Essential Question #1 Why is the use of inductive reasoning important to understanding mathematics?

Geometry developed when people began recognizing and describing patterns. Studying patterns is very useful because you can recognize a pattern and make an educated prediction (conjecture).

Describe the pattern you see below. What would be the next figure?

Find the next three numbers is the sequence and explain what the pattern is. 1, 4, 16, 64, _____, _____, _____

Reasoning that uses a number of specific examples to arrive at a logical conclusion. 1.Look for a pattern 2.Make a conjecture 3.Verify the conjecture

Complete the conjecture below: The product of any two consecutive positive integers is ___________________.

For points A, B, and C, AB = 10, BC = 8 and AC = 5. Make a conjecture about points A, B, and C.

We want to find a conjecture that is ALWAYS true. ONE EXAMPLE THAT PROVES A CONJECTURE IS FALSE IS CALLED A COUNTER EXAMPLE.

Find a counter example for this conjecture: The difference between two positive numbers is always positive.

Given that points P, Q, and R are collinear Joel made a conjecture that Q is between P and R. Determine if his conjecture is true or false. If it is false you must provide a counterexample.

Essential Question #2 Compare and contrast the reflexive, symmetric, and transitive properties.

Statements in the form If-Then. Example: If you are 13 years old, then you are a teenager. Hypothesis Conclusion

Identify the hypothesis and conclusion for each conditional statement. If you are an NBA basketball player, then you are at least 5’2”. If 3x – 5 = -11, then x = -2.

Write each statement in the conditional form. “A Champion is afraid of losing” Perpendicular lines form right angles.

Solve the following equation: 5 = 3x - 4

Reflexive Property For every number a, a = a Symmetric Property For every number a and b, if a = b, then b = a Transitive Property For all numbers a, b, and c, if a = b and b = c, then a = c.

Add/Sub Property For all numbers a, b, and c, if a = b then a + c = b + c Multi/Div Property For all numbers a, b, and c, if a = b then a(c) = b(c) Substitution Property For all numbers a and b, if a = b, then a may be replaced by b for any equation. Distributive Property For all numbers a, b, and c, a(b + c) = ab + ac

Name the property of equality that justifies each step 1.If AB + BC = DE + BC, then AB = DE 2.m ∠ ABC = m ∠ ABC 3.If XY = PQ and XY = RS, then PQ = RS. 4.If 1/3x = 5, then x = 15 5.If 2x = 9, then x = 9/2

A written record of the steps to completing a problem and reasons why each step works. Two Column Proof A proof that lists statements in one column and reasons for those statements in another column.

Justify each step in solving. StatementsReasons x + 5 = x = 9 5. x = 3

Justify the steps for the proof. If PR = QS, then PQ = RS. Given: Prove: P Q R S StatementsReasons 1.PR = QS 2.PQ + QR = PR QR + RS = QS 3.PQ + QR = QR + RS 4.PQ = RS

With a partner complete: Page 94 #10 and 12.

Essential Question #3 Explain why many true examples of a statement are not sufficient evidence to conclude that a statement is true for all cases. Why are proofs necessary?

Name the property of equality that justifies each step 1.AB = AB 2.If CD = XY, then XY = CD 3.If RS = JK and JK = PQ, then RS = PQ

Since segments with equal length are congruent, then the properties that apply to segment length also apply to segment congruence. Congruence of segments is reflexive, symmetric, and transitive.

Justify the steps for the proof. Given: P, Q, R, and S are collinear Prove: PQ = PS - QS P Q R S StatementsReasons 1.P, Q, R, S are collinear 2.PS = PQ + QS 3.PS – QS = PQ 4.PQ = PS - QS

A B C Given: Prove: D E F StatementsReasons 1. 2.AC = DF, AB = DE 3.AC – AB = DF – DE 4.AC = AB + BC DF = DE + EF 5.AC – AB = BC DF – DE = EF 6.DF – DE = BC 7.BC = EF 8.

With a partner complete: Page 103 #11 and 12

Write a reason for each statement. If then m ∠ ABC = m ∠ DEF. m ∠ ABC = m ∠ ABC m ∠ ABC = m ∠ DEF and m ∠ DEF = m ∠ PQR, then m ∠ ABC = m ∠ PQR

Since angles with equal measure are congruent, then the properties that apply to angle measure also apply to angle congruence. Congruence of angles is reflexive, symmetric, and transitive.

If two angles form a linear pair, then they are supplementary angles.

Given: m ∠ ABC = m ∠ DFE m ∠ 1 = m ∠ 4 Prove: m ∠ 2 = m ∠ 3 StatementsReasons B A C 1 2 F D E 4 3

Given: ∠ 1 ≌ ∠ 2 Prove: ∠ 3 ≌ ∠ 4 StatementsReasons