3-1 Definitions
Parallel lines
l
• Parallel lines (║): coplanar
lines that do not intersect
A
B
C
n
D
l and n are parallel
lines
Skew lines
k
• Skew lines: noncoplanar lines
• Neither ll nor intersecting
• Look perpendicular but not
because in 2 different planes
j
j and k are skew lines
S
• Segments and rays contained
in ║ lines are also ║
• Parallel planes: do not intersect
• A line and a plane are ║ if they do not intersect
R
P
Q
PQ and RS do not intersect, but
they are parts of lines PQ and
RS that do intersect.
Thus PQ is NOT parallel to RS
•Theorem 3-1: If two 3-1
parallel planes are cut by a
third plane, then the lines of
intersection are parallel.
X
l
Theorem 3-1
n
Y
Z
Given: Plane X ║ plane Y; plane Z intersects
X in line l; plane Z intersects Y in line n.
Prove: l ║ n
Reasons
Statements
1.
2.
3.
4.
5.
l is in Z; n is in Z
l and n are coplanar
l is in X; n is in Y; X ║ Y
l and n do not intersect.
l║n
1.
2.
3.
4.
Given
Definition of coplanar
Given
Parallel planes do not
intersect. (Definition of ║
planes)
5. Definition of ║lines (2,4)
3-1
• Transversal: a line that intersects two or
more coplanar lines in different points.
– Interior angles: angles 3, 4, 5, 6
– Exterior angles: angles 1, 2, 7 ,8
h
• Alternate interior angles: two
nonadjacent interior angles on opposite
sides of the transversal
–
3 and
6
4 and
5
–
3 and
5
4 and
6
• Corresponding angles: two angles in
corresponding positions relative to the
two lines
–
1 and 5
4 and 8
2 and
6
3
k
5
7
• Same-side interior angles: two interior
angles on the same side of transversal
3 and
7
t
6
8
1 2
4
3-2 Properties of Parallel Lines
• Postulate 10: If two ║ lines are cut by a
transversal, then corresponding angles are
congruent.
•Theorem 3-2: If two ║
lines are cut by a
transversal, then
alternate interior angles
are congruent.
Theorem 3-2
t
k
3
1
n
2
3-2
Theorem 3-3
• Theorem 3-3: If two ║ lines
are cut by a transversal,
then same-side interior
angles are supplementary.
• Theorem 3-4: If a
transversal is
perpendicular to one of two
║lines, then it is
perpendicular to the other
one also.
t
k
n
1
4
2
Theorem 3-4
t
l
n
1
2
3-2
This type of arrow usage shows that the
lines are parallel (2 arrows show its
talking about a different line)
3-3 Proving Lines Parallel
• Postulate 11: If two lines are cut by a transversal
and corresponding angles are congruent, then the
lines are ║. (the following theorems of this section 35,3-6, and 3-7 can be deducted from this postulate)
The following theorems in section 3-3 are the converses of the theorems
in section 3-2
•Theorem 3-5: If two lines
are cut by a transversal
and alternate interior
angles are congruent, then
the lines are ║.
Theorem 3-5
t
k
3
2
n
1
3-3
• Theorem 3-6: If two lines
are cut by a transversal
and same-side interior
angles are supplementary,
then the lines are ║.
• Theorem 3-7: In a plane
two lines perpendicular to
the same line are ║.
Theorem 3-6
t
k
1
n
2
3
Theorem 3-7
t
k
1
n
2
3-3
• Theorem 3-8: Through a point outside a line,
there is exactly one line ║ to the given line.
• Theorem 3-9: Through a point outside a line,
there is exactly one line perpendicular to the
given line.
Theorem 3-10: Two lines
║ to a third line are ║ to
each other.
Theorem 3-10
k
l
n
3-3
Ways to Prove Two Lines Parallel
show that a pair of corresponding angles are congruent
show that a pair of alternate interior angles are congruent
show that a pair of same-side interior are supplementary
in a plane show that both lines are perpendicular to a 3rd line
show that both lines are parallel to a 3rd line
3-4 Angles of a Triangle
• Triangle: the figure formed by three segments
joining three noncollinear points
• Each point is a vertex (plural form is vertices)
• Segments are sides of triangle
Triangles can be classified by number of congruent sides
Scalene Triangle
No sides congruent
Acute∆
Three acute
Isosceles Triangle
Equilateral Triangle
At least two sides congruent
s
Obtuse∆
One obtuse
Right∆
One right
All sides congruent
Equiangular∆
All s congruent
3-4
• Auxiliary line: a line (or ray or segment) added to
a diagram to help in a proof
Auxiliary lines allow you to add additional things to your
pictures in order to help in proofs-this becomes very
important in proofs (in the following diagram the auxiliary
line is shown as a dashed line)
• Theorem 3-11: The sum of the measures of the
angles of a triangle is 180.
D
B
4
A
1
2
5
3
C
Theorem 3-11
and auxiliary
line usage
3-4
• Corollary: a statement that can be proved easily by
applying a theorem.
• Like theorems, corollaries can be used as reasons for proofs
• The following 4 corollaries are based off of theorem 3-11: the
sum of the measures of the angles of a triangle is 180
• Corollary 1: If two angles of one triangle are congruent to
two angles of another triangle, then the third angles
are congruent.
• Corollary 2: Each angle of an equiangular triangle has
measure 60.
• Corollary 3: In a triangle there can be at most one right
angle or obtuse angle.
• Corollary 4: The acute angles of a right triangle are
complementary.
3-4
• Theorem 3-12: The measure of an
exterior angle of a triangle equals the sum
of the measures of the two remote interior
angles.
B
30 °
Exterior angle
Remote interior
angles
150 °
30°
C
Theorem 3-12
120 °
A
3-5 Angles of a Polygon
•
Polygon: “many angles”
– Formed by coplanar segments such that
1. Each segment intersects exactly two other
segments, one at each endpoint.
2. No two segments with a common endpoint are
collinear.
•
Convex polygon: a polygon such that no
line containing a side of the polygon
contains a point in the interior of the
polygon
Number of Sides
3
4
5
6
7
8
9
10
11
12
13
14
15
n
3-5
Name
–
–
–
–
–
–
–
–
–
–
–
–
–
–
triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
decagon
undecagon
dodecagon
tridecagon
tetracagon
pentadecagon
n-gon
3-5
• When referring to polygons, list consecutive vertices in order
• Diagonal: a segment joining two nonconsecutive vertices (indicated
by dashes)
• Finding sum of measures of angles of a polygon:
• draw all diagonals from one vertex to divide polygon into
triangles
Diagonals:
4 sides
2 triangles
Angle sum= 2(180)
5 sides
3 triangles
Angle sum= 3(180)
6 sides
4 triangles
Angle sum= 4(180)
• Theorem 3-13: The sum of the measures of the angles of a convex
polygon with n sides is (n - 2)180
3-5
• Theorem 3-14: The sum of the measures of the exterior
angles of any convex polygon, one angle at each vertex, is
360
• Regular Polygon: must be both equiangular and equilateral
120°
120 °
120 °
This is a hexagon that is
neither equiangular nor
equilateral.
Not a regular
polygon
120 °
120°
120 °
120°
120 °
120 °
Equiangular hexagon
Not a regular polygon
120°
120°
Equilateral triangle
Not a regular polygon
120°
Regular hexagon
3-6 Inductive Reasoning
Deductive Reasoning
Conclusion based on
accepted statements
(definitions, postulates,
previous theorems,
corollaries, and given
info)
Inductive Reasoning
Conclusion based on
several past observations
Conclusion is
PROBABLY true, but
necessarily true
Conclusion MUST be true
if the hypothesis is true
So far we have only used deductive reasoning