chapter 2: basic concepts and proof · 2016. 9. 21. · notes pd3 september 21, 2016 chapter 2 -...

notes pd3 September 21, 2016

Chapter 2 - Openers and notesResources (1) Theorem/postulate packet (2) Vocabulary - download from website (3) Answer key to chapter 2 Review exercises (4) Homework 2.1

Describe how to find the shortest distance from a point to a line?

In algebra you studied relationships between points when you graphed on a coordinate system. How are the axes related on the coordinate plane?

In the diagram, name the right angles

E

GF

H

Chapter 2: Basic Concepts and Proof

2.1 perpendicularity


A

B

CD

E

F1 2 3 4

Angles 1,2,3 and 4 are in the ratio of 2:1:1:2, find the measure of <EBC


GIVEN: <ACB=90,

Prove: <C <DA B

C

D


Chapter 2: Basic Concepts and Proof

2.1 perpendicularityFind the area of rectangleABCD

Find coordinates of D

If the coordinates of B are not given, could you answer the first two questions?


Explain perpendicularity:

What would be a good definition?

What does the word oblique mean?


GIven: a τ bProve: < 1 ≅ < 2


A ( 3, 2) Where is A ' if it is a result of rotating A 90 degrees about the origin?

WHat is the result of rotating B 90 degrees about the origin if B is at (0, -3)?


LIne a is perpendicular to line b. The angle formed by their intersection is trisected. One of the new angles then is also trisected. One of these newer angles is bisected. How large is the smallest angle?


What is the relationship algebraically between perpendicular lines?

A(1, 1)

B(7, 5)

C(9, 3)

Can you find the equation of the perpendicular line through B to side AC ?


2.2

x

y

(1,8)(-2,8)

(-2,-3) ( , )

A B

C D

what are the coordinates of D ?

What is the area and perimeter of the rectangle

Complementary and Supplementary Angles


If two angles are complementary then ________________________________________

________________________________________

If two angles are supplementary then _________________________________________

_________________________________________


Find the complement and the supplement of

The supplement of an angle is five times the complementof the angle. Find the measures of the original angle, the complement and the supplement:

let x be the angle180 - x is the supplement? is the complement. Now translate the problem into an equation


A

CTS

given: <CAS and <TAS are complementaryProve:


2.3 Drawing Conclusions

It is important as a student of Geometry to analyze and hypothesize and draw conclusions based on your observations, theorems, postulates and definitions.

Can you draw a conclusion based on the information?

given: <1 <2conclusion:

Be CAREFUL to not ASSUME !


Given: M is the midpoint of GHConclusion:

G

H

MP

Q

Given: <1 <2 <3

Conclusion:


Our task is to solve the problem, but before we just jump in...let usanalyze. We can use definitions, theorems & postulates and then draw conclusions.


Once again, can we draw conclusions based on the information given?Let's proceed...


A C M

E

What type of 'assumptions' or conclusions can we make?


2.4 Congruent Supplementsand congruent Complements

suppose <1 is supplementary to <2and also suppose that <2 is supplementary to <3, what conclusion can you draw?


Now suppose that <1 is supplementary to <2 an<3 is supplementary to <4 and also that <2 <4,what conclusion can you draw?


suppose that <1 is complementary to <2 andthat <3 is complementary to <2, what conclusion can be drawn?

Suppose that <1 is complementary to <2 <3 is complementary to <4 and also that <1 <3What conclusion can be drawn?


theorems:

If two angles are supplementary to the same angle then theyare congruentIf two angles are supplementary to congruent angles then they are congruent

If two angles are complementary to the same angle then theyare congruentIf two angles are complementary to congruent angles then they are congruent


Given: diagram as shownProve <1 <3

now, do #1-8 in BIG BLUE and #19


2.5 Addition and Subtraction Properties

A B C M

Let AB =CM , can you draw any conclusions?

W

I

N S

T

Let <WIN <SIT,can you draw any conclusions?


N

E

L

SCan you conclude anything about <SNL and <SLN?


Addition Properties for segments and anglestheorems 8-11Subtraction Properties for segments and anglestheorems 12-13


A

B C D E F

Given <BAD <FAD, AD bisects <CAEProve: <BAC <FAE


2.6 Multiplication and Division Properties

M N O P Q R S T

Let the points N and O be trisection points ANDR and S be trisection points.ALSO, MN=QR. can you conclude anything?


X

Y

Z

W

Ray XY and Ray ZW are angle bisectors, if <1 = <3, what can be said of <AXE and <PZT ?

AE

P

T

Multiplication and Division Theorems 14 and 15

Using the theorems 14 & 151) Look for double use of the words bisect or trisect of midpoint2) use the Mult. Prop. if segments or angles are greater in the prove statement3) use the Div Prop. if the segments or angles are smaller in the prove statement.


practice


2.7 transitive and Substitution properties

If <A =<B and <B = <C, is <A = <C ?

Does this sound like a familiar property?

Theorems 16 & 17


P

Q R

S T

G: Q and R are midpoints and PS=PTP: QS = RT


2.8 Vertical Angles

Name the vertical angle pairs

A

B

C

D

E

G

H

J

K

Which angle is a vertical angle to <GBK ?

to <EBA ?

to <ABG ?


G: GD bisects <CBE

P: <1 <2


If a pair of vertical angles are complementary, what are their measures?

If a pair of vertical angles are supplementary, what are their measures?

chapter 2: basic concepts and proof · 2016. 9. 21. · notes pd3 september 21, 2016 chapter 2 -...

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