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3-1 Lines and Angles
3-2 Angles Formed by Parallel Lines and Transversals
3-3 Proving Lines Parallel
3-4 Perpendicular Lines
3-5 Slopes of Lines
3-6 Lines in the Coordinate Plane
3-1 Lines and AnglesParallel Lines (||): Coplanar lines that do not
intersect.Perpendicular Lines: ( ): Lines that intersect
in a 90 degree angle.
3-1 Lines and AnglesSkew Lines: Not coplanar. Neither parallel and
do not intersectParallel Planes: planes that do not intersect.
3-1 Lines and AnglesTransversal: line that intersects two or more
coplanar lines in different points.
h
k
t
Two coplanar lines
Transversal- the line of intersection
3-1 Lines and Angles
h
k
t
1 23 4
5 6
7 8
Interior Angles-
Angles 3, 4, 5, 6
Exterior Angles-
Angles 1, 2, 7, 8
h
k
t
1 23 4
5 6
7 8
Same side interior angles:∠3 and ∠5, ∠4 and ∠6
Alternate interior angles:∠3 and ∠6, ∠4 and ∠5
Corresponding Angles∠1 and ∠5, ∠2 and ∠6, ∠3 and ∠7, ∠4 and ∠8
3-1 Lines and Angles
Alternate exterior angles:∠2 and ∠7, ∠1 and ∠8
3-1 Lines and Angles
3-2 Angles Formed by Parallel Lines and Transversals
3-3 Proving Lines Parallel
3-4 Perpendicular Lines
3-5 Slopes of Lines
3-6 Lines in the Coordinate Plane
3-2 Angles Formed by Parallel Lines and TransversalsPostulate 3-2-1: Corresponding Angles
PostulateIf two parallel lines are cut by a transversal,
then the pairs of corresponding angles are congruent.
k
n
t
3-2 Angles Formed by Parallel Lines and TransversalsTheorem 3-2-2: Alternate Interior Angles
TheoremIf two parallel lines are cut by a transversal,
then alternate interiors angles are congruent.
k
n
t
3-2 Angles Formed by Parallel Lines and TransversalsTheorem 3-2-3: Alternate Exterior Angles
TheoremIf two parallel lines are cut by a transversal,
then alternate exterior angles are congruent.k
n
t
3-2 Angles Formed by Parallel Lines and TransversalsTheorem 3-2-4: Same-Side Interior Angles
TheoremIf two parallel lines are cut by a transversal,
then same side interior angles are supplementary. k
n
t
3-1 Lines and Angles
3-2 Angles Formed by Parallel Lines and Transversals
3-3 Proving Lines Parallel
3-4 Perpendicular Lines
3-5 Slopes of Lines
3-6 Lines in the Coordinate Plane
3-3 Proving Lines ParallelTheorems: Write the converse of each of the four if-then
statements we created last period. If two parallel lines are cut by a transversal, then the pairs of corresponding
angles are congruent. If two parallel lines are cut by a transversal, then alternate interiors angles are
congruent. If two parallel lines are cut by a transversal, then alternate exterior angles are
congruent. If two parallel lines are cut by a transversal, then same side interior angles are
supplementary.What do these things mean to you? This is how we SHOW that any two lines could be/are parallel.
3-1 Lines and Angles
3-2 Angles Formed by Parallel Lines and Transversals
3-3 Proving Lines Parallel
3-4 Perpendicular Lines
3-5 Slopes of Lines
3-6 Lines in the Coordinate Plane
3-4 Perpendicular LinesPerpendicular Bisector: a line or segment that
is perpendicular and goes through the midpoint of a segment.
A B
3-4 Perpendicular LinesConstruction: Perpendicular through a point
outside the line.
C D
X
3-4 Perpendicular LinesWrite an If-then statement based on info in the
table.Hypothesis Conclusion Conditional Statement
a ⊥ b
h ⊥ d
j || k
3-4 Perpendicular Lines
Theorem 3-4-1: If two intersecting lines form congruent angles, then the lines are perpendicular.
3-4 Perpendicular Lines
Theorem 3-4-2: In a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line.
3-4 Perpendicular Lines
Theorem 3-4-3: If two coplanar lines are perpendicular to the same line, then the two lines are parallel.
3-1 Lines and Angles
3-2 Angles Formed by Parallel Lines and Transversals
3-3 Proving Lines Parallel
3-4 Perpendicular Lines
3-5 Slopes of Lines
3-6 Lines in the Coordinate Plane
3-5 Slopes of LinesSlope : steepness of a line
Find the slope between the
points (5,4) and (-7, -3)
3-5 Slopes of LinesWhen is slope Positive, Negative, Zero, or Undefined?
3-5 Slopes of LinesLines are parallel if their slopes are
_______________
3-5 Slopes of LinesLines are perpendicular if their slopes _____________
3-1 Lines and Angles
3-2 Angles Formed by Parallel Lines and Transversals
3-3 Proving Lines Parallel
3-4 Perpendicular Lines
3-5 Slopes of Lines
3-6 Lines in the Coordinate Plane
3-6 Lines in the Coordinate Plane How do we identify one line
from another line?
The equation of a line helps us identify different types of lines.
X
Y
3-6 Lines in the Coordinate Plane Slope
10
20
30
40
50
60
70
1 2 3 4 5
3-6 Lines in the Coordinate Plane Point Slope Form y – y1 = m(x - x1)
10
20
30
40
50
60
70
1 2 3 4 5
3-6 Lines in the Coordinate Plane Slope – intercept Form y = mx + b
10
20
30
40
50
60
70
1 2 3 4 5
3-6 Lines in the Coordinate PlaneA line with slope 3, that
goes through (3, -4) in point-slope form.
3-6 Lines in the Coordinate PlaneA line through (-1,0) and
(1,2) in slope-intercept form.
3-6 Lines in the Coordinate PlaneA line with x – intercept at 2
and y – intercept 3 in point-slope form.
