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