# Chapter 3 Guided Notes Parallel and Perpendicular ?· Chapter 3 Guided Notes Parallel and Perpendicular…

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• 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 ()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. kP

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 ()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 ()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