Geometry Winter Semester (2012-13)
Name: ______________
Chapter 3 Guided Notes
Parallel and Perpendicular Lines
Chapter Start Date:_____
Chapter End Date:_____ Chapter Test Date:_____
CH.3 Guided Notes, page 2
3.1 Identify Pairs of Lines and Angles
Term Definition Example
parallel lines (// or ||)
parallel to (// or ||)
not parallel to
(// or ||)
skew lines
parallel planes
What is a ‘Named Theorem’ ?
Postulate 13 Parallel Postulate
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.
Postulate 14 Perpendicular
Postulate
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.
transversal
The lines the transversal intersects do not
need to be parallel; the transversal can also
be a ray or line segment.
CH.3 Guided Notes, page 3
Special Angles formed by Transversals
exterior angles
interior angles
corresponding angles
alternate interior angles
alternate exterior angles
consecutive (same-side)
interior angles
consecutive (same-side)
exterior angles
CH.3 Guided Notes, page 4
3.2 Use Parallel Lines and Transversals
Term Definition Example
Postulate 15 Corresponding
Angles Postulate (not ‘named’)
If two parallel lines are cut by a transversal, then
the pairs of corresponding angles are congruent.
Proof Abbrieviation:
Theorem 3.1 Alternate
Interior Angles Theorem
(not ‘named’)
If two parallel lines are cut by a transversal, then
the pairs of alternate interior angles are
congruent.
Proof Abbrieviation:
Theorem 3.2 Alternate
Exterior Angles Theorem
(not ‘named’)
If two parallel lines are cut by a transversal, then
the pairs of alternate exterior angles are
congruent.
Proof Abbrieviation:
Theorem 3.3 Same-Side
Interior Angles Theorem
(not ‘named’)
If two parallel lines are cut by a transversal, then
the pairs of Same-Side Interior (AKA Consecutive
Interior) angles are supplementary.
Proof Abbrieviation:
BONUS Theorem Same-Side
Exterior Angles Theorem
(not ‘named’)
If two parallel lines are cut by a transversal, then
the pairs of Same-Side Exterior angles are
supplementary.
Proof Abbrieviation:
CH.3 Guided Notes, page 5
3.3 Prove Lines are Parallel
Term Definition Example
Postulate 16 Corresponding
Angles Converse (not ‘named’)
If two lines are cut by a transversal so the
corresponding angles are congruent, then the lines
are parallel.
Proof Abbrieviation:
Theorem 3.4 Alternate
Interior Angles Converse
(not ‘named’)
If two lines are cut by a transversal so the
alternate interior angles are congruent, then the
lines are parallel.
Proof Abbrieviation:
Theorem 3.5 Alternate
Exterior Angles Converse
(not ‘named’)
If two lines are cut by a transversal so the
alternate exterior angles are congruent, then the
lines are parallel.
Proof Abbrieviation:
Theorem 3.6 Same-Side
Interior Angles Converse
(not ‘named’)
If two lines are cut by a transversal so the Same-
Side (Consecutive) Interior angles are
supplementary, then the lines are parallel.
Proof Abbrieviation:
BONUS Theorem Same-Side
Exterior Angles Converse
If two parallel lines are cut by a transversal, then
the pairs of Same-Side Exterior angles are
supplementary.
Proof Abbrieviation:
paragraph proof
CH.3 Guided Notes, page 6
Theorem 3.7 Transitive Property of Parallel Lines
If two lines are parallel to the same line, then they
are parallel to each other.
CH.3 Guided Notes, page 7
3.4 Find and Use Slopes of Lines
Term Definition Example
slope
positive slope
negative slope
zero slope (slope of zero)
(no slope)
A horizontal line.
undefined slope
A vertical line.
Postulate 17 Slopes of
Parallel Lines
In a coordinate plane, two nonvertical lines
are parallel if and only if they have the
same slope.
Any two vertical lines are parallel!
Postulate 18 Slopes of
Perpendicular Lines
In a coordinate plane, two nonvertical lines
are perpendicular if and only if the product
of their slopes is -1.
The slopes of the two lines that are
perpendicular are negative reciprocals of
each other. Horizontal lines are
perpendicular to vertical lines.
“if and only if” form (iff)
The form used when both a conditional and
its converse are true.
CH.3 Guided Notes, page 8
3.5 Write and Graph Equations of Lines
Term Definition Example
slope-intercept
form
standard form
x-intercept
y-intercept
CH.3 Guided Notes, page 9
Chap3 Constructing Parallel & Perpendicular Lines Remember that the complete construction guide (all 7 Basic constructions) has been posted online at www.behmermath.weebly.com 4. Construct the perpendicular bisector of a line segment.
Or, construct/find the midpoint of a line segment.
1. Begin with line segment XY. YX
2. Place the compass at point X. Adjust the compass radius so that it is more than (½)XY. Draw two arcs as shown here.
YX
3. Without changing the compass radius, place the compass on point Y. Draw two arcs intersecting the previously drawn arcs. Label the intersection points A and B.
A
B
YX
4. Using the straightedge, draw line AB. Label the intersection point M. Point M is the midpoint of line segment XY, and line AB is perpendicular to line segment XY.
X Y
A
B
M
CH.3 Guided Notes, page 10 5. Given a point (P) ON a line (k), construct a line through P, perpendicular to k.
1. Begin with line k, containing point P. k
P
2. Place the compass on point P. Using an arbitrary radius, draw arcs intersecting line k at two points. Label the intersection points X and Y.
k
P YX
3. Place the compass at point X. Adjust the compass radius so that it is more than (½)XY. Draw an arc as shown here. k
P YX
4. Without changing the compass radius, place the compass on point Y. Draw an arc intersecting the previously drawn arc. Label the intersection point A. k
A
P YX
5. Use the straightedge to draw line AP. Line AP is perpendicular to line k.
k
A
X YP
CH.3 Guided Notes, page 11 6. Given a point (R), NOT ON a line (k), construct a line through R, perpendicular to k.
1. Begin with point line k and point R, not on the line. k
R
2. Place the compass on point R. Using an arbitrary radius, draw arcs intersecting line k at two points. Label the intersection points X and Y.
kR
YX
3. Place the compass at point X. Adjust the compass radius so that it is more than (½)XY. Draw an arc as shown here.
kYX
R
4. Without changing the compass radius, place the compass on point Y. Draw an arc intersecting the previously drawn arc. Label the intersection point B.
k
B
YX
R
5. Use the straightedge to draw line RB. Line RB is perpendicular to line k.
k
B
YX
R
CH.3 Guided Notes, page 12 7. Given a line & a point not on the line, construct a line through the point, parallel to the given line.
1. Begin with point P and line k.
k
P
2. Draw an arbitrary line through point P, intersecting line k. Call the intersection point Q. Now the task is to construct an angle with vertex P, congruent to the angle of intersection.
kQ
P
3. Center the compass at point Q and draw an arc intersecting both lines. Without changing the radius of the compass, center it at point P and draw another arc.
kQ
P
4. Set the compass radius to the distance between the two intersection points of the first arc. Now center the compass at the point where the second arc intersects line PQ. Mark the arc intersection point R. k
R
Q
P
5. Line PR is parallel to line k.
k
R
Q
P
CH.3 Guided Notes, page 13
3.6 Prove Theorems about Perpendicular Lines
Term Definition Example
Theorem 3.8 (not ‘named’)
If two lines intersect to form a linear pair
of congruent angles, then the lines are
perpendicular.
Theorem 3.9 (not ‘named’)
If two lines are perpendicular, then they
intersect to form four right angles.
Theorem 3.10 (not ‘named’)
If two sides of two adjacent acute angles
are perpendicular, then the angles are
complementary.
Theorem 3.11 Perpendicular Transversal Theorem
(not ‘named’)
If a transversal is perpendicular to one of
two parallel lines, then it is perpendicular to
the other.
Theorem 3.12 Lines
Perpendicular to a Transversal
Theorem (not ‘named’)
In a plane, if two lines are perpendicular to
the same line, then they are parallel to each
other.
distance from a point to a line
distance between two parallel lines
