Class Notes

Ch. 2 Introduction toLogical Reasoning,

Algebraic and Geometric Proofs,and Angle Conjectures

Ch. 4 Triangle Conjectures

Warm UpWrite a step-by-step solution for the linear equation, by writing

these equations in order.3x - 12 + 5 = 17

3x = 243x - 7 = 17

3(x - 4) + 5 = 17x = 8

Deductive Reasoning – Coming up with a conclusion by logical steps, using evidence or facts to support the argument.

Algebraic Proof – Applying deductive reasoning to solving equations, using Algebraic Properties to justify each step.

Solve this equation for a.

2(a +1) = -6

Algebraic Properties of EqualityAddition Property of Equality

If a = b, then a + c = b + c.Subtraction Property of Equality

If a = b, then a - c = b - c.Multiplication Property of Equality

If a = b, then ac = bc.Division Property of Equality

If a = b and c ≠ 0, then _a_ = _b_ . c c

Algebraic Properties of EqualityReflexive Property of Equality a = aSymmetric Property of Equality

If a = b, then b = a.Transitive Property of Equality

If a = b and b = c, then a = c.Substitution Property of EqualityIf a = b, then b can be substituted for a in any expression.

Properties of ArithmeticThe associative propertyIn addition and multiplication, terms may be arbitrarily associated with each other through the use of parentheses: a + ( b + c ) = (a + b ) + c a(bc) = (ab)cThe commutative propertyIn addition and multiplication, terms may be arbitrarily interchanged, or commutated:

ab = ba a + b = b + aThe distributive property

a(b + c) = ab + ac

Solve this equation for y.Provide justifications for each step.Refer to the Properties of Equality and Properties of Arithmetic.

y + 4 = 3 7

Algebraic Proof

Justification

1)

3)

4)

5)

6)

7)

2)

of Arithmetic

Complete pp. 1-2 of Handout

Complete pp. 3-4 of Handout

Do NowTwo ‘Challenge’ proofs page 5

of handout with partner

(Math looks hard, but samejustifications we already know)

Pre-Quiz Practice

• Solve for a and justify each step.5[1(a + 10)] = 5(-3) 5

QUIZ – Algebraic Proofs

Solve the equation in your notebook.Write a justification for each step on the right.

Left Table Partner Right Table Partner z – 5 = -2 6r – 3 = -2(r + 1) 6

DoNow (10 min) work with partner

1. Draw three angles: BAD, DAC, and CAB.mBAD = 120° mDAC = 158° mCAB = 92°2. Label them as acute, right, or obtuse. 3. Bisect each angle with a compass & measure the newly formed ‘daughter’ angles.4. Complete the conjecture as a full sentence.“If an obtuse angle is bisected, then the two newly formed congruent angles are _________________.”

Geometric Proofs

• Conjecture – An educated guess in geometry. Like a hypothesis in science. Discovered inductively by observing a pattern.

• Postulate – Definitions or facts we assume are true without proof. The building blocks for proving a conjecture.

• Theorem – A proven conjecture supported by step-by-step deductive logic. Each statement is supported with a fact or postulate.

Geometric ProofConjecture: If an obtuse angle is bisected, then the two newly formed congruent angles are acute.

1) Every angle measure is less than 180°.2) If m is the measure of an obtuse angle, then m < 180°.3) When you bisect an angle, the two newly formed angles each measure half of the original angle, or m.4) If m < 180°, then m < (180), so < 90°. The two angles are each less than 90°, so they are acute.

HW out for check

Make a Conjecture & Prove It

Linear Pair ConjectureIf two angles are linear (share vertex & middle side, outside rays form a line), then they are _________ .

A Pair of Linear Angles are:

Linear Pair ConjectureIf two angles are linear, then they are supplementary (add to 180o). (Write in notes.)

A Pair of Linear Angles are Supplementary & Adjacent:

Vertical Angle ConjectureIf two angles are vertical (opposite across line intersection, share only a vertex), then they are _________ .

Vertical Angle ConjectureIf two angles are vertical, then they are congruent . (Write this in notes.)

Proof of Vertical Angles Conjecture

Get out HW for check.Do Now (10 min):

• Using a compass and straight edge, draw two parallel lines. Make them big and well-separated.

• Draw three lines crossing these parallel lines at different angles. These are called transversals.

• Measure and label all the angles formed in the picture (at least 24 of ‘em!)

• Write conjectures based on the patterns you see.

Special Angle Conjectures (in notes)

• Corresponding Angle Conjecture CA’s are congruent.

• Alternate Interior Angle Conjecture

AIA’s are congruent.

• Alternate Exterior Angle Conjecture

AEA’s are congruent.

Special Angle Conjectures (in notes)

• These three conjectures are only true when a transversal crosses two parallel lines.

• Corresponding Angle Conjecture• Alternate Interior Angle Conjecture• Alternate Exterior Angle Conjecture

“Exterior” = outside parallels“Interior” = between parallels“Alternate” = both sides of transversal“Concurrent” = same side of transversal

5 Angle Conjectures

• Linear Pair Conjecture• Vertical Angle Conjecture• Corresponding Angles Conjecture• Alternate Interior Angle

Conjecture• Alternate Exterior Angle

Conjecture

Warm Up / Do Now (10 min)

1. Draw three triangles with geometry tools:–Acute–Right–Obtuse

2. Measure and label all nine angles.3. Write a conjecture based on the pattern

you see in all three triangles.

Exterior Angles

Exterior Angle Sum Conjecture

1. Find the exterior angles of the three triangles you made for the Do Now.

2. Measure and label all nine exterior angles.

3. Write a conjecture based on the pattern of exterior angles in all three triangles.

5 Triangle Conjectures

• Triangle Sum Conjecture• Third Angle Conjecture• Isosceles Angle Conjecture• Converse of Isosceles Angle

Conjecture• Exterior Angle Sum Conjecture

Triangle Sum Conjecture

• If a shape is a triangle, then its interior angles sum to 180o.

Exterior Angle Sum Conjecture

• If a shape is a triangle, then its exterior angles sum to 360o.

Third Angle Conjecture

• If two different triangles have the same measures of two angles, then their third angles are also the same.

Isosceles Triangle Conjecture

• If a triangle is isosceles, then two of its angles are congruent.

Converse of theIsosceles Triangle Conjecture

• If a triangle has two congruent interior angles, then it is isosceles.

Using 5 Triangle Conjectures

In class:•As a group: p. 202 # 9.•With partners: pp. 201-202

# 2-8

Tues 12/31. Between last night’s HW and today’s in-class

partner practice, you should have answered:• Ch. 4.4 # 1 - 9• Ch. 4.6 # 1 – 9

2. You also have HW due tomorrow:Ch 4 Review # 1 - 20

3. Also write down conjectures from sections 4.3 -4.6 in your notes. You may use notes on Friday’s test. Only useful if complete and organized.

Wed 12/41. Because we’ve seen so many conjectures,

you may use your math notebook on Friday’s test. The more organized and complete it is, the more useful.

2. Goals for today: Review triangle congruence ‘shortcuts’. Four of ‘em.

3. Use all conjectures to prove examples on handouts & HW. Flowchart and two-column proofs.

4. Note: Proofs are an A level topic on the test.

Friday 12/61. Have math notebook and several pencils out.2. May use math notes on this assessment of your

understanding. You may not use anything else.3. I will check HW while you take this assessment.4. This assessment will go in the grade book as a

quiz. We will take another assessment on Angles, Triangles and Proofs next Friday. The better of two grades will count as your unit test. So if you do well today, it counts as a test. If not, just a quiz. Same in other geometry classes.

5. We will have new partners on Monday.

Tuesday 12/10

Practice Worksheet Day3 Worksheets

Show first correct to get next.Save these and your corrected test:

ticket to retake test on Thursday.Work with table partner.

Thursday 12/12

• Have out your corrected test and practice handouts on table for check. These are your tickets to a retake.• Will take best of two grades for unit

test.• No cell phones.• You may use a calculator.