 * 15 = 90 use inductive reasoning to make a conjecture Mr. Chin-Sung Lin ERHS Math Geometry Analyzing Reasoning Example What conclusion can you make about the product of an even integer and any other integer 2 * 5 = 10 (-4) * (-7) = 28 2 * 6 = 12 6 * 15 = 90 use inductive reasoning to make a conjecture Conjecture: Even integer * Any integer = Even integer Mr. Chin-Sung Lin ERHS Math Geometry Analyzing Reasoning Example Use deductive reasoning to show that a conjecture is true Conjecture: Even integer * Any integer = Even integer Let n and m be any integer 2n is an even integer since any integer multiplied by 2 is even (2n)m represents the product of an even interger and any integer (2n)m = 2(nm) is the product of 2 and an integer nm. So, 2nm is an even integer Mr. Chin-Sung Lin ERHS Math Geometry Deductive Reasoning Deductive reasoning, or deduction, is using facts, definitions, accepted properties, and the laws of logic to form a logical argument While inductive reasoning is using specific examples and patterns to form a conjecture Mr. Chin-Sung Lin ERHS Math Geometry Definitions as Biconditionals Mr. Chin-Sung Lin ERHS Math Geometry Definitions as Biconditionals • Right angles are angles with measure of 90 • Angles with measure of 90 are right angles • When a conditional and its converse are both true: Mr. Chin-Sung Lin ERHS Math Geometry Definitions as Biconditionals • Right angles are angles with measure of 90 If angles are right angles, then their measure is 90 p q (T) • Angles with measure of 90 are right angles • When a conditional and its converse are both true: Mr. Chin-Sung Lin ERHS Math Geometry Definitions as Biconditionals • Right angles are angles with measure of 90 If angles are right angles, then their measure is 90 p q (T) • Angles with measure of 90 are right angles If measure of angles is 90, then their are right angles q p (T) • When a conditional and its converse are both true: Mr. Chin-Sung Lin ERHS Math Geometry Definitions as Biconditionals • Right angles are angles with measure of 90 If angles are right angles, then their measure is 90 p q (T) • Angles with measure of 90 are right angles If measure of angles is 90, then their are right angles q p (T) • When a conditional and its converse are both true: Angles are right angles if and only if their measure is 90 q p (T) Mr. Chin-Sung Lin ERHS Math Geometry Deductive Reasoning Mr. Chin-Sung Lin ERHS Math Geometry Proofs • A proof is a valid argument that establishes the truth of a statement • Proofs are based on a series of statements that are assume to be true • Definitions are true statements and are used in geometric proofs • Deductive reasoning uses the laws of logic to link together true statements to arrive at a true conclusion Mr. Chin-Sung Lin ERHS Math Geometry Proofs of Euclidean Geometry • given: The information known to be true • prove: Statements and conclusion to be proved • two-column proof: • In the left column, we write statements that we known to be true • In the right column, we write the reasons why each statement is true * The laws of logic are used to deduce the conclusion but the laws are not listed among the reasons Mr. Chin-Sung Lin ERHS Math Geometry Two Column Proof Example Given: In ΔABC, AB BC Prove: ΔABC is a right triangle Proof: Statements 1.AB BC Reasons 1. Given. Mr. Chin-Sung Lin ERHS Math Geometry Two Column Proof Example Given: In ΔABC, AB BC Prove: ΔABC is a right triangle Proof: Statements Reasons 1.AB BC 1. Given. 2.ABC is a right angle. 2. If two lines are perpendicular, then they intersect to form right angles. Mr. Chin-Sung Lin ERHS Math Geometry Two Column Proof Example Given: In ΔABC, AB BC Prove: ΔABC is a right triangle Proof: Statements Reasons 1.AB BC 1. Given. 2.ABC is a right angle. 2. If two lines are perpendicular, then they intersect to form right angles. 3.ΔABC is a right triangle. 3. If a triangle has a right angle then it is a right triangle. Mr. Chin-Sung Lin ERHS Math Geometry Paragraph Proof Example Given: In ΔABC, AB BC Prove: ΔABC is a right triangle Proof: We are given that AB BC. If two lines are perpendicular, then they intersect to form right angles. Therefore, ABC is a right angle. A right triangle is a triangle that has a right angle. Since ABC is an angle of ΔABC, ΔABC is a right triangle. Mr. Chin-Sung Lin ERHS Math Geometry Two Column Proof Example Given: BD is the bisector of ABC. Prove: mABD = mDBC Proof: Statements Reasons Mr. Chin-Sung Lin ERHS Math Geometry Two Column Proof Example Given: BD is the bisector of ABC. Prove: mABD = mDBC Proof: Statements 1.BD is the bisector of ABC. Reasons 1. Given. Mr. Chin-Sung Lin ERHS Math Geometry Two Column Proof Example Given: BD is the bisector of ABC. Prove: mABD = mDBC Proof: Statements 1.BD is the bisector of ABC. 2.ABD ≅ DBC Reasons 1. Given. 2. The bisector of an angle is a ray whose endpoint is the vertex of the angle and that divides the angle into two congruent angles. Mr. Chin-Sung Lin ERHS Math Geometry Two Column Proof Example Given: BD is the bisector of ABC. Prove: mABD = mDBC Proof: Statements 1.BD is the bisector of ABC. 2.ABD ≅ DBC 3.mABD = mDBC Reasons 1. Given. 2. The bisector of an angle is a ray whose endpoint is the vertex of the angle and that divides the angle into two congruent angles. 3. Congruent angles are angles that have the same measure. Mr. Chin-Sung Lin ERHS Math Geometry Two Column Proof Exercise Given: M is the midpoint of AMB. Prove: AM = MB Proof: Statements Reasons Mr. Chin-Sung Lin ERHS Math Geometry Two Column Proof Exercise Given: M is the midpoint of AMB. Prove: AM = MB Proof: Statements 1.M is the midpoint of AMB. Reasons 1. Given. Mr. Chin-Sung Lin ERHS Math Geometry Two Column Proof Exercise Given: M is the midpoint of AMB. Prove: AM = MB Proof: Statements 1.M is the midpoint of AMB. 2.AM ≅ MB Reasons 1. Given. 2. The midpoint of a line segment is the point of that line segment that divides the segment into congruent segments. Mr. Chin-Sung Lin ERHS Math Geometry Two Column Proof Exercise Given: M is the midpoint of AMB. Prove: AM = MB Proof: Statements 1.M is the midpoint of AMB. 2.AM ≅ MB 3.AM = MB Reasons 1. Given. 2. The midpoint of a line segment is the point of that line segment that divides the segment into congruent segments. 3. Congruent segments are segments that have the same measure. Mr. Chin-Sung Lin ERHS Math Geometry Direct and Indirect Proofs Mr. Chin-Sung Lin ERHS Math Geometry Direct Proof A proof that starts with the given statements and uses the laws of logic to arrive at the statement to be proved is called a direct proof In most direct proofs we use definitions together with the Law of Detachment to arrive at the desired conclusion All of the proofs we have learned so far are direct proofs Mr. Chin-Sung Lin ERHS Math Geometry Indirect Proof A proof that starts with the negation of the statement to be proved and uses the laws of logic to show that it is false is called an indirect proof or a proof by contradiction An indirect proof works because the negation of the statement to be proved is false, then we can conclude that the statement is true Mr. Chin-Sung Lin ERHS Math Geometry Indirect Proof Let p be the given and q be the conclusion 1. Assume that the negation of the conclusion (~q) is true 2. Use this assumption (~q is true) to arrive at a statement that contradicts the given statement (p) or a true statement derived from the given statement 3. Since the assumption leads to a condiction, it (~q)must be false. The negation of the assumption (q), the desired conclusion, must be true Mr. Chin-Sung Lin ERHS Math Geometry Direct Proof Example Given: mCDE ≠ 90 Prove: CD is not perpendicular to DE Proof: Statements Reasons Mr. Chin-Sung Lin ERHS Math Geometry Direct Proof Example Given: mCDE ≠ 90 Prove: CD is not perpendicular to DE Proof: Statements 1.mCDE ≠ 90 Reasons 1. Given. Mr. Chin-Sung Lin ERHS Math Geometry Direct Proof Example Given: mCDE ≠ 90 Prove: CD is not perpendicular to DE Proof: Statements 1.mCDE ≠ 90 Reasons 1. Given. 2.CDE is not a right angle. 2. If the degree measure of an angle is not 90, then it is not a right angle. Mr. Chin-Sung Lin ERHS Math Geometry Direct Proof Example Given: mCDE ≠ 90 Prove: CD is not perpendicular to DE Proof: Statements 1.mCDE ≠ 90 Reasons 1. Given. 2.CDE is not a right angle. 2. If the degree measure of an angle is not 90, then it is not a right angle. 3.CD is not perpendicular to DE 3. If two intersecting lines do not form right angles, then they are not perpendicular. Mr. Chin-Sung Lin ERHS Math Geometry Indirect Proof Example Given: mCDE ≠ 90 Prove: CD is not perpendicular to DE Proof: Statements Reasons Mr. Chin-Sung Lin ERHS Math Geometry Indirect Proof Example Given: mCDE ≠ 90 Prove: CD is not perpendicular to DE Proof: Statements 1.CD is perpendicular to DE Reasons 1. Assumption. Mr. Chin-Sung Lin ERHS Math Geometry Indirect Proof Example Given: mCDE ≠ 90 Prove: CD is not perpendicular to DE Proof: Statements 1.CD is perpendicular to DE 2.CDE is a right angle. Reasons 1. Assumption. 2. If two intersecting lines are perpendicular, then they form right angles. Mr. Chin-Sung Lin ERHS Math Geometry Indirect Proof Example Given: mCDE ≠ 90 Prove: CD is not perpendicular to DE Proof: Statements 1.CD is perpendicular to DE 2.CDE is a right angle. 3.mCDE = 90. Reasons 1. Assumption. 2. If two intersecting lines are perpendicular, then they form right angles. 3. If an angle is a right angle, then its degree measure is 90 Mr. Chin-Sung Lin ERHS Math Geometry Indirect Proof Example Given: mCDE ≠ 90 Prove: CD is not perpendicular to DE Proof: Statements Reasons 1.CD is perpendicular to DE 2.CDE is a right angle. 3.mCDE = 90. 4.mCDE ≠ 90 1. Assumption. 2. If two intersecting lines are perpendicular, then they form right angles. 3. If an angle is a right angle, then its degree measure is 90 4. Given Mr. Chin-Sung Lin ERHS Math Geometry Indirect Proof Example Given: mCDE ≠ 90 Prove: CD is not perpendicular to DE Proof: Statements Reasons 1.CD is perpendicular to DE 2.CDE is a right angle. 1. Assumption. 2. If two intersecting lines are perpendicular, then they form right angles. 3.mCDE = 90. 4.mCDE ≠ 90 3. If an angle is a right angle, then its degree measure is 90 4. Given 5.CD is not perpendicular to 5. Contradiction in 3 and 4. Therefore, DE the assumption is false and its negation is true. Mr. Chin-Sung Lin ERHS Math Geometry Postulates, Theorems, and Proof Mr. Chin-Sung Lin ERHS Math Geometry Postulate (or Axiom) A postulate (or axiom) is a statement whose truth is accepted without proof Mr. Chin-Sung Lin ERHS Math Geometry Theorem A theorem is a statement that is proved by deductive reasoning Mr. Chin-Sung Lin ERHS Math Geometry Theorems and Geometry applications theorems undefined terms defined terms postulates Mr. Chin-Sung Lin ERHS Math Geometry Postulates for Proofs Basic Properties of Equality • Reflexive Property • Symmetric Property • Transitive Property Substitution Postulate Partition Postulate Mr. Chin-Sung Lin ERHS Math Geometry Postulates for Proofs Addition Postulate Subtraction Postulate Multiplication Postulate Division Postulate Power Postulate Roots Postulate Mr. Chin-Sung Lin ERHS Math Geometry Reflexive Property of Equality A quantity is equal to itself a=a Algebraic example: x=x Mr. Chin-Sung Lin ERHS Math Geometry Reflexive Property of Equality Geometric example: The length of a segment is equal to itself AB = AB A B Mr. Chin-Sung Lin ERHS Math Geometry Symmetric Property of Equality An equality may be expressed in either order If a = b, then b = a Algebraic example: x=5 then 5=x Mr. Chin-Sung Lin ERHS Math Geometry Symmetric Property of Equality Geometric example: If the length of AB is equal to the length of CD, then the length of CD is equal to the length of AB AB = CD then CD = AB A B C D Mr. Chin-Sung Lin ERHS Math Geometry Transitive Property of Equality Quantities equal to the same quantity are equal to each other If a = b and b = c, then a = c Algebraic example: x = y and y = 4 then x=4 Mr. Chin-Sung Lin ERHS Math Geometry Transitive Property of Equality Geometric example: If the lengths of segments are equal to the length of the same segment, they are equal to each other AB = EF then AB = CD and EF = CD A B E F C D Mr. Chin-Sung Lin ERHS Math Geometry Substitution Postulate A quantity may be substituted for its equal in any statement of equality Algebraic example: x + y = 10 and y = 4x then x + 4x = 10 Mr. Chin-Sung Lin ERHS Math Geometry Substitution Postulate Geometric example: If the length of a segment is equal to the length of another segment, it can be substituted by that one in any statement of equality AB = XY and AB + BC = 10 then XY + BC = 10 A B X C Y Mr. Chin-Sung Lin ERHS Math Geometry Partition Postulate A whole is equal to the sum of all its parts • • A segment is congruent to the sum of its parts An angle is congruent to the sum of its parts Algebraic example: 2x + 3x = 5x Mr. Chin-Sung Lin ERHS Math Geometry Partition Postulate Geometric example: The sum of all the parts of a segment is congruent to the whole segment AB + BC = AC AB + BC = AC A B C Mr. Chin-Sung Lin ERHS Math Geometry Addition Postulate If equal quantities are added to equal quantities, the sums are equal • • If congruent segments are added to congruent segments, the sums are congruent If congruent angles are added to congruent angles, the sums are congruent If a = b and c = d, then a + c = b + d Algebraic example: x - 5 = 10 then x = 15 Mr. Chin-Sung Lin ERHS Math Geometry Addition Postulate Geometric example: If the length of a segment is added to two equal-length segments, the sums are equal AB ≅ CD and BC ≅ BC then A B C D AB + BC ≅ CD + BC Mr. Chin-Sung Lin ERHS Math Geometry Subtraction Postulate If equal quantities are subtracted from equal quantities, the differences are equal • • If congruent segments are subtracted to congruent segments, the differences are congruent If congruent angles are subtracted to congruent angles, the differences are congruent If a = b and c = d, then a - c = b - d Algebraic example: x + 5 = 10 then x = 5 Mr. Chin-Sung Lin ERHS Math Geometry Subtraction Postulate Geometric example: If a segment is subtracted from two congruent segments, the differences are congruent AC ≅ BD and BC ≅ BC A then AC - BC ≅ BD - BC B C D Mr. Chin-Sung Lin ERHS Math Geometry Multiplication Postulate If equal quantities are multiplied by equal quantities, the products are equal • Doubles of equal quantities are equal If a = b, and c = d, then ac = bd Algebraic example: x = 10 then 2x = 20 Mr. Chin-Sung Lin ERHS Math Geometry Multiplication Postulate Geometric example: If the lengths of two segments are equal, their like multiples are equal AO = CP then 2AO = 2CP A O B C P D Mr. Chin-Sung Lin ERHS Math Geometry Division Postulate If equal quantities are divided by equal nonzero quantities, the quotients are equal • Halves of equal quantities are equal If a = b, and c = d, then a / c = b / d (c ≠ 0 and d ≠ 0) Algebraic example: 2x = 10 then x=5 Mr. Chin-Sung Lin ERHS Math Geometry Division Postulate Geometric example: If the lengths of two segments are congruent, their like divisions are congruent AB = CD then ½ AB = ½ CD A O B C P D Mr. Chin-Sung Lin ERHS Math Geometry Powers Postulate The squares of equal quantities are equal If a = b, and a2 = b2 Algebraic example: x = 10 then x2 = 100 Mr. Chin-Sung Lin ERHS Math Geometry Powers Postulate Geometric example: If the lengths of two hypotenuses are equal, their powers are equal AB = XY then AB2 = XY2 A C X B Z Y Mr. Chin-Sung Lin ERHS Math Geometry Root Postulate Positive square roots of positive equal quantities are equal If a = b, and a > 0, then √a = √b Algebraic example: x = 100 then √x = 10 Mr. Chin-Sung Lin ERHS Math Geometry Root Postulate Geometric example: If the squares of the lengths of two hypotenuses are equal, their square roots are equal AB2 = XY2 then AB = XY A C X B Z Y Mr. Chin-Sung Lin ERHS Math Geometry Identify Postulates for Proofs Mr. Chin-Sung Lin ERHS Math Geometry Postulates for Proofs Which postulate tells us that if the measures of two angles are equal to a third angle’s, then they are equal to each other? Mr. Chin-Sung Lin ERHS Math Geometry Postulates for Proofs Which postulate tells us that if the measures of two angles are equal to a third angle’s, then they are equal to each other? Transitive Property Mr. Chin-Sung Lin ERHS Math Geometry Postulates for Proofs Which postulate tells us that the measure of an angle is equal to itself? Mr. Chin-Sung Lin ERHS Math Geometry Postulates for Proofs Which postulate tells us that the measure of an angle is equal to itself? Reflexive Property Mr. Chin-Sung Lin ERHS Math Geometry Postulates for Proofs If BD is equal to AE, and AE is equal to EC, how do we know that BD is equal to EC? Mr. Chin-Sung Lin ERHS Math Geometry Postulates for Proofs If BD is equal to AE, and AE is equal to EC, how do we know that BD is equal to EC? Transitive Property Mr. Chin-Sung Lin ERHS Math Geometry Postulates for Proofs How do we know that BAF + FAC is equal to BAC? B F A C Mr. Chin-Sung Lin ERHS Math Geometry Postulates for Proofs How do we know that BAF + FAC is equal to BAC? B F A C Partition Postulate Mr. Chin-Sung Lin ERHS Math Geometry Postulates for Proofs If BA is equal to FA, and EC is equal to AD, how do we know BA / EC = FA / AD? Mr. Chin-Sung Lin ERHS Math Geometry Postulates for Proofs If BA is equal to FA, and EC is equal to AD, how do we know BA / EC = FA / AD? Division Postulate Mr. Chin-Sung Lin ERHS Math Geometry Postulates for Proofs If AF is equal to AC, how do we know that AF - BD = AC - BD? Mr. Chin-Sung Lin ERHS Math Geometry Postulates for Proofs If AF is equal to AC, how do we know that AF - BD = AC - BD? Subtraction Postulate Mr. Chin-Sung Lin ERHS Math Geometry Postulates for Proofs If segment AB is congruent to segment XY and segment BC is congruent to segment YZ, how do we know that AB + BC = XY + YZ Mr. Chin-Sung Lin ERHS Math Geometry Postulates for Proofs If segment AB is congruent to segment XY and segment BC is congruent to segment YZ, how do we know that AB + BC = XY + YZ Addition Postulate Mr. Chin-Sung Lin ERHS Math Geometry Apply Postulates for Proofs Mr. Chin-Sung Lin ERHS Math Geometry Apply Postulates for Proofs If AB BC and LM MN, prove mABC = mLMN Mr. Chin-Sung Lin ERHS Math Geometry Apply Postulates for Proofs If AB BC and LM MN, prove mABC = mLMN Given: AB BC and LM MN Prove: mABC = mLMN L A B C M N Mr. Chin-Sung Lin ERHS Math Geometry Apply Postulates for Proofs Given: AB BC and LM MN Prove: mABC = mLMN Proof: Statements Reasons Mr. Chin-Sung Lin ERHS Math Geometry Apply Postulates for Proofs Given: AB BC and LM MN Prove: mABC = mLMN Proof: Statements 1.AB BC and LM MN Reasons 1. Given. Mr. Chin-Sung Lin ERHS Math Geometry Apply Postulates for Proofs Given: AB BC and LM MN Prove: mABC = mLMN Proof: Statements Reasons 1.AB BC and LM MN 1. Given. 2.ABC and LMN are right angles. 2. Perpendicular lines are two lines intersecting to form right angles. Mr. Chin-Sung Lin ERHS Math Geometry Apply Postulates for Proofs Given: AB BC and LM MN Prove: mABC = mLMN Proof: Statements Reasons 1.AB BC and LM MN 1. Given. 2.ABC and LMN are right angles. 2. Perpendicular lines are two lines intersecting to form right angles. 3. 3. A right angle is an angle whose degree measure is 90 mABC = 90, mLMN = 90 Mr. Chin-Sung Lin ERHS Math Geometry Apply Postulates for Proofs Given: AB BC and LM MN Prove: mABC = mLMN Proof: Statements Reasons 1.AB BC and LM MN 1. Given. 2.ABC and LMN are right angles. 2. Perpendicular lines are two lines intersecting to form right angles. 3. 3. A right angle is an angle whose degree measure is 90 mABC = 90, mLMN = 90 4.90 = mLMN 4. Symmetric property of equality Mr. Chin-Sung Lin ERHS Math Geometry Apply Postulates for Proofs Given: AB BC and LM MN Prove: mABC = mLMN Proof: Statements Reasons 1.AB BC and LM MN 1. Given. 2.ABC and LMN are right angles. 2. Perpendicular lines are two lines intersecting to form right angles. 3. mABC = 90, mLMN = 90 3. A right angle is an angle whose degree measure is 90 4. 90 = mLMN 4. Symmetric property of equality 5. mABC = mLMN 5. Transitive property of equality Mr. Chin-Sung Lin ERHS Math Geometry Apply Postulates for Proofs If AB = 2 CD, and CD = XY, prove AB = 2 XY Mr. Chin-Sung Lin ERHS Math Geometry Apply Postulates for Proofs If AB = 2 CD, and CD = XY, prove AB = 2 XY Given: AB = 2 CD, and CD = XY Prove: AB = 2 XY A B C D X Y Mr. Chin-Sung Lin ERHS Math Geometry Apply Postulates for Proofs Given: AB = 2 CD, and CD = XY Prove: AB = 2 XY Proof: Statements Reasons Mr. Chin-Sung Lin ERHS Math Geometry Apply Postulates for Proofs Given: AB = 2 CD, and CD = XY Prove: AB = 2 XY Proof: Statements 1.AB = 2 CD, and CD = XY Reasons 1. Given. Mr. Chin-Sung Lin ERHS Math Geometry Apply Postulates for Proofs Given: AB = 2 CD, and CD = XY Prove: AB = 2 XY Proof: Statements Reasons 1.AB = 2 CD, and CD = XY 1. Given. 2.AB = 2 XY 2. Substitution postulate Mr. Chin-Sung Lin ERHS Math Geometry Apply Postulates for Proofs If ABCD are collinear, AB = CD, prove AC = BD Mr. Chin-Sung Lin ERHS Math Geometry Apply Postulates for Proofs If ABCD are collinear, AB = CD, prove AC = BD Given: ABCD and AB = CD Prove: AC = BD A B C D Mr. Chin-Sung Lin ERHS Math Geometry Apply Postulates for Proofs Given: ABCD and AB = CD Prove: AC = BD Proof: Statements Reasons Mr. Chin-Sung Lin ERHS Math Geometry Apply Postulates for Proofs Given: ABCD and AB = CD Prove: AC = BD Proof: Statements 1.AB = CD Reasons 1. Given. Mr. Chin-Sung Lin ERHS Math Geometry Apply Postulates for Proofs Given: ABCD and AB = CD Prove: AC = BD Proof: Statements Reasons 1.AB = CD 1. Given. 2.BC = BC 2. Reflexive property of equality Mr. Chin-Sung Lin ERHS Math Geometry Apply Postulates for Proofs Given: ABCD and AB = CD Prove: AC = BD Proof: Statements Reasons 1.AB = CD 1. Given. 2.BC = BC 2. Reflexive property of equality 3.AB + BC = CD + BC 3. Addition postulate Mr. Chin-Sung Lin ERHS Math Geometry Apply Postulates for Proofs Given: ABCD and AB = CD Prove: AC = BD Proof: Statements Reasons 1.AB = CD 1. Given. 2.BC = BC 2. Reflexive property of equality 3.AB + BC = CD + BC 4.AB + BC = AC 3. Addition postulate 4. Partition postulate CD + BC = BD Mr. Chin-Sung Lin ERHS Math Geometry Apply Postulates for Proofs Given: ABCD and AB = CD Prove: AC = BD Proof: Statements Reasons 1.AB = CD 1. Given. 2.BC = BC 2. Reflexive property of equality 3.AB + BC = CD + BC 4.AB + BC = AC 3. Addition postulate 4. Partition postulate CD + BC = BD 5. AC = BD 5. Substitution postulate Mr. Chin-Sung Lin ERHS Math Geometry Q&A Mr. Chin-Sung Lin ERHS Math Geometry The End Mr. Chin-Sung Lin