Bell Work• 1) Name the congruent triangles and the congruence shortcut

that verifies their congruence:

• 2) Use segment addition to find x• AB = x + 11; BC = 2x + 5; AC = 22

• 3) Angle A and Angle B are complementary: Angle A = 3x + 5 Angle B = 5x + 5. Find x

• 4) Find the value of x and y in the triangle:

AGENDA

• Please turn in your Triangle Drawings!• Chapter 4- Practice Test- To gauge where

you are on those skills. We will go over these on Weds.

• Skill 1-13 Review- You will fill in the sheet as we go along and work on the practice problems.

• Begin Study Guides- Chapter 1 DUE Wednesday.

Outcomes

• I will be able to:• 1) Use and understand skills 1 - 13

Chapter 4 Practice Test

• It is important you try your best to see how you are doing on these skills.

• Like a regular test, this is to be done silently, independently and with shown work.

• You are to find something to do silently or work on your study guide if finished early.

Take Out Your Targeted Review Sheet

• We will do examples and practice problems.

• Please record your answers in the targeted review sheet.

Skill #1 Inductive Reasoning• Inductive Reasoning – Observing data,

recognizing patterns, and making generalizations about that data

• 3 Stages of Inductive Reasoning• 1) Look for a pattern – Look at examples and

use diagrams, tables, and pictures to help discover a pattern.

• 2) Make a conjecture - Use your observations to make “guess” about the pattern.

• 3) Verify the conjecture - Use logical reasoning skills to decide if your conjecture is valid.

Skill #1 Inductive ReasoningPractice

• 1) Find the next 3 numbers in the sequence and describe the pattern:

• -5, 10, -20, 40…• A) -70, 110, -160• B) -80, 160, -320• C) -50, 60, -70• D) -60, 90, -120

• 2) Find the next figure

• A) B)

• C) D)

Skill #2 Writing Conjectures• Conjecture – an unproven statement

based on observations. Conjectures can be modified until they are concrete.

• ***The process of describing what is being observed

• Make a conjecture about the sums of any two odd numbers.

• 1+1 = 2 3 + 7 = 10 • 1 + 3 = 4 5 + 9 = 14• 3 + 5 = 8 7 + 9 = 16

Conjecture: If two odd numbers are added, thenthe result is an even number.

Skill #2 Writing ConjecturesPractice

• 3) Write a conjecture based upon the pattern seen below:

• 4) Write a conjecture based upon the pattern seen below:

Skill #3 Recognizing Points, Lines, and Planes

• You need to be able to use the symbols for points, lines, rays, segments, and planes

• Symbols:• Point: O• Line: PR• Ray: NR• Segment: MN• Plane: STO(must contain at least 3 points

that are on the plane). Unless the plane has a name.

X

Skill #3 Recognizing Points, Lines and Planes

• Collinear: 3 or more points on the same line

• Coplanar: 3 or more points on the same plane

Skill #3 Recognizing Points, Lines, and Planes(Problems)

• 5) Name a ray

• 6) Name 3 coplanar points

• 7) Name 3 noncollinear points

• 8) Name a plane

Skill #4 Counterexamples

• Not all conjectures are true

• Counterexample – an example that shows that a conjecture is false

• Example: If two numbers are positive, then their difference is always positive.

• Counterexample: 1 – 2 = -1

Skill #4 CounterexamplePractice

• 9) Find a counterexample for the following conjecture:

• If a number is prime, then it is odd

• 10) Find a counterexample for the following conjecture:

• If you square root a number, then it is always less than the number

Skill #5 Segment Addition• Segment Addition: Adding two smaller

segments together to get a larger segment.

• Example: If HI = 2x + 3 and IJ = 4x + 1and HJ = 16, find x.

• It may help to draw a picture

• 2x + 3 + 4x + 1 = 16• 6x + 4 = 16• x = 2

Skill #5 Segment AdditionPractice

• 11) AB = 12; BC = 24; AC = 3x. Find x.• 12) EF = 3x – 1; FG = 2x + 6; EG = 25. Find x.

Skill #6 Distance Formula• Distance Formula: Used to determine the

distance of points in the coordinate plane.

• Distance formula =

• Example: A is at (-2, 3) and B is at (6, 9). Find AB.

• 10

212

212 )()( yyxx

22 )39()26( 22 )6()8(

Skill #6 Distance Formula Practice

• 13) Find CD if C is at (5, 1) and D is at (1, 4)

Skill #7 Simplifying Radicals

• If a perfect square exists inside a radical(square root sign), then we can simplify the radical by creating a factor tree and pulling one of the numbers from each pair outside the square root sign.

• Example: 175

25 7

5 5 75

Skill #7 Simplifying Radicals Practice

• 14) If F is at (7, -2) and G is at (3, 10). Find FG. Remember to simplify any radicals.

16014416

104

Skill #8 Angle Measure and Classification

• There were four different classifications for individual angles: acute, right, obtuse, straight.

• Acute – any angle less than 90°• Right – any angle exactly 90°• Obtuse – any angle greater than 90° but

less than 180°• Straight – any angle exactly 180°

Skill #8 Angle Measure and Classification Practice

• Find the angle measure and classify it• 15) Angle EAD

• 16) Angle CAF

• 17) Angle EAB

Skill #9 Midpoint• Midpoint – The point that cuts a line into

two congruent pieces• Example: If N is the midpoint of MO and

MN = 2x + 3 and NO = x + 7. Find x. It may help to draw a picture.

• 2x + 3 = x + 7• x = 4

Skill #9 Midpoint

• Midpoint Formula – a formula used to find the midpoint when points are in the coordinate plane

• Midpoint Formula = • Example: A is at (-2, 2) and B is at (4, 4).

Find the midpoint

• (1, 3)

2,

22121 yyxx

2

42,

2

42

Skill #9 MidpointPractice

• 18) B is the midpoint of AC. Find x• AB = 4x + 5• BC = x + 14

• 19) Find the midpoint between C and D if:• C(5, 1) and D(1, 8)

Skill #10 Angle Bisector

• Angle Bisector – a ray, line, or segment that cuts an angle into two congruent smaller angles

• Example: EF is an angle bisector, find x.• 42 = 2x + 12• x = 15

Skill #10 Angle Bisector Practice

• 20)

Skill #11 Angle Pair Relationships• There are 4 angle relationships that

involve two lines, rays or segments, intersecting each other.

• Vertical Angles – Two angles that share a vertex, are opposite each other, and made from opposite rays

• Linear Pair – Two angles that form a straight line

• Supplementary – Two angles whose sum is 180°

• Complementary – Two angles whose sum is 90°

Skill #11 Angle Pair Relationships Examples

• Vertical Angles

• Linear Pair

• Supplementary

• Complementary

Skill #11 Angle Pair Relationship Practice

• 21) The two angles below form a linear pair. Find x.

• 22) The two angles below are complementary. Find x.

Skill #12 Conditional Statements

• Conditional Statements: Statements that have a hypothesis and a conclusion.

• To be true, both the hypothesis and the conclusion must be true.

• May be written symbolically, where p is the hypothesis and q is the conclusion.

• Example: If it is sunny, then it is warm.• Hypothesis: it is sunny(p)• Conclusion: it is warm(q)• In symbols: p --> q

Skill #13 Converse Statements

• Converse Statement – Switching the hypothesis and the conclusion of a conditional statement

• Conditional: If it is sunny, then it is warm.• Converse: • If it is warm, then it is sunny.

Skill #14 Inverse Statements

• Inverse Statement – negating both the hypothesis and the conclusion of a conditional statement.

• Conditional: If it is sunny, then it is warm.• Inverse: If it is not sunny, then it is not

warm.

Skill #15 Contrapositive Statements

• Contrapositive Statement – Negating both the hypothesis and the conclusion of a converse statement

• Conditional: If it is sunny, then it is warm.• Converse: If it is warm, then it is sunny.• Contrapositive: If it is not warm, then it is

not sunny.

Skill #’s 12-15 Practice

• Conditional statement:• If it is freezing outside, then there is snow

on the ground.• 23) Write the converse of the statement• 24) Write the inverse of the statement• 25) Write the contrapositive of the

statement

Skill #16 Biconditional Statements• Biconditional Statement – Any statement with

the phrase “if and only if” in it. • To verify if a biconditional is true, break it down

to check the validity of the conditional statement and it’s converse.

• Example: Two triangles are congruent if and only if their corresponding parts are congruent.

• Conditional: If two triangles are congruent, then their corresponding parts are congruent.

• Converse: If their corresponding parts are congruent, then two triangles are congruent.

Skill #16 BiconditionalPractice

• Determine if the following is a biconditional statement. If it is a biconditional, is it true?

• 26) A polygon is a square if and only if it has 4 sides.

Study Guide Check Wednesday

• Chapter 1 and 2 of the study guide should be completed