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Transcript
Aim #5: How do we do write proofs of angle theorems?
Do Now:
1. Solve for the variables if AB ll CD.
A
720
a
B
b
e d c
f g
C
h 1120
i
j
l
k
CC Geometry H
a =_____
b =_____
c =_____
d =_____
e =_____
f =_____
g =_____
h =_____
i =_____
j =_____
k =_____
l =______
D
2. How many lines are there passing through A parallel to CD?_____
Complete: Through a point not on a line, there is _________________________
_____________________________________________________
(This is known as thePARALLEL POSTULATE.)
A postulate is a mathematical statement accepted as true without proof.
A theorem is a mathematical statement that can be proven.
Once a theorem has been proven, it can be added to our list of known facts and
used in proofs.
We previously accepted the following:
• Vertical angles are of equal measure. (theorem)
• If a transversal intersects two parallel lines, alternate interior angles are of
equal measure. (postulate)
Let's prove:
If transversal intersects two parallel lines, corresponding angles are congruent.
Given: AB ll CD
Prove: x = w
Statements
Reasons
You have available the following facts:
• Vertical angles are equal in measure. (theorem)
• Alternate interior angles are equal in measure if lines are parallel. (postulate)
• Corresponding angles are equal in measure if lines are parallel. (theorem)
Let's prove:
If a transversal intersects two parallel lines, same-side
interior angles are supplementary.
Given: AB ll CD, transveral EF
Prove: m ≮1 + m≮2 = 180
G
1
H
Statements
Let's prove:
2
3
Reasons
The three angles of a triangle sum to 180°.
You will need to draw an auxiliary line parallel to one of the triangle‛s sides and
passing through the vertex opposite that side. Add any necessary labels
and write out your proof.
yxoo
Given: Triangle ABC
o
Prove: x + y + z = 180
yxoo
Statements
Reasons
zo
Each of the three parallel line theorems has aCONVERSE (or reversing)
theorem as follows:
Original
If two parallel lines are cut by
a transversal, then alternate
interior angles are congruent.
a
1
2
2
b
If a ll b, then _______
Original
If two parallel lines are cut by
a transversal, then corresponding
angles are congruent.
a
1
2
b
If two lines are cut by a transversal
such that alternate interior angles
are congruent, then the lines are parallel.
a
1
b
Converse
If m 1 = m 2, then __________
Converse
If two lines are cut by a transversal
such that corresponding angles are
congruent, then the lines are parallel.
a
1
2
b
If a ll b, then _______
If m 1 = m 2, then __________
Original
If two parallel lines are cut by
a transversal, then interior angles
on the same side of the
transversal add up to 1800.
Converse
If two lines are cut by a transversal
such that interior angles on the same side
of the transversal add up to 1800, then the
lines are parallel.
a
a
1
1
b
2
If a ll b, then ______________
b
2
If m 1 + m 2 = 180, then ____________
Let's prove:
If two lines are perpendicular to the same line, they are parallel to each other.
State the given facts and the conjecture to be proven:
Given:
Prove:
Statements
a
b
l
Reasons
Prove the "converse" of the theorem we proved on page 1:
If a transversal intersects two lines so that corresponding angles are equal,
then the lines are parallel.
Given: x = y
Prove: AB ll EF
zo
Statements
Reasons
Let's Sum it up!!
If a transversal intersects 2 parallel lines,
180
supplementary
• same side interior angles are ___________________
or add up to ______
equal
• alternate interior angles are _______________
equal
• corresponding angles are ________________.
0
.
CC Geometry H
Name ______________________
Date ______________________
HW #5
1) Is it possible to prove lines a and b parallel? Explain your answer.
b) a
b
a)
a
b
114
0
48
66
c) a
50
70
b
d)
0
a
0
120
0
0
29
0
0
b
60
0
C
B
2) Given: m≮C + m≮D = 180
m≮B = m≮D
Prove: AB ll CD
D
A
OVER
0
3) Prove: d + e - a = 180
(Suggested auxiliary lines have been drawn.)
Statements
Reasons
1. d + y = 180
1.
2. e = x + y
2.
b0
a0
x00
0
c
d
3. x = x
3.
4. y = e - x
4.
5. d + e - x = 180
5.
6. a = x
6.
7. d + e - a = 180
7. 0
y
0
e
4) Prove: If a line is perpendicular to one of two parallellines and intersects the other,
n
then it is perpendicular to the otherparallel line.
Complete.
Given:
Prove:
Statements
Review:
6) Find the measure of the smaller angle.
a
b
Reasons
7) Find m≮x and m≮y.
26
(6x + 4)
0
2x
o
y
116
0
0
38
0
x