Download 3.1 Lines and Angles Parallel Lines

3.1 Lines and Angles
Parallel Lines - Coplanar lines that do not intersect
Skew Lines - Are NOT coplanar and do not intersect
Parallel Planes - Planes that do not intersect
Lines m and n are parallel (m‖ n)
Lines m and k are skew lines
Planes T and U are parallel planes (T ‖ U)
Line m is parallel to plane U. (m ‖ U)
Lines k and n are intersecting lines. There is a plane
(not shown) that contains them.
Perpendicular lines – 2 lines that intersect at a right angle
Transversal – Line that intersects two or more coplanar lines at distinct points
Interior Angles – the angles that lie between the lines
Exterior Angles – the angles that lie outside of the
lines
Corresponding Angles- lie on the same side of the transversal and are in corresponding positions (same spot).
Ex: <1 and <5 , <2 and <6 , <3 and <7 , <4 and <8
Alternate Exterior Angles- nonadjacent exterior angles that lie on the opposite sides of the transversal.
Ex: <1 and <7 , <2 and <8
Alternate Interior Angles - nonadjacent interior angles that lie on opposite sides of the transversal.
Ex: <4 and <6 , <3 and <5
Same-side Interior Angles - interior angles that lie on the same side of the transversal.
Ex: <3 and <6 , <4 and <5
Ex: Name a pair of
a.
b.
c.
d.
e.
f.
Alternate interior angles: <2 and<6
Alternate exterior angles: <1 and <5
Same-side interior angles: <2 and <3
Corresponding angles: <1 and <3
Vertical Angles: <1 and <7
Supplementary angles: <1 and <2
3.2 Properties of Parallel Lines
Same-Side Interior Angles Theorem
If two parallel lines are cut by a tranversal, then sameside interior angles are supplementary.
m<3 +m<6 = 180
m<4 + m<5 = 180
Alternate Interior Angles Theorem
If two parallel lines are cut by a transversal, then
alternate interior angles are congruent.
< 4 ≅< 6
<3≅<5
Alternate Exterior Angles Theorem
If two parallel lines are cut by a transversal, then
alternate exterior angles are congruent.
< 2 ≅ <8
<1≅<7
Corresponding Angles Theorem
If two parallel lines are cut by a transversal, then
corresponding angles are congruent.
Identify all the numbered angles that are congruent to the given angles. Justify.
< 3 – vertical angles
<5 – corresponding angles
<7 – Alternate exterior angles
Find x and the value of the angles.
3x – 5 = x + 55
2x =60
X = 30 (Angle = 85º)
Find m<1 and m<2.
< 1 ≅ <5
<2≅<6
<3≅<7
<4≅<8
m<1 = 73 (Alt. Interior angles)
m<2 = 61 (Same-side interior angles)
Prove Alt. Int. Angles Theorem
Given: x y
Prove: 2  3
Statements
Reasons
1. x y
1. Given
2. m<1 + m<2 = 180
2. Supplementary angles
3. m<1 + m<3 = 180
3. Same-side interior angles postulate
4. m<1 + m<2 = m<1 + m<3
4. Transitive Property of =
5. m<2 = m<3
5. Subtraction Property of =
6. 2  3
6. Definition of Congruence
3.3 Proving Lines Parallel
Converse of the Same-Side Interior Angles
Theorem
If two lines are cut by a tranversal and the same-side
interior angles are supplementary, then the lines are
parallel.
If m<3 + m<6 = 180,
then l and m are
parallel.
Converse of the Alternate Interior Angles Theorem
If two lines are cut by a transversal and the alternate
interior angles are congruent, th en th e lines are
parallel.
If <4 ≅ <6, then l and
m are parallel.
Converse of the Alternate Exterior Angles
Theorem
If two lines are cut by a transversal and the alternate
exterior angles are congruent, then th e lines are
parallel.
If <1≅ <7, then l and
m are parallel.
Converse of the Corresponding Angles Theorem
If two lines are cut by a transversal and the
corresponding angles are congruent, the lines are
congruent.
If <2 ≅ <6, then l and m
are parallel.
Which lines/segments are parallel?
⃡𝐵𝐸 𝑎𝑛𝑑 ⃡𝐶𝐺 by the converse of corresponding angles
⃡𝐶𝐴 𝑎𝑛𝑑 ⃡𝐻𝑅 by the converse of corresp. Angles
55 = x + 25
X = 30
95 = 2x - 5
100 = 2x
X = 50
3.4 Parallel and Perpendicular Lines
3.5 Parallel lines and triangles
Triangle Angle-Sum Theorem: The sum of the measures of the angles of a triangle is 180.
Ex: Find m<1
M<1 =25
m<1 =60
Ex.: Find the value of each variable.
X= 59, y = 81, z = 99
Interior angles – angles inside a triangle (when we
extend the sides out) Ex: <2, <3, <4
-Remote interior: the 2 nonadjacent interior
angles of a specific extrerior angle
Ex: <3 and <4 are remote interior angles to <1
Exterior angles – angles that form linear pairs with
the interior angles (when we extend the sides out)
Ex: Find m<1.
m<1 = 80 + 18 = 98
m<1 = 38 + 70 =108
m<2 = 72
Ex: Find the value of the variables and the angles.
X = 36
36, 66, 78
3.7 Equations of Lines in the Coordinate Plane
Find the slope of the line:
Slope of line a:
3−7
−4
4
m = 2−5 = −3 = 3
Slope of line b:
−2−2
m = 4−−1 =
−4
5
Slope of line c:
7−7
0
m = 5−1 = 4 = 0
Slope of line d:
m=
Slope Intercept Form:
y = mx + b
m = slope
b = y-intercept
Point-Slope Form:
y - 𝑦1 = m(x - 𝑥1 )
m = slope
𝑥1 = x-coordinate
𝑦1 = y-coordinate
−2−0
4−4
=
−2
0
= undefined
Graph:
y=
−1
3
y + 2 = 3(x – 2)
𝑥+4
y = -4
x=3
Writing Equations of lines:
2
Ex: Slope = 3 and y-intercept = - 4
Ex: Point: (-2, 3) and slope = 2
2
y – 3= 2(x- -2)
y – 3 = 2x + 4
y = 2x + 7
y = 3𝑥 −4
Using two points to write an equation:
1. Find the slope.
2. Use the slope and one point for point-slope form
3. Distribute/Simplify to slope intercept form (y =)
Ex: (-4, 4) and (2,10)
10−4
6
m = 2−−4 = 6 = 1
Ex: (6, 2) and (2,4)
4−2
2
m = 2−6 = −4 =
y – 4 = 1(x - - 4)
y–2=
y–4=x+4
y – 2=
y=x+8
y=
−1
2
−1
2
−1
(x – 6)
2
−1
2
x+3
x +5
3.8 Slopes of Parallel and Perpendicular Lines
Parallel Lines: Have the same slope
Ex: Are the lines parallel?
y – 4 = 3x
2y – 6x = -6
−2−1
𝑙1 m = 0−−6 =
Parallel
1−3
𝑙2 m = 4−0 =
−3
6
−2
4
1
= −2
1
= −2
Parallel
−4−2
𝑙2 m =
Writing Equations of parallel lines:
1. Find the slope (same slope).
2. Use the slope and a point in point-slope form.
3. Distribute and solve for y.
Ex: Write an equation parallel to y = -2x + 1 that goes through (0, 3).
Parallel slope = -2
y – 3 = -2(x- 0)
y – 3 = -2x
y = -2x + 3
Ex: Write an equation parallel to y = − 2x + 6 that goes through (6, -2).
3
Parallel slope = − 2
3
y - - 2 = − 2 (x – 6)
3
y + 2 = −2 x + 9
3
y = −2 x + 7
−3−3
2−5
Not parallel
3
−6
3
𝑙1 m = −4−0 = −4 = 2
−6
= −3 = 2
Perpendicular Lines: Slopes are opposite reciprocals.
Ex: Are the lines perpendicular?
3
y = 4𝑥 −3
4
y + 3 𝑥 = −7
−2−4
𝑙1 m = 1−−3 =
Perpendicular
0−4
−6
3
4
= −2
−4
2
−3−4
𝑙1 m = 3−−4 =
−4
Perpendicular
Not perpendicular
1. Find the slope (opposite reciprocal slope).
2. Use the slope and a point in point-slope form.
3. Distribute and solve for y.
Ex: What is an equation perpendicular to y = -3x – 5 that goes through (-3, 7).
1
Perpendicular slope = 3
1
y – 7 = 3 (x- -3)
1
y–7=3x+1
1
y=3x+8
1
Ex: What is an equation perpendicular to y = 2 x – 5 and goes through (4, 0).
Perpendicular slope = -2
y = -2x + 8
= −1
𝑙2 m =
1−6
=
7
𝑙2 m = −4−2 = −6 = 3
Writing Equations of perpendicular lines:
y – 0 = -2 (x – 4)
−1−3
−7
5