Chapter 3
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Transcript of Chapter 3
3-1 Definitions• Parallel lines (║): coplanar
lines that do not intersect
• Skew lines: noncoplanar lines• Neither ll nor intersecting • Look perpendicular but not
because in 2 different planes
• Segments and rays contained in ║ lines are also ║
Parallel linesA B
C D
l
n
l and n are parallel lines
Skew linesk
j
j and k are skew lines
P Q
R
S
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
• Parallel planes: do not intersect• A line and a plane are ║ if they do not intersect
3-1
Statements1. l is in Z; n is in Z
2. l and n are coplanar
3. l is in X; n is in Y; X ║ Y
4. l and n do not intersect.
5. l ║ n
Reasons1. Given2. Definition of coplanar3. Given4. Parallel planes do not
intersect. (Definition of ║ planes)
5. Definition of ║lines (2,4)
X
Y
Z
n
lTheorem 3-1
•Theorem 3-1: If two parallel planes are cut by a third plane, then the lines of intersection are parallel.
Given: Plane X ║ plane Y; plane Z intersects X in line l; plane Z intersects Y in line n.
Prove: l ║ n
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
• Alternate interior angles: two nonadjacent interior angles on opposite sides of the transversal– 3 and 6 4 and 5
• Same-side interior angles: two interior angles on the same side of transversal– 3 and 5 4 and 6
• Corresponding angles: two angles in corresponding positions relative to the two lines– 1 and 5 2 and 6 3 and 7
4 and 8
h t
k
1 234
5 6
7 8
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
k
t
n
1
2
3
3-2
• 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.
Theorem 3-3
kt
n 1
24
Theorem 3-4
l
t
n
1
2
This type of arrow usage shows that the lines are parallel (2 arrows show its talking about a different line)
3-2
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 3-5,3-6, and 3-7 can be deducted from this postulate)
•Theorem 3-5: If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are ║.
The following theorems in section 3-3 are the converses of the theorems in section 3-2
Theorem 3-5
kt
n 1
2
3
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 ║.
k
n
1
2 3
Theorem 3-6t
Theorem 3-7
k
t
n
1
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 ParallelWays to Prove Two Lines Parallel
show that a pair of corresponding angles are congruentshow that a pair of corresponding angles are congruent
show that a pair of alternate interior angles are congruentshow that a pair of alternate interior angles are congruent
show that a pair of same-side interior are supplementary show that a pair of same-side interior are supplementary
in a plane show that both lines are perpendicular to a 3in a plane show that both lines are perpendicular to a 3rdrd line line
show that both lines are parallel to a 3show that both lines are parallel to a 3rdrd line 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 Isosceles Triangle Equilateral TriangleNo sides congruent At least two sides congruent All sides congruent
Acute∆ Obtuse∆ Right∆ Equiangular∆Three acute s One obtuse One right 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.
CA
B
1
2
3
4 5
D
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.
A
B
C
120 150
30
° °
°
Theorem 3-12
Exterior angle
Remote interior angles
30°
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
3-5Number of Sides
3456789
101112131415n
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
• Theorem 3-13: The sum of the measures of the angles of a convex polygon with n sides is (n - 2)180
4 sides 2 triangles
Angle sum= 2(180)
Diagonals:
5 sides 3 triangles
Angle sum= 3(180)
6 sides
4 triangles
Angle sum= 4(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
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 Equilateral triangle
Not a regular polygon
120° 120°
120°
120° 120°
120°
Regular hexagon
3-6 Inductive Reasoning
Deductive Reasoning
Conclusion based on accepted statements (definitions, postulates, previous theorems, corollaries, and given info)
Conclusion MUST be true if the hypothesis is true
Inductive Reasoning
Conclusion based on several past observations
Conclusion is PROBABLY true, but necessarily true
So far we have only used deductive reasoning