1. Parallel Lines and Planes
Chapter 3
Parallel Lines and Planes
By: Jessica House and Yana Feldman

2. 3-1 Definitions
Parallel lines A B C D l n
l and n are parallel lines
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 ║
Skew lines k 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. X Y Z n l
Theorem 3-1
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
Statements
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
Reasons
Given
Definition of coplanar
Parallel planes do not intersect. (Definition of ║ planes)
Definition of ║lines (2,4)

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
Alternate interior angles: two nonadjacent interior angles on opposite sides of the transversal
3 and and
Same-side interior angles: two interior angles on the same side of transversal
3 and and 6
Corresponding angles: two angles in corresponding positions relative to the two lines
1 and and and and 8 h t k 1 2 3 4 5 6 7 8

5. 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

6. 3-2
Theorem 3-3 k t n 1 2 4
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 3-4 l t n 1 2

7. 3-2
This type of arrow usage shows that the lines are parallel (2 arrows show its talking about a different line)

8. 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)
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 k t n 1 2 3

9. 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 3-6 t
Theorem 3-7 k t n 1 2

10. 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

11. 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

12. 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
No sides congruent
At least two sides congruent
All sides congruent
Scalene Triangle Isosceles Triangle Equilateral Triangle
Acute∆ Obtuse∆ Right∆ Equiangular∆
Three acute s One obtuse One right All s congruent

13. 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. A B 1 2 3 4 5 D
Theorem 3-11 and auxiliary line usage C

14. 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.

15. 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

16. 3-5 Angles of a Polygon Polygon: “many angles”
Formed by coplanar segments such that
Each segment intersects exactly two other segments, one at each endpoint.
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

17. 3-5 Number of Sides Name 3 4 5 6 7 8 9 10 11 12 13 14 15 n triangle
quadrilateral
pentagon
hexagon
heptagon
octagon
nonagon
decagon
undecagon
dodecagon
tridecagon
tetracagon
pentadecagon
n-gon

18. 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
Diagonals:
4 sides triangles Angle sum= 2(180)
5 sides triangles Angle sum= 3(180)
6 sides
4 triangles
Angle sum= 4(180)

19. 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
This is a hexagon that is neither equiangular nor equilateral.
Not a regular polygon
Equilateral triangle Not a regular polygon
120
Equiangular hexagon Not a regular polygon
Regular hexagon

20. So far we have only used deductive reasoning
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

21. The End