5.5 ­ 5.6 ­ Indirect Proof and Inequalities in One and Two Triangle.notebook

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November 29, 2011

School

Triangle Inequalities...this concept is a part of the ACT math test!

Lesson 5.5/5.6 Indirect Proof and Inequalities in One (and Two) Triangles

November 29/30, 2011

Check for Understanding3108.1.13 Use proofs to further develop and deepen the understanding of the study of geometry (e.g. two­column, paragraph, flow, indirect,

coordinate) 3108.4.11 Use the triangle inequality theorems (e.g., Exterior Angle Inequality Theorem, Hinge Theorem, SSS Inequality Theorem, Triangle

Inequality Theorem) to solve problems

Nov 13­7:56 PM

Type: Indirect Proof

Idea: Assume the contradiction or conclusion is false!

Once you assume it is false, you will show that the assumption leads to a contradiction.

This type of proof is alsocalled a proof by contradiction.

The types of proofs we have used previously have been written using direct reasoning. We began with a true hypothesis and built

a logical argument to show the conclusion was true.

New Type of Proof!

Nov 18­2:39 PM

Start with something we know:

Prove that ∠ADB is not a straight angle.

Given: AD is perpendicular to BC.

Note: most indirect proofs are written in paragraph form, but to visualize we will use a

two column now.

1. ∠ADB is a straight angle.

2. m∠ADB=180o

3. AD is perpendicular to BC

4. ∠ADB is a right angle.

5. m∠ADB=90o

6. m∠ADB cannot equal 90o and 180o at the same time.

7. ∠ADB is not a straight angle.

1. Assume Opposite

2. Definition of straight angle.

3. Given

4. Definition of perpendicular lines.

5. Definition of a right angle.

6. Contradiction!

7. Proof by contradiction.

Statements ReasonsB C

D

A

Nov 18­11:54 AM

Consider this statement: "Two acute angles do not form a linear pair"Steps:1. Identify the conjecture to be proven.2. Assume the opposite of the conclusion is true.3. Use direct reasoning to show that the assumption leads to a contradiction.4. Conclude that the assumption is false and therefore the original conjecture must be true.

Example for the above statement:I. Given: <1 and <2 are acute angles

Prove: <1 and <2 do not form a linear pair.

2. Assume <1 and <2 form a linear pair.

3. m<1+m<2=180 (def of linear pair) Since m<1 < 90o and m<2 < 90o m<1 + m<2 < 180. This is a contradiction.

4. The assumption that <1 and <2 form a linear pair is false. Therefore <1 and <2 do not form a linear pair.

Draw a diagram to visual!

Nov 18­12:01 PM

Steps:1. Identify the conjecture to be proven.

2. Assume the opposite of the conclusion. Write this assumption.

3. Use direct reasoning to show a contradiction.

4. What can you conclude?

You try:

Use the following statement and answer 1­4.

"An obtuse triangle cannot have a right angle."

Nov 18­12:06 PM

Let's work through our next objective: Apply inequalities in one triangle.

Using your paper and your geometer:1. Sketch triangle #2 using your geometer and label the vertices as A, B, and C.

2. Using the cm ruler on your geometer, measure each side of the triangle and write those measurements on your triangle.

3. Using your protractor on your geometer, measure each interior angle of your triangle and write those degrees on your triangle for each angle.

4. Write the sides of your triangle in order from largest to smallest.

5. Write the angles of your triangle in order from largest to smallest.

What do you notice about the smallest side and smallest angle?

Is this true for the longest side and largest angles as well?

5.5 ­ 5.6 ­ Indirect Proof and Inequalities in One and Two Triangle.notebook

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November 29, 2011

Nov 18­2:36 PM

You have now discovered Angle­Side Relationships in Triangles !

Nov 18­2:42 PM

Let's try again:Write the angles in order from smallest to largest.

FH

G

19.617.2

20.4

Write the sides in order from shortest to longest.

PR

Q72o

60o

Nov 18­2:45 PM

Can any three segment lengths form a triangle?

A triangle is formed by three segments, but not every set of three segments form a triangle!

Nov 18­2:47 PM

Let's Apply the Triangle Inequality Theorem:Tell whether a triangle can have sides with lengths: 7, 10, 19

Tell whether a triangle can have sides with lengths: 2.3, 3.1, 4.6

Tell whether a triangle can have sides with lengths: (n+6), (n2­1), 3n... when n=4

1.

2.

3.

No; 7+10 ≯ 19 even though 10+19 >7 and 7+19 > 10

Yes: 2.3 + 3.1 > 4.6 ; 3.1 + 4.6 > 2.3 ; 2.3 + 4.6 > 3.1

Yes; (n+6) + (n2 ­ 1) > 3n (n+6) + 3n > (n2 ­ 1) 3n + (n2 ­ 1) > (n+6) 10 + 15 > 12 10 + 12 > 15 12 + 15 > 10

Nov 18­2:55 PM

Key ACT Concept!!!!!

The lengths of two sides of a triangle are 6 centimeters and 11 centimeters. Find the range of possible lengths for the third side.

Set up each triangle inequality: Let x = the unknown side length

Add together:

Unknown and Smallest Given: Unknown and Largest Given:

x + 6 > 11 x + 11 > 6 x > 5 x > ­5

Two Givens

6 + 11 > x 17 > x

Combine the inequalities ! So... ­5 < x < 17

Nov 18­2:58 PM

The lengths of two sides of a triangle are 8 inches and 13 inches. Find the range of possible lengths for the third side.

You try!

5.5 ­ 5.6 ­ Indirect Proof and Inequalities in One and Two Triangle.notebook

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November 29, 2011

Nov 28­11:40 AM

Moving onto two triangles.....

Nov 28­3:22 PM

Example 1A: Using the Hinge Theorem and Its Converse

Compare m∠BAC and m∠DAC.

Are there two pairs of congruent sides?

6 ≠ 7, therefore what is true?

By the Converse of the Hinge Theorem, m∠BAC > m∠DAC.

Nov 28­3:23 PM

Compare EF and FG.

By the Hinge Theorem, EF < GF.

Example 1B: Using the Hinge Theorem and Its Converse

Are there two pairs of congruent sides?

m∠FHG =

Therefore compare EF and FG.

Nov 28­3:28 PM

Check It Out!

Compare m∠EGH and m∠EGF.

By the Converse of the Hinge Theorem, m∠EGH < m∠EGF.

Compare BC and AB.

By the Hinge Theorem, BC > AB.

Nov 28­3:33 PM

What is the range of values for k?Compare the side lengths in ∆MLN and ∆PLN.

The angle must be greater than what degree?

Zero5k – 12 > 0k > 2.4

The angle must be less than what degree? 38ο5k – 12 < 38

k < 10

Combine the two inequalities.

The range of values for k is 2.4 < k < 10.

Tap the stars!

Nov 28­3:40 PM

When the swing ride is at full speed, the chairs are farthest from the base of the swing tower. What can you conclude about the angles of the swings at full

speed versus low speed? Explain.

Application:

The ∠ of the swing at full speed is greater than the ∠ at low

speed because the length of the triangle on the opposite side is the greatest at full swing.

5.5 ­ 5.6 ­ Indirect Proof and Inequalities in One and Two Triangle.notebook

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November 29, 2011

Nov 28­3:45 PM

Hinge Theorem and Proofs? SAYITISNOTSO!

Write a two­column proof.

Given: C is the midpoint of BD.m∠1 = m∠2 m∠3 > m∠4

Prove: AB > ED

Statements Reasons

Nov 28­3:50 PM

Statements Reasons

On Your Own!

Nov 28­3:52 PM

Assignment:

Page 336

#'s 18 ­ 20, 25, 28, 33, 34, 37, 38, 41, 42, 44, 45, 48, 50, 51, 70 ­ 72 (19 problems)

Page 343

#'s 10 ­ 13, 17, 18, 21, 24 ­ 27, 31, 32 (12 problems)