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

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