PROJECT Inequalities in Geometry Chapter 6 - beginning on page 202 Student Notes

Project Grading Notes………...(10 points per page = 70 points) Quizzes (2)… (40 points each = 80 points) Homework (5) (10 points each = 50 points) Total Project (worth 200 points)

Please take note! 1)The test will be taken after the projects have all been returned and reviewed. The quizzes are done by YOU! 2)Staple all the homework assignments, in order, to the end of this project. 3)Points will be deducted for incorrect spelling. No abbreviations! 4)20 points will be deducted for every day this is late.

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Lesson 6-1: Inequalities (page 117) Properties of Inequality If a > b and c ≥ d, then a + c > b + d. If a > b and c > 0, then ac > bc & a / c > b / c. If a > b and c < 0, then ac < bc & a / c < b / c. If a > b and b > c, then a > c. If a = b + c and c > 0, then a > b.

The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle. Theorem 6-1 The Exterior Angle Inequality Theorem D E F 1

Assignment Written Exercises on pages 206 & to 11 odd numbers (check your answers in your textbook)

Lesson 6-2 Inverses and Contrapositives (page 208) Review conditional statements and converses from your previous notes. INVERSE (of a conditional): the inverse of the statement, “If p, then q,” is the statement: “If not p, then not q.” CONTRAPOSITIVE (of a conditional): the contrapositive of the statement, “If p, then q,” is the statement: “If not q, then not p.” NOTE: The symbol for “not p” is ~ p.

VENN DIAGRAM: a diagram that may be used to represent a conditional. LOGICALLY EQUIVALENT STATEMENTS: statements that are either both true or both false. AB U

Summary of Related If-Then Statements Given Statement: If p, then q. p ⇒ q Contrapositive: If not q, then not p.~ q ⇒ ~ p Converse: If q, then p. q ⇒ p Inverse: If not p, then not q.~ p ⇒ ~ q

Note: 1.A statement and its contrapositive are logically equivalent. 2.A statement is not logically equivalent to its converse or its inverse.

GET THIS NOTE on VENN DIAGRAMS! The conclusion is the BIG circle and the hypothesis is the small circle. conclusion hypothesis U

If p, then q. Circle all statements the Venn Diagram represent. q p Conditional Converse Inverse Contrapositive U p ⇒ q

U If p, then q. Circle all statements the Venn Diagram represent. q p Conditional Converse Inverse Contrapositive ~ q ⇒ ~ p

If p, then q. Circle all statements the Venn Diagram represent. p q Conditional Converse Inverse Contrapositive U q ⇒ p

U If p, then q. Circle all statements the Venn Diagram represent. p q Conditional Converse Inverse Contrapositive ~ p ⇒ ~ q

Conditional If today is Tuesday, then it is a weekday. Today is a weekday Today is Tuesday U

Conditional If today is Tuesday, then it is a weekday. …TRUE Converse If it is a weekday, then today is Tuesday. … FALSE Inverse If today is not Tuesday, then it is not a weekday. … FALSE Contrapositive If it is not a weekday, then today is not Tuesday. … TRUE

Conditional If today is Tuesday, then tomorrow is Wednesday. … TRUE Converse If tomorrow is Wednesday, then today is Tuesday. … ???? Inverse If today is not Tuesday, then tomorrow is not Wednesday. … ???? Contrapositive If tomorrow is not Wednesday, then today is not Tuesday. … ????

Conditional If an angle is acute, then its measure is more than 90º. … FALSE Converse If an angle measure is more than 90º, then it is acute. … ???? Inverse If an angle is not acute, then its measure is not more than 90º. … ???? Contrapositive If an angle measure is not more than 90º, then it is not acute. … ????

Assignment Written Exercises on pages 210 to to 21 odd numbers (check your answers in your textbook)

Lesson 6-3: Indirect Proof (page 214) Most of the proofs you have seen or written have been direct proofs. You reasoned directly from the given to the conclusion using definitions, postulates, and theorems. Sometimes it is difficult or even impossible to find a direct proof. In this case it may possible to reason indirectly. Indirect reasoning is used in everyday life.

INDIRECT PROOF … a proof in which you assume temporarily that the conclusion is not true, and then deduce a contradiction. An indirect proof is usually written in paragraph form. After making the temporary assumption, you reason logically until you reach a contradiction of a known fact.

… also Indirect proofs are often used to prove a conclusion that is a negation of a known fact. ie. not equal or not parallel. Basically, you must prove the contrapositive of the conditional statement.

Example: Write the first sentence of an indirect proof of each conditional shown. 1) If AB = BC, then ∆ABC is not scalene. Assume temporarily that ∆ABC ___________.

Example: Write the first sentence of an indirect proof of each conditional shown. Assume temporarily that ______________.

3) If AC ≠ BD, then ABCD is not a rectangle. Assume temporarily that ______________.

How to Write an Indirect Proof. 1) Assume temporarily that the conclusion is not true. 2) Reason logically until you reach a contradiction of a known fact. 3) Point out that the temporary assumption must be false and that the conclusion must then be true.

example: Given: AC = RT AB = RS m ∠ A ≠ m ∠ R Prove:BC ≠ ST C A B T S R

Given: AC = RT AB = RS m ∠ A ≠ m ∠ R Prove: BC ≠ ST Proof: C A B T S R Assume temporarily that _________, Then ∆ABC ≅ ∆RST by ______ Postulate, and ∠ A ≅ ∠ R by __ __ __ __ __. But this contradicts the given information that m ∠ A ≠ m ∠ R. Therefore the temporary assumption that BC = ST must be _______. It follows that __________.

Assignment Written Exercises on page to 9 odd numbers (check your answers in your textbook) Challenge on page 217 (this is NOT a bonus question) Take the quiz on Lessons 6-1 to 6-3: Inequalities and Indirect Proof (worth 40 points)

Lesson 6-4 Inequalities for One Triangle (page 219)

If one side of a triangle is longer than a 2 nd side, then the angle opposite the 1 st side is larger than the angle opposite the 2 nd side. Theorem 6-2 R TS

If one side of a triangle is longer than a 2 nd side, then the angle opposite the 1 st side is larger than the angle opposite the 2 nd side. Theorem 6-2 R TS 1 st side opposite angle

If one side of a triangle is longer than a 2 nd side, then the angle opposite the 1 st side is larger than the angle opposite the 2 nd side. Theorem 6-2 R TS 2 nd side opposite angle

If one side of a triangle is longer than a 2 nd side, then the angle opposite the 1 st side is larger than the angle opposite the 2 nd side. Theorem 6-2 R TS

Conversely to theorem 6-2 is theorem 6-3.

If one angle of a triangle is larger than a 2 nd angle, then the side opposite the 1 st angle is longer than the side opposite the 2 nd angle. Theorem 6-3 R TS

If one angle of a triangle is larger than a 2 nd angle, then the side opposite the 1 st angle is longer than the side opposite the 2 nd angle. Theorem 6-3 R TS opposite side 1 st angle

If one angle of a triangle is larger than a 2 nd angle, then the side opposite the 1 st angle is longer than the side opposite the 2 nd angle. Theorem 6-3 R TS opposite side 2 nd angle

If one angle of a triangle is larger than a 2 nd angle, then the side opposite the 1 st angle is longer than the side opposite the 2 nd angle. Theorem 6-3 R TS

The perpendicular segment from a point to a line is the shortest segment from the point to the line. Corollary 1

The perpendicular segment from a point to a plane is the shortest segment from the point to the plane. Corollary 2

If sum of the lengths of any two sides of a triangle is greater than the length of the third side. Theorem 6-4 C A B NOTE: This is used to determine whether 3 lengths will determine a triangle.

Examples: State whether or not the given lengths are sides of a triangle. (1) 2, 3, 4 YES NO ? 4 5 > 4

Examples: State whether or not the given lengths are sides of a triangle. (2) 5, 6, 7 YES NO ? 7 11 > 7

Examples: State whether or not the given lengths are sides of a triangle. (3) 8, 9, 17 YES NO YOU CAN DO THIS!

Assignment Written Exercises on pages 222 & to 19 odd numbers (check your answers in your textbook)

Lesson 6-5 Inequalities for Two Triangles (page 228)

If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the 1 st triangle is larger than the included angle of the 2 nd, then the 3 rd side of the 1 st triangle is longer than the 3 rd side of the 2 nd triangle. Theorem 6-5 SAS Inequality Theorem A B C D E F

Theorem 6-5 SAS Inequality Theorem A B C D E F 3 rd side If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the 1 st triangle is larger than the included angle of the 2 nd, then the 3 rd side of the 1 st triangle is longer than the 3 rd side of the 2 nd triangle.

Theorem 6-5 SAS Inequality Theorem A B C D E F 3 rd side If two sides of one triangle are congruent to two sides of another triangle, but the included angle of the 1 st triangle is larger than the included angle of the 2 nd, then the 3 rd side of the 1 st triangle is longer than the 3 rd side of the 2 nd triangle.

And in addition to theorem 6-5 is theorem 6-6.

If two sides of one triangle are congruent to two sides of another triangle, but the 3 rd side of the 1 st triangle is longer than the 3 rd side of the 2 nd triangle, then the included angle of the 1 st triangle is larger than the included angle of the 2 nd. Theorem 6-6 SSS Inequality Theorem A B C D E F

Theorem 6-6 SSS Inequality Theorem A B C D E F If two sides of one triangle are congruent to two sides of another triangle, but the 3 rd side of the 1 st triangle is longer than the 3 rd side of the 2 nd triangle, then the included angle of the 1 st triangle is larger than the included angle of the 2 nd. 3 rd side

Theorem 6-6 SSS Inequality Theorem A B C D E F If two sides of one triangle are congruent to two sides of another triangle, but the 3 rd side of the 1 st triangle is longer than the 3 rd side of the 2 nd triangle, then the included angle of the 1 st triangle is larger than the included angle of the 2 nd. 3 rd side

Assignment Written Exercises on pages 231 & to 11 odd numbers and 15 (check your answers in your textbook) Take the quiz on Lessons 6-4 to 6-5: Inequalities in Triangles (worth 40 points)

You are now ready to take the test on Chapter 6 Inequalities in Geometry (worth 50 points)