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Geometry
Parallel and
Perpendicular Lines
2013-08-08
www.njctl.org
Angles and Parallel Lines
Slopes of Parallel Lines
Slopes of Perpendicular Lines
Proofs Involving Parallel and Perpendicular Lines
Constructing Parallel Lines
Constructing Perpendicular Lines
Table of Contents
Parallel Lines - using Menu Options
Perpendicular Lines - with Compass and Straight Edge
Perpendicular Lines - using Menu Options
Videos-Table of Contents
1
Angles and Parallel Lines
of Contents
Angles & Parallel Lines
Parallel, Perpendicular and Skew
How many different ways can you draw 2 lines in a plane?
Click
Click
Jun 29-11:48 AM
Parallel, Perpendicular and Skew
never meet are called
parallel lines
point are called
intersecting lines
Angles & Parallel Lines
2
Parallel, Perpendicular and Skew
Term
Definition
Diagram
2 lines that
lie in the
Parallel lines same plane
and never
intersect
2 lines that
lie in the
Intersecting
same plane
lines
and meet in
one point
2 lines that
lie in the
Perpendicular same plane
lines
and meet at
4 right
angles
Symbol
k || m
no symbol
a
b
a b
Jun 29-12:19 PM
Parallel, Perpendicular and Skew
Using the following diagram, name a line that does not lie in
the same plane with HG and does not intersect HG.
Jun 29-12:08 PM
Parallel, Perpendicular and Skew
Lines that do not intersect and do not lie in the same plane are
called skew.
HG is skew with EA
HG is skew with BF
HG is skew with AB
HG is skew with AC
Jul 3-3:08 PM
3
1 Are lines a and b skew?
Yes
No
Jul 17-11:23 AM
If there is a line and a point not on the line, then there is exactly one
line through the point parallel to the given line.
P
k
There is only 1 line through
point P parallel to line k.
Jul 3-3:17 PM
If there is a line and a point not on the line, then there is exactly one
line through the point perpendicular to the given line.
P
k
There is only 1 line through
point P perpendicular to line k.
Jul 3-3:24 PM
4
How many lines can be drawn to fit the description?
through C parallel to AB?
B
A
C
through C perpendicular to AB?
Jul 3-3:25 PM
2
Name all lines parallel to EF.
Angles & Parallel Lines
3
AB
Angles & Parallel Lines
5
4
Angles & Parallel Lines
5
Angles & Parallel Lines
6
Two skew lines are __________ parallel.
Angles & Parallel Lines
6
: A line that intersects two or more coplanar
lines at different points.
Angles & Parallel Lines
When a transversal intersects two lines, eight angles
Angles & Parallel Lines
2. Name the interior angles.
Angles & Parallel Lines
7
Nov 27-3:22 PM
Nov 27-3:22 PM
Nov 27-3:22 PM
8
Nov 27-3:22 PM
Angles & Parallel Lines
Angles & Parallel Lines
9
Jun 20-3:16 PM
7 <3 and < 6 are _____.
A
B
C
D
Corresponding Angles
Alternate Exterior Angles
Same-Side Exterior Angles
Vertical Angles
Jun 30-10:27 AM
8 <1 and <3 are _____.
A
B
C
D
Alternate Interior Angles
Corresponding Angles
Linear Pair of Angles
Same-Side Interior Angles
B
A
2
1
3
4
C
D
AB || CD and AD || BC
Jun 30-10:41 AM
10
9 <3 and <4 are _____.
A
B
C
D
Corresponding Angles
Same-Side Interior Angles
Alternate Interior Angles
Same-Side Exterior Angles
B
A
2
1
3
4
C
D
AB || CD and AD || BC
Jun 30-10:44 AM
10 <3 and <6 are _____.
A
B
C
D
Corresponding Angles
Same-Side Exterior Angles
Alternate Interior Angles
Alternate Exterior Angles
B
A
2
1
6
3
4
5
C
D
AB || CD and AD || BC
Jun 30-10:56 AM
11 <2 and <6 are ____.
A
B
C
D
E
Corresponding
Alternate Exterior Angles
Vertical Angles
Same-Side Exterior Angles
None of the above
2
3
4
1
5
8
12
6
7
9
10
11
Jun 30-11:12 AM
11
12 Name all angles corresponding with <4.
A
B
C
D
E
<7
<10
<8
<9
Both A and B
2
1
3
4
5
8
12
6
7
9
10
11
Jul 6-12:08 PM
There are several theorems and postulates related to parallel lines. At
this time, please go to the lab titled, "Properties of Parallel Lines".
Click here to go to the lab titled, "Properties
of Parallel Lines"
Lab 1
If two parallel lines are cut by
a transversal, then the corresponding angles are congruent.
According to the Corresponding Angles Postulate what
angles are congruent?
Angles & Parallel Lines
12
Angles & Parallel Lines
If two parallel lines are cut by a
transversal, then the alternate interior angles are congruent.
According to the Alternate Interior Angles Theorem what
angles are congruent?
Angles & Parallel Lines
Angles & Parallel Lines
13
If two parallel lines are cut by
a transversal, then the alternate exterior angles are congruent.
According to the Alternate Exterior Angles Theorem
what angles are congruent?
Angles & Parallel Lines
Angles & Parallel Lines
If two parallel lines are cut by a
transversal, then the same-side angles are supplementary.
According to the Same-Side Angles Theorem which pairs of
angles are supplementary?
Angles & Parallel Lines
14
Angles & Parallel Lines
1. Name all of the angles congruent to <1.
2. Name all of the angles supplementary to <1.
Jun 30-4:45 PM
13 Find all of the angles congruent to <5.
A
B
C
D
<1
<4
<8
all of the
above
Jun 24-7:29 PM
15
14 Find the value of x.
Jun 24-7:32 PM
15 Find the value of x.
Jun 24-7:36 PM
16 If the m<4 = 1160 then m<9 = _____0?
n || p
Jun 24-7:40 PM
16
17
If the m<15 = 570, then the m<2 = _____0.
A
B
C
D
57
123
33
none of the above
n || p
Jun 25-3:32 PM
Extending Lines to Make Transversals
1310
1
0
41
Angles & Parallel Lines
Extending Lines to Make Transversals
131 0
Extend the line that
would create a
transversal.
Input angle
measures
accordingly and
solve for the
missing angle.
1
41
0
131 0
1
41
131
0
0
Nov 27-3:45 PM
17
+ 12) = 180
4y + 144 = 180
click = 36
4y
y=9
click
4y + 12=x
4(9)+12 = x
36 + 12 = x
click
x = 48
click
click
click
click
click
Angles & Parallel Lines
Transversals and Perpendicular Lines
Angles & Parallel Lines
18 Find the m<1.
Jun 25-3:40 PM
18
19 Find the value of x.
A
B
C
D
12
54
42
18
Jun 25-3:43 PM
20 Find the value of x.
Jun 25-3:45 PM
21 Find the value of x.
Jun 25-3:49 PM
19
Proving Lines are Parallel
In the preceding section you saw that when two lines are
parallel, you can conclude that certain angles created by the
transversal are congruent or supplementary.
parallel.
Nov 27-4:44 PM
Proving Lines are Parallel
If two lines are cut by a transversal AND the corresponding
angles are congruent, then the lines are parallel.
or
or
or
k || m
then click
Remember
If two parallel lines are cut by
a transversal, then the corresponding angles are congruent.
Angles & Parallel Lines
Proving Lines are Parallel
Nov 27-4:05 PM
20
Proving Lines are Parallel
If two parallel lines are cut by a transversal and the alternate
interior angles are congruent, then the lines are parallel.
click
If two parallel lines are cut by a
transversal, then the alternate interior angles are congruent.
Angles & Parallel Lines
Proving Lines are Parallel
Nov 27-4:15 PM
Proving Lines are Parallel
Theorem
If two parallel lines are cut by a transversal and the alternate
exterior angles are congruent,then the lines are parallel.
click
If two parallel lines are cut by
a transversal, then the alternate exterior angles are congruent.
Angles & Parallel Lines
21
Proving Lines are Parallel
Nov 27-4:37 PM
Proving Lines are Parallel
Theorem
If two parallel lines are cut by a transversal and the same-side
interior angles are supplementary, then the lines are parallel.
m<3 + m<6 = 1800
m<4 + m<5 = 1800
click
If two parallel lines are cut by a
transversal, then the same-side angles are supplementary.
Angles & Parallel Lines
Proving Lines are Parallel
Nov 27-4:59 PM
22
22 Which statement below would make lines k and m parallel?
A
B
C
D
m<2 = m<4
m<5 + m<6 = 180
m<3 = m<5
m<1 + m<5 = 90
Jun 25-4:10 PM
23 Based on the diagram below which of the following is true?
A
B
C
D
e || f
f || g
h || i
e || g
Jun 25-4:06 PM
24 What theorem would NOT prove that lines a and b are parallel?
A
If lines a and b are cut by a transversal such that
cooresponding angles are congruent.
B
If lines a and b are cut by a transversal such that
alternate interior angles are congruent.
C
If lines a and b are cut by a transversal such that sameside interior angles are complementary.
D
If lines a and b are cut by a transversal such that sameside interior angles are supplementary.
Jun 25-4:13 PM
23
25 Find the value of x for which a || b.
x
Jul 15-2:40 PM
26 Find the value of x which makes a || b.
Jul 15-2:47 PM
27 Find the value of x for which m || n.
Jun 25-4:25 PM
24
Slopes of Parallel Lines
of Contents
Slopes of Parallel Lines
Vertical Change
=
Rise
Run
Slopes of Parallel Lines
Horizontal Change can be written as the difference of
the x-coordinates:
Slope
=
Slopes of Parallel Lines
25
Types of slopes.
Undefined
Nov 30-2:58 PM
Slope of Red Line:
3-1
= 1
0 - (-2)
click
Dec 3-10:41 AM
Slope of Red Line:
2-1
=
0-3
click
-1
3
Dec 3-10:41 AM
26
Red Line:
3-3
= 0
1 - (-2)
click
Slopes of Parallel Lines
Slope of Red Line:
-2 - 3
=
2-2
click
-5
0
= undefined
Slopes of Parallel Lines
Evaluate the slope for the given line.
: Identify two points on the
given line
(-2, -3) and (1, 3)
Nov 30-1:38 PM
27
: Evaluate the slope
using the slope formula.
: Whatever point you start with
for the y-value, you
start with
the same point for the x-value.
Slope(m) =
-3 - 3
-2
click- 1
=
-6
=2
-3
Nov 30-1:46 PM
Evaluate the slope of the given line.
Slopes of Parallel Lines
28 How can you determine if a line has a negative slope when
looking at its graph?
The line goes from the bottom left to the top right of the
graph.
The line goes from the top left to the bottom right of the
B
graph.
C The line is horizontal.
D The line is vertical.
A
Jun 25-4:37 PM
28
29 Choose the graph(s) with a positive slope.
A.
B.
A
B
C
D
D.
C.
Jul 17-12:28 PM
30 Determine the slope of the given line.
Jun 25-4:42 PM
31 Determine the slope of the given line.
A
B
C
D
0
1
-1
undefined
Jun 25-4:44 PM
29
32 Evaluate the slope of the line containing (-2, 5) and (-5, 2).
Jun 25-4:47 PM
33 What is the slope of the line that goes through
(1, -3) and (2, -6)?
Jun 25-4:49 PM
34 What is the slope of the line that contains the points
(6, -9), and (4, -9)?
Jun 25-4:52 PM
30
Writing Equations of Lines
Linear equations can be written in several different forms.
Slope-Intercept Form of a linear equation provides the slope and
-intercept of the line.
m = the slope of the line
-intercept of the line
Point Slope Form of a linear equation provides the slope and a
specific point on the line.
m = the slope of the line
) = point on the line
Slopes of Parallel Lines
Writing Equations of Lines
Standard Form of a linear equation is most useful when
you want to:
-find the x & y intercepts
Ax + By = C
(A, B & C are not fractions or decimals)
Slopes of Parallel Lines
Writing Equations of Lines
Rewrite the following equation in slope-intercept form:
The equation is in standard form and we must solve for y to
put it in slope-intercept form.
Subtract 2x from both sides of the = sign
Divide both sides by 5.
Dec 3-4:25 PM
31
Writing Equations of Lines
Identify the slope and -intercept for the following linear equation:
Hint: The equation is in standard form, if we solve it for y we
will
click easily be able to identify the slope and the y-intercept.
Dec 3-4:25 PM
Writing Equations of Lines
Write the equation of a line that has a slope of 8 and passes
through the point (-6, 7) in point-slope form.
click
click
Rewrite the same equation in slope-intercept form.
click
click
click
Dec 3-4:25 PM
35 What is the slope of the line with the given equation
5x - 2y = 15 ?
A
B
C
D
3
-5/2
5/2
-3
Jun 25-5:40 PM
32
36 Choose the equation of the line in point-slope form that has a
slope of 1/4 and contains the point (-2, 7).
A
B
C
D
y + 7 = 1/4 (x + 2)
y = 1/4 x + 7.5
y - 7 = 1/4 (x - 2)
y - 7 = 1/4 (x + 2)
Jun 25-5:44 PM
37 Write the equation of the line in slope-intercept form
that contains the following points: (4,3) and (-8,6)
A
B
C
D
y = -1/4 x - 1
y = -1/4 x + 2
y=-1/4 x +4
y = -1/4 x + 7
Jul 2-3:46 PM
38 Determine the equation of the line in slope-intercept form.
A
B
C
D
y = 3/4 x + 3
y = 3/4 x - 4
3x - 4y = -12
y = 4/3 x + 3
Jun 25-5:48 PM
33
Slopes of Parallel Lines
To investigate slopes of parallel lines go to the lab
titled,"Slopes of Parallel Lines".
Click here to go to Slopes of
Lab 2
Slopes of Parallel Lines
Write a conjecture about slopes of parallel lines.
Jul 3-1:13 PM
Slopes of Parallel Lines
Two lines have the same slope if and only if they are parallel.
certain they are parallel, we
must evaluate the slope of
each line.
Red Line: (1, 2) and (5, 4)
m = 4 - 2 = 1/2
click
5-1
Blue Line: (0, -2) and (4, 0)
0 - (-2)
m=
= 1/2
click
4-0
The slopes of the two lines are the same.
Therefore they are parallel.
Slopes of Parallel Lines
34
Slopes of Parallel Lines
Slopes of Parallel Lines
Slopes of Parallel Lines
1. Determine whether LM and NO are parallel given the
following
information:
L: (3,-5) M: (-6,1) N: (4,-5) O: (7,-7)
: Evaluate the slope for each line
1 - (-5)
= -2/3
-6 - 3
click
m=
m=
-7 - (-5)
7
-4
click
= -2/3
: Compare the slopes of each line to determine if they
are parallel.
Slopes of Parallel Lines
Slopes of Parallel Lines
: Identify the information given in the problem.
: Identify what information you still need to create the
equation and choose the method to obtain it.
The slope: Use the equation of the parallel line to determine the
click
click
click
Therefore m = 3/2
Slopes of Parallel Lines
35
Slopes of Parallel Lines
: Create the equation.
click
Point-Slope Form
click
click
Slope-Intercept Form
The correct solution to the original problem is either form of the
equation. Point-Slope Form and Slope-Intercept Form are two
ways to write the same linear equations.
Dec 3-4:18 PM
39 What is the equation of the line passing through (6, -2) and
parallel to the line whose equation is y = 2x - 3?
A
B
C
D
y = 2x + 2
y = -2x + 10
y = 1/2 x - 5
y = 2x - 14
Jun 25-5:56 PM
40 Which is the equation of a line parallel to the line represented
by: y = -x - 22 ?
A
B
C
D
x - y = 22
y - x = 22
y + x = -17
2y + x = -22
Jun 25-5:58 PM
36
41 Two lines are represented by the equation:
-3y=12x-14 and y=kx+14
For which value of k will the lines be parallel?
A
B
C
D
12
-14
3
-4
Jun 25-6:02 PM
42 Which equation represents a line parallel to the line whose
equation is:
3y + 4x = 21
A
B
C
D
12y + 16x = 12
3y - 4x = 22
3y = 4x + 21
4y + 3x = 21
Jun 25-6:06 PM
43
Slopes of Parallel Lines
37
44
What is an equation of the line that passes
through the point (5,-2) and is parallel to the
Slopes of Parallel Lines
Perpendicular Lines
of Contents
Perpendicular Lines & Angles
Perpendicular Lines
Two lines are perpendicular if and only if they form 4
right angles when they intersect.
Perpendicular lines are also coplanar.
Dec 3-5:01 PM
38
Perpendicular Lines
In a plane, if a line is perpendicular to one of two parallel lines,
then it is perpendicular to the other.
click
click
Perpendicular Lines & Angles
45
Perpendicular Lines & Angles
46
Perpendicular Lines & Angles
39
47
Perpendicular Lines & Angles
48
Perpendicular Lines & Angles
49
AB || DF
Perpendicular Lines & Angles
40
Slopes of Perpendicular Lines
of Contents
Slopes of Perpendicular Lines
Dec 4-1:09 PM
Slopes of Perpendicular Lines
Negative Reciprocals combine opposites and reciprocals.
Finish filling out the table below.
Original
Number
Opposite
A.
-1/4
B.
D.
Negative
Reciprocal
Reciprocal
E.
C.
F.
H.
2/3
I.
Dec 4-1:15 PM
41
50
Slopes of Perpendicular Lines
51
Slopes of Perpendicular Lines
Slopes of Perpendicular Lines
To investigate slopes of perpendicular lines go to the lab
titled,"Slopes of Perpendicular Lines".
Click here to go to Slopes of
Perpendicular Lines
Lab 3
42
Slopes of Perpendicular Lines
Write a conjecture on slopes of perpendicular lines.
Jul 3-2:27 PM
Slopes of Perpendicular Lines
red
Perpendicular lines have negative
reciprocal slopes.
Observations
blue
-2/3
Slope of Blue Line = click
The red line rises, so it has a
positive slope.
click
Slope of Red Line = 3/2
What do you notice about the lines?
Slopes of Perpendicular Lines
Slopes of Perpendicular Lines
red
blue
their
Slope of Blue Line = click
1
Slope of Red Line = -1
click
Slopes of Perpendicular Lines
43
Slopes of Perpendicular Lines
blue
red
Slope of Blue Line = 4/5
click
Slope of Red Line = -2/3
click
Dec 4-1:54 PM
Slopes of Perpendicular Lines
Any horizontal line and vertical
are always perpendicular
because they form 4 right angles
at their point of intersection.
Slopes of Perpendicular Lines
52
Are these two lines perpendicular?
Slopes of Perpendicular Lines
44
53
Slopes of Perpendicular Lines
54
Slopes of Perpendicular Lines
55
Slopes of Perpendicular Lines
45
Writing Equations of Lines
Facts about slope that can assist with writing linear equations:
-Parallel Lines have theclick
SAME slope
click
click
click
-Perpendicular Lines have NEGATIVE
RECIPROCAL slopes
click
one another
Dec 4-2:09 PM
Writing Equations of Lines
: Identify the slope according to the given equation.
Given equation: m = click
1/2 Perpendicular Line: m = click
-2
y - 5 = -2 (x + 2)
click
Point-Slope Form
y - 5 = -2x - 4
click
yclick
= -2x + 1
Slope-Intercept Form
Slopes of Perpendicular Lines
Writing Equations of Lines
Dec 4-2:22 PM
46
56
the line whose equation is 4
Slopes of Perpendicular Lines
57
What is an equation of the line that contains the
equation is
Slopes of Perpendicular Lines
58
What would be the best statement to describe
these two lines?
=15
5(
Slopes of Perpendicular Lines
47
Proofs Involving Parallel
Perpendicular Lines
of Contents
Proofs
Proofs in Geometry
A proof is a logical list of steps that is used to reach a
conclusion. Each step is supported by a theorem, postulate,
definition, or property.
There are three types of proof.
1. 2-column proof
2. flowchart proof
3. paragraph proof
Jul 3-3:56 PM
Proofs in Geometry
Example of a 2 column proof
Proof of Same-Side Interior Angles Theorem.
a
b
Given: a || b
Prove: <2 and <3 are supplementary
1
2
3
NOTE: If we are going to prove Same-Side Interior Angles Theorem we cannot
use it as a reason in our proof.
Reason
Statement
1. a || b
2. <1 and <2 are
supplementary
3. m<1+m<2=180 0
4. <1 ≅<3
5. m<1=m<3
6. m<3+m<2=1800
7. <3 and <2 are
supplementary
1. Given
2. A linear pair of angles is supplementary. (Linear Pair
Postulate)
3. Def of supplementary
4. Alternate Interior Angles Theorem
5. Def. of Congruent Angles
6. Substitution
7. Definition of supplementary
Jul 3-4:46 PM
48
Proofs in Geometry
a
Example of a flowchart proof
Proof of Same-Side Interior Angles Theorem.
b
1
2
3
Given: a || b
Prove: <2 and <3 are supplementary
NOTE: If we are going to prove Same-Side Interior Angles Theorem we cannot
use it as a reason in our proof.
0
m<1+m<2=180
a || b
given
<1 and <2 are
supplementary
Def. of
Supplementary
Linear Pair
Postulate
<3 and <2 are
supplementary
0
m<3+m<2=180
Substitution
Def. of
supplementary
m<1=m<3
<1≅<3
Def. of congruent
angles
Alt. Int. <'s
Theorem
Jul 5-11:02 AM
Proofs in Geometry
a
Example of a paragraph proof
Proof of Same-Side Interior Angles Theorem.
b
1
2
3
Given: a || b
Prove: <2 and <3 are supplementary
NOTE: If we are going to prove Same-Side Interior Angles Theorem we cannot
use it as a reason in our proof.
Because a || b we know that <1≅<3 by alternate interior angles
theorem, which also makes m<1 = m<3 by def. of congruent angles.
We also know that <1 and <2 are supplementary by linear pair
postulate, which implies that m<1 + m<2 = 180 by definition of
supplementary. By substitution m<3 + m<2 = 180. Therefore, <3 and
<2 are supplementary by definition of supplementary.
0
0
Jul 5-11:02 AM
Proofs in Geometry
The justifications in a proof are made up of definitions,
theorems, postulates, and properties.
Theorems and Postulates
Vertical Angles Theorem
Segment Addition Postulate
Linear Pair Postulate
Angle Addition Postulate
Parallel and Perpendicular Postulate Congruent supplements / compliments
thm
Corresponding Angles Theorem
Converse of Corresponding Angles Thm
Alternate Interior Angles Theorem
Converse of Alternate Interior Angles Thm
Same-Side Angles Theorem
Converse of Same-Side Angles Thm.
Alternate Exterior Angles Theorem
Converse of Exterior Angles Thm.
Jul 5-11:31 AM
49
Proofs in Geometry
Proofs
Proofs in Geometry
Substitution Property
, then either a or b may be
substituted for the other in any equation/inequality
Reflexive Property
Symmetric Property
Distributive Property
Dec 5-9:45 AM
Proofs in Geometry
Reflexive Property:
Symmetric Property:
Transitive Property:
If
, then
If
, then
If
and
, then
If
and
, then
Proofs
50
Proofs in Geometry
Complete the proof.
Given: k || m
Prove: Alternate Interior Angles Theorem
NOTE: If we are going to prove Alternate Interior Angles Theorem we cannot use it
as a reason in our proof.
Reason
3)
4) Transitive Property
Dec 5-10:07 AM
Proofs in Geometry
Unscramble the list of reasons in the following proof.
0
Reasons
a) Same-Side Angles Thm.
b) Substitution Property
c) Given
0
d) Vertical Angles Thm.
0
e) Def. of Congruent Angles
Proofs
Proofs in Geometry
Supply the missing reasons in the following proof.
1
4
Prove vertical angles theorem.
2
3
NOTE: If we are going to prove Vertical Angles Theorem we cannot use it as a
reason in our proof.
<1 and <2 form
a linear pair
<1 and <2 are
supplementary
1. _____________
3. ____________
m<1+m<2 = 1800
5. ____________
m<1= m<3
7. ____________
<2 and <3 form
a linear pair
2. ____________
<2 and <3 are
supplementary
4. ____________
m<2+m<3=180
0
6. ____________
Jul 5-12:13 PM
51
Proofs in Geometry
Fill in the blanks in the following paragraph proof.
0
Given: m<2 = 125, m<4 = 55 0
Prove: k || m
0
0
It is given that m<2 = 125 and m<4 = 55 and we know that 1250 +
0
0
55 = 180 which implies that m<2 + m<4 = 1800 by a)____. If m<2 +
m<4 = 180 then <2 and <4 are supplementary by definition of
b._____. If <2 and <4 are supplementary them k || m by the
converse of c._____.
Jul 5-12:30 PM
Proofs in Geometry
Unscramble the reasons in the following proof.
Given: <A
Prove: ABCD is a parallelogram
Reasons
Statements
1. <A≅<C; <B≅<D
2. m<A=m<C; m<B=m<D
3. m<A+m<B+m<C+m<D=
0
360
4. m<A+m<B+m<A+m<B=
0
360
5. 2(m<A+m<B) = 3600
i. given
j. def. of congruence
h. The sum of the interior angles of a
quad is 3600
g. Substitution property
d. Distributive Property
6. m<A+m<B = 1800
b. Division property of equality
7. <A and <B are supplementary
f. def. of supplementary angles
8. <C and <D are supplementary
a. substitution property
e. converse of the same side angles theorem.
9. BC || AD; AB || CD
10. Quad ABCD is a parallelogram c. Definition of parallelogram
Jul 9-11:08 AM
Proofs in Geometry
Supply the missing reasons in the following proof.
Given: BD || AC
0
Prove: m<2+m<4+m<5 = 180
Statements
Reasons
__ __
1. BD || AC
~
~
2. <1=<4;
<5=<3
1.
2.
3. m<1=m<4; m<5=m<3
4. m<1+m<2+m<3=180
5. m<4+m<2+m<5=180
3.
0
0
4.
5.
Jul 9-11:08 AM
52
59 If a || b, how can we prove m<1=m<4?
A
B
C
D
Corresponding angles postulate
Converse of corresponding angles postulate
Alternate Interior angles theorem
Converse of alternate interior angles theorem
b
a
3
2 1
c
4
Jul 5-3:20 PM
60 If m<1=m<3, how can we prove a || b?
A
B
C
D
Corresponding angles postulate
Converse of corresponding angles postulate
Alternate Interior angles theorem
Converse of alternate interior angles theorem
b
a
3
2 1
c
4
Jul 5-3:27 PM
61 What is the justification for the missing component
in the provided proof?
Given:
Prove:
, AB ||DE
Justification
1)
, AB ||DE
1)
2)
2) Alternate Interior Angles Theorem
3)
3) Corresponding Angles Postulate
4)
4) Transitive
5)
5) Transitive
6)
6) Def. of bisector
of BDC
A Alternate Interior
Angles Theorem
B Same-Side Interior
Angles Theorem
C Corresponding
Angles Theorem
D Given
Dec 5-10:51 AM
53
62 What is the justification for the missing component
in the provided proof?
Justification
vertical angles thm
Transitive
Angles
Converse Theorem
Angles
Converse Theorem
Angles
Converse Theorem
Definition of
Supplementary Angles
Dec 5-10:36 AM
63 What is the missing statement in the provided proof?
Given:
Prove: k || m
Justification
1) Given
Def. perpendicular Lines & Def. of
right angle
Def. perpendicular Lines & Def. of
right angle
Substitution
5)
5) Def. of Congruent Angles
6) k || m
6) Corresponding Angles Converse
Theorem
A
B
C
D
Dec 5-10:47 AM
64 Given m<1=m<2, m<3=m<4, what can we prove?
a
A
B
C
D
a || b
c || d
line a is perpendicular to line c
line b is perpendicular to line d
b
1
2
4
3
5
c
d
Jul 5-1:59 PM
54
65 Given c || d, what can we prove?
a
A
B
C
D
m<1 = m<2
m<4 = m<5
m<2 = m<3
m<2 + m<5 = 180
b
1
2
4
3
5
c
d
Jul 5-2:05 PM
Constructing Parallel Lines
of Contents
Constructing Parallel Lines
Parallel Line Construction
Constructing geometric figures means you are constructing lines,
angles, and figures with basic tools accurately.
We use a compass, and straightedge for constructions, but we also
use some paper folding techniques.
Construction by: MathIsFun
Jul 9-4:37 PM
55
Parallel Line Construction
Have students do constructions with the following slides.
Jul 9-4:45 PM
Parallel Line Construction
When asked to construct a line through a point parallel to a
given line, there are
Method 1
Given
Constructing Parallel Lines
Construction Continued
0°
34
0°
24
Dec 5-11:11 AM
56
0°
24
: Mark the arc intersection
point E and use a ruler to join C
therefore
Constructing Parallel Lines
Video Demonstrating Constructing
Parallel Lines with Corresponding Angles
using Dynamic Geometric Software
Click here to see video
Video
Given
Constructing Parallel Lines
57
0°
24
0°
24
Dec 5-11:43 AM
0°
24
: Mark the arc intersection point E
and use a ruler to join C and E.
therefore
Constructing Parallel Lines
Video Demonstrating Constructing
Parallel Lines with Alternate Interior Angles
using Dynamic Geometric Software
Click here to see video
Video
58
Method 3
Given
Constructing Parallel Lines
24
0°
0°
24
Dec 5-11:51 AM
24 0°
: Mark the arc
intersection point E and use a
ruler to join C and E.
therefore
Dec 5-11:51 AM
59
Video Demonstrating Constructing
Parallel Lines with Alternate Exterior
Angles using Dynamic Geometric Software
Click here to see video
Video
Parallel Line Construction Using Patty Paper
Step 1: Draw a line on your patty paper. Label the line g. Draw
a point not on line g and label the point B.
Step 2: Fold your patty paper so that the two parts of line g lie
exactly on top of each other and point B is in the crease.
Jun 26-12:52 PM
Parallel Line Construction Using Patty Paper
Step 3: Open the patty paper and draw a line on the crease.
Label this line h.
Step 4: Through point B, make another fold that is perpendicular
to line h.
Jun 26-1:09 PM
60
Step 5: Open the patty paper and draw a line on the crease.
Label this line i.
Because lines i and g are perpendicular to line h they are
parallel to each other. Therefore line i || line g.
Jun 26-3:21 PM
Video Demonstrating Constructing a
Parallel Line using Menu Options of
Dynamic Geometric Software
Click here to see video 1
Click here to see video 2
Video
66
The lines in the diagram below are parallel because
Alternate Interior Angles Theorem
Alternate Exterior Angles Theorem
Same-Side Angles Theorem
Corresponding Angles Postulate
Dec 5-12:02 PM
61
67
The lines in the diagram below are parallel because
Alternate Interior Angles Theorem
Alternate Exterior Angles Theorem
Same-Side Angles Theorem
Corresponding Angles Postulate
Dec 5-12:02 PM
68
The lines in the diagram below are parallel because
Alternate Interior Angles Theorem
Alternate Exterior Angles Theorem
Same-Side Angles Theorem
Corresponding Angles
Postultate
Dec 5-12:09 PM
Constructing
Perpendicular Lines
of Contents
Constructing Perpendicular Lines
62
Perpendicular Line Construction
Have students do constructions with the following slides.
Construction by: MathIsFun
Jul 9-4:53 PM
Constructing a Perpendicular Line
Construct a line through a point perpendicular to a given line.
0°
118
Constructing Perpendicular Lines
Construction Continued
0°
118
Dec 4-3:30 PM
63
Name the point of intersection of the two arcs as F.
Constructing Perpendicular Lines
Example
Constructing Perpendicular Lines
Example Continued
: From points A and B, draw 2 intersecting arcs below line
using the same compass width.
Dec 4-3:56 PM
64
Dec 4-3:51 PM
Video Demonstrating Constructing a
Perpendicular Line with a Compass and
Straightedge using Dynamic Geometric
Software
Click here to see video
Video
Constructing a Perpendicular Line Using
Patty Paper
Step 1: Draw a line on your patty paper. Label the line g. Draw a
point not on line g and label the point B.
Step 2: Fold your patty paper so that the two parts of line g lie
exactly on top of each other and point B is in the crease.
Jun 26-3:30 PM
65
Step 3: Open the patty paper and draw a line on the crease. Label
this line h.
Line h is perpendicular to line line g.
Jun 26-3:34 PM
Video Demonstrating Constructing a
Perpendicular Line using Menu Options of
Dynamic Geometric Software
Click here to see video
Video
Constructions
Try This:
0
1. Construct a 60 angle
Click here to view the
animated construction of
a 60 angle.
Construction by: Mathisfun
2. Construct a regular hexagon in a circle.
Click here to view the
animated construction of
a hexagon inscribed in a circle.
Construction by: MathOpenReference
Jul 9-6:00 PM
66
Videos Demonstrating Constructing a 60
Degree Angles and a Hexagon using
Dynamic Geometric Software
Click here to see
60 degree angle video
Click here to see
hexagon video
Video
69
Which diagram represents the construction of the
perpendicular bisector of MN?
Constructing Perpendicular Lines
67