1 34 common core math 3 notes - unit 2 day 1 introduction...
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CP Math 3 Page 1 of 34
Common Core Math 3
Notes - Unit 2 Day 1
Introduction to Proofs
Properties of Congruence
Reflexive A A
Symmetric If A B , then B A
Transitive If A B and B C
then A C
Properties of Equality
Reflexive A = A
Symmetric If A = B, then B = A
Transitive If A = B and B = C,
then A = C
Distributive a(b + c ) = ab + ac
Substitution If a + b = c and b = f,
Then a + f = c.
Addition If a = b, then a + c = b +c
Subtraction If a = b, then a – c = b – c
Multiplication If a = b, then 3a = 3b.
Division If a = b, then a/3 = b/3.
CP Math 3 Page 2 of 34
Proofs – YOU MUST SHOW ALL THE STEPS IN AN ALGEBRAIC PROOF!!!!
Examples:
1. Given: 3(x – 2) = 42
Prove: x = 16
Statements Reasons
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
2. Given: 7m + 3 = 6
4
Prove: m = 3
Statements Reasons
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
7. 7.
CP Math 3 Page 3 of 34
Practice:
3. Given: -3(a + 3) + 5(3-a) = -50
Prove: a = ?
Statements: Reasons:
4. Given:
Prove: p = ?
Statements: Reasons:
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5. Given:
Prove: a = ?
Statements: Reasons:
CP Math 3 Page 5 of 34
Common Core Math 3
Notes - Unit 2 Day 2
Segments
What’s the difference???? Geometry: It’s all about the SYMBOLS and the PICTURES!!!
AB
AB AB
Example:
Line Segment – What do you notice about the segments?
●Definition of Equality: ●Definition of Congruence:
●Definition of Midpoint – midpoint of a segment is the point halfway between the endpoints of the segment
If X is the midpoint of AB , then AX = XB
●Midpoint Theorem – If X is the midpoint of AB , then AX XB
Ex: X is the midpoint of AB Hints:
Draw and label picture
AX = 3y + 7 Make sure you answer the question.
XB = 4y – 2
Find AB.
5units
M
A T H
5units
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●Definition of Segment Bisector- any segment, line or plane that intersects a segment at its midpoint
To “bisect” a segment means to
Sketch each of the following; be sure to include appropriate “marks.”
CD bisects AB at x AB bisects CD at x AB and CD bisect each other at x
●Definition of BETWEEN - refers to points on a line, ray or segment
T is between S and U
1. Find x if T is between S and U and ST = 7x, SU = 45, TU = 5x – 45.
2. A is between B and C. BA = x2, AC = 6x + 10, and BC = 17. Find x and the length of each segment.
A
A A
B
B B
Draw & Label picture!
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3. L is between K and M. KL = x2 – 10, LM = 5x + 4, and KM = 2x
2 – 42. Find x.
Segment Addition Postulate -
Ex 1: Given AC
AC = 3y + 1
AB = 2y
BC = 21
Find AB.
Find the value of the variable and LM if L is between points N and M.
Ex 2: NL = 5x, LM = 3x, NL = 15 Ex 3: NL = 5x – 3, NM = 2x + 6, LM = x – 7
D O G
CP Math 3 Page 8 of 34
Common Core Math 3
Notes - Unit 2 Day 3
Angles
Definition of Equality: Definition of Congruence:
Definition of Linear Pair – a pair of adjacent angles whose non-common sides are opposite rays
Definition of Vertical Angles – two nonadjacent angles formed by two intersecting lines. Notice that the definition just describes the picture of the angles, not the relationship.
Vertical Angles Theorem - If two angles are vertical angles, then they are congruent. Notice that the theorem describes the relationship between the 2 angles.
Ways to name angles:
1.
2.
3.
4.
1 2
CP Math 3 Page 9 of 34
Angle Addition Postulate –
If A is on the interior of <DOG then, __________ + ____________ = ____________
So if you see…… You must be able to write…….
Angle Bisector 1. Label <CAT
2. Draw and label AB such that it bisects <CAT
3. Place the appropriate “marks” to indicate the bisector
4. Write an equality statement about the two angles.
CAREFUL!!! These are not the same! Solve for x in each.
1. PN bisects <MPR 2. <CAT = 10x 3. <CAT is a right angle
<MPN = 2x + 14 <CAD = 7x <CAD = 7x
<NPR = x + 34 <DAT = 15 <DAT = 15
O L
E
V
D D C C
A T
T A
Hmmmm…same picture……..as above!
L
CP Math 3 Page 10 of 34
Definition of Complementary Angles– Definition of Supplementary Angles–
2 angles whose sum is _________________
<A is comp to <B
2 angles whose sum is ________________
<J is supp to <D
Statement: Statement:
Ex: The supplement of an angle measures 78 degrees less than the measure of the angle. What are the
measures of the angle and the supplement?
4. 5.
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Supplement Postulate – Complement Postulate
If two angles form a linear pair, then they are
supplementary
If the non-common sides of two adjacent angles for a
right angle, then the angles are complementary.
If I see this picture I can write……
If I see this picture I can write……
Supplementary Angles Theorem Complementary Angles Theorem Angles supplementary to the same angle Angles complementary to the same angle
or to congruent angles are congruent or to congruent angles are congruent.
Given: <A is sup to <C and A B Given: <D is comp to <Y and A D
Prove: <B is sup to <C Prove: <A is comp to <Y
Statement Reason Statement Reason
1 1. 1. 1.
2.
2. 2. 2.
3. 3. 3. 3.
4. 4. 4. 4.
5. 5. 5. 5.
6. 6. 6. 6.
A
B
C
D
A Y
non-common side
non
-com
mo
n sid
e
CP Math 3 Page 12 of 34
Definition of Perpendicular Lines: Lines that form right angles
Sketch: Symbol: Slopes:
Theorem: Perpendicular lines intersect to form 4 right angles
Theorem: All right angles are congruent.
If you know that <A is a right angle and <B is right angle, prove that <A is congruent to <B.
Statement Reason
1. 1.
2. 2.
3. 3.
4. 4.
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Common Core Math 3
Unit 2 Day 4
Segment Proofs!!!
Getting Ready for Proofs
For each of the following “givens” state the conjecture and the reason.
# Given Conjecture Reason
1. <A is complementary to <B
2. T is the midpoint of BE
3. T is the midpoint of BE
4. <A is supplementary to <B
5. ID= OL
6. ID = OL
7. ID = OL
OL = ME
8. m<1 = m<2
m<4 +m <2 = 180
9. <A is a right angle
10. AB is perpendicular to BC
11.
AB ≅ CD
12. C H S
13. B
U
G Y
CP Math 3 Page 14 of 34
N
M
P
Q
R S
14. TA is an angle bisector
M
A
T H
15.
1 2
16.
1 2
17.
1 2
18. S is between P an U
19. AB BC
20. <3 and <4 are vertical
PROOFS!!!
#1.
Given: MN PQ ; PQ RS
Prove: MN RS
Statements Reasons
1. MN PQ 1.
2. PQ RS 2.
3. MN RS 3.
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E
D
A
C
B
PS
QR
#2.
Given: BC = DE
Prove: AB + DE = AC
Statements Reason
1. 1.
2. 2.
3. 3.
.
#3.
Given: Q is between P and R
R is between Q and S
PR = QS
Prove: PQ = RS
Statements Reason
1. 1.
2. 2.
3. 3.
4.
4.
5.
5.
6.
6.
7.
7.
.
CP Math 3 Page 16 of 34
PS
QR
A
BM
#4.
Given: PR QS
Prove: PQ RS
Statements Reason
1. 1.
2. 2.
3. 3.
4.
4.
5.
5.
6.
6.
#6.
Given: M is the midpoint of AB
Prove: 2AM = AB
Statements Reason
1. 1.
2. 2.
3. 3.
4.
4.
5.
5.
CP Math 3 Page 17 of 34
A
ED
B
C
Common Core Math 3
Unit 2 Day 5
Angle Proofs
Steps for parts of some proofs:
To prove two angles = 180⁰ To prove angles = 90⁰
1. Prove LP
2. Prove Supp
3. If Supp then = 180
1. Prove perpendicular
2. If perpendicular then right <
3. If right < then = 90⁰
1.
Given: AEC BED
Prove: AEB CED
Statements Reasons
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
7. 7.
2.
Given: AEB CED
Prove: AEC BED
Statements Reasons
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
7. 7.
8. 8.
CP Math 3 Page 18 of 34
Proof of the Vertical Angle Theorem:
The Vertical Angle Theorem states: ________________________________________________
#3.
Given: 2 intersecting lines
Prove: <1 ≅ <3
Statements Reasons
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
7. 7.
8. 8.
9. 9.
10. 10.
#4.
Given: <1 and <3 are supplementary
<3 and <4 are linear pairs
Prove: <1 ≅ <4 4 3 2 1
Statements Reasons
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
7. 7.
8. 8
9. 9.
10. 10.
1 3 2
CP Math 3 Page 19 of 34
#5.
Given: <1 and <2 form a linear pair
<2 ≅ <1
Prove: <1 and <2 are right angles
Statements Reasons
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
7. 7.
8. 8.
9. 9.
10. 10.
11. 11.
12. 12.
CP Math 3 Page 20 of 34
Common Core Math 3
Unit 2 Day 6 Parallel Lines
Definition:
Sketch: Symbol: Slopes:
Skew Lines –
Transversal –
Angles formed by 2 lines and a transversal:
Alternate interior
Alternate exterior
Consecutive or same side interior
Corresponding
8 7 6 5
2 1 4 3
CP Math 3 Page 21 of 34
INVESTIGATION – You will need two different colored highlighters.
Investigation 1:
1. Trace with your highlighter. What letter do you see?
2. Name the typs of angle created?
3. What is the relationshp between those two angles?
Investigation 2:
4. Trace with your highlighter. What letter do you see?
5. Name the typs of angle created?
6. What is the relationshp between those two angles?
Investigation 3:
7. Trace with your highlighter. What letter do you see?
8. Name the typs of angle created?
9. What is the relationshp between those two angles
CP Math 3 Page 22 of 34
Investigation 4:
10. Trace with your highlighter. What letter do you see?
11. Name the typs of angle created?
12. What is the relationshp between those two angles?
Investigation 5:
13. Trace with your highlighter. What letter do you see?
14. Name the typs of angle created?
15. What is the relationshp between those two angles?
PRACTICE!!
1. Name alt int <s using line x as the transversal:
2. Name s.s. int <s using line y as the transversal:
3. Name corr <s using line z as the transversal:
4. Name alt ext <s using line y as the transversal:
5. Name alt int <s using line z as the transversal:
6. Name s.s. int <s using line z as the transversal:
CP Math 3 Page 23 of 34
CP Math 3 Page 24 of 34
3. Write an equation and solve for the unknown. State the theorem used to make the equation.
a)
b)
c)
d)
Use the diagrams to find x and y:
e. f.
:
g.
CP Math 3 Page 25 of 34
Common Core Math 3
Unit 2 Day 7 Proofs with Parallel Lines
Theorem: If given a line and a point not on the line, then there exists exactly one line through the point
that is parallel to the given line.
Example: Find the measure of the angle noted by the “ ? ”
Practice:
1. Solve the crook problem to find the missing angle:
2. Solve the crook problem to find the missing angle:
65˚
60˚
?
CP Math 3 Page 26 of 34
Using the theorems presented previously, prove that the sum of the degrees of the angles in a triangle
measure 180 degrees.
Given: Triangle CHS
Prove: m<C + m<H + m<S = 180o
Statements Reasons
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
Theorem – Perpendicular Transversal Theorem – In a plane, if a line is perpendicular to one of two
parallel lines, then it is perpendicular to the other.
CP Math 3 Page 27 of 34
m 1 2
3 4
6
7 8
5 p
z
m 1 2
3 4
6
7 8
5 p
z
1. Given: 1 5;
Prove:4 is supplementary to 6
m 1 2
3 4
6
7 8
5p
z
2. Given: 4 is supplementary to 6
Prove 3 7
3. Given:3 is supplementary to 8
Prove: 4 5
CP Math 3 Page 28 of 34
m 1
2
3
4
6
7
8
5
p
z
9
10
11
12
13
14
15
16
y
m 1
2
3
4
6
7
8
5
p
z
9
10
11
12
13
14
15
16
y
4. Given: 5 13, 10 15
Prove: 2 12
5. Given: 2 is supplementary to 3,
1 is supplementary to 13
Prove: 4 10
CP Math 3 Page 29 of 34
Common Core Math 3
Unit 2 Day 8
Proving Triangles Congruent
Side–Side–Side Congruence:
If the sides of one triangle are congruent to the sides
of a second triangle, then the triangles are
congruent.
Side–Angle–Side Congruence:
If two sides and the included angle of one
triangle are congruent to two sides and the
included angle of another triangle, then the
triangles are congruent.
Angle–Side–Angle Congruence: If two angles and the included side of one
triangle are congruent to two angles and
the included side of another triangle, then
the triangles are congruent.
Angle–Angle–Side Congruence:
If two angles and a non-included side of one
triangle are congruent to the corresponding
two angles and a side of a second triangle,
then the two triangles are congruent.
Hypotenuse-Leg Congruence:
If the hypotenuse and one leg of a right triangle are
congruent to the hypotenuse and the corresponding
leg of another triangle then the triangles are
congruent.
Abbreviation:
Not all triangles are congruent:
What about AAA?? What about SSA??
CP Unit 2 Notes Page 30
A
B
C
D
S
R
V
T
U
Proving “Triangles congruent”
Statements Reasons
1) 1) Given
2) angle or side stated 2)
3) angle or side stated 3)
4) angle or side stated 4)
5) Δ Δ 5) SSS, SAS, ASA, AAS, or HL
_________________________________________________________________________________________________
Congruent Triangle Proof Examples
1. Given: ACD ACB
CD CB
Prove: ACD ACB
Statements Reasons
1.
ACD ACB
CD CB
1.
2. 2.
3. ACD ACB 3.
------------------------------------------------------------------------------------------------------------
2. Given: RS US
ST SV
Prove: RSV UST
Statements Reasons
1.
RS US
ST SV
1.
2. 2.
3. RSV UST 3.
Picture/Diagram
required! Include tic
marks, arc marks, color,
“A” or “S” label.
CP Unit 2 Notes Page 31
M
L
P
N
Q O
3. Given: LP NP
NQP LOP
Prove: NQP LOP
Statements Reasons
1. LP NP
NQP LOP
1.
Redraw each triangle and label 2. 2.
3. NQP LOP 3.
Proving “Triangle PARTS”
C P C T C
Given:
Prove: part part
Statements Reasons
1) 1) Given
2) angle or side stated 2)
3) angle or side stated 3)
4) angle or side stated 4)
5) Δ Δ 5) SSS, SAS, ASA, AAS, or HL
6) part part 6) CPCTC
CP Unit 2 Notes Page 32
A E
GB
F
A
B C
D
Congruent “Triangle PARTS” Proof Examples
4. Given: || and ||BC AD AB CD
Prove: <B <D
Statements Reasons
1) 1)
2) 2)
3) 3)
4) 4)
5) 5)
6) 6)
5. Given:
AB FG
AB FG
B G
Prove: AF FE
Statements Reasons
1) 1)
2) 2)
3) 3)
4) 4)
5) 5)
6) 6)
CP Unit 2 Notes Page 33
G
F
O N
M
6. Given: FON MNO
FO MN
Prove: F M
Statements Reasons
1)
; FON MNO FO MN 1)
2) 2)
3) 3)
4) 4)
Isosceles Triangle Theorem - If two sides of a triangle are congruent, then the angles opposite
those sides are congruent
Statement Reason
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
7. 7.
8. 8.
Given: BCAC
Prove: BA
(hint: make CD an angle bisector)
CP Unit 2 Notes Page 34