Parallel and Perpendicular Lines Note Book Name: Class Period: Teacher’s Name:
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Lines and Angles / Properties of Parallel Lines D A
C B
H E
G F
Parallel Lines:
Skew Lines:
Parallel Planes:
Transversal:
t l m
2
Angel Pairs Formed By Transversals: Alternate Interior Angles:
t 1 4
2
l
3
5 6 8 7
m
Same-Side Interior Angles:
t 1 4
2
l
3
5 6 8 7
m
Alternate Exterior Angles:
t 1 4
2
l
3
5 6 8 7
m
Same-Side Exterior Angles:
t 1 4
2
l
3
5 6 8 7
m
Corresponding Angles:
t 1 4
3
5 6 8 7
2
l
3 m
t 1 4
2
l
3
5 6 8 7
m
If m∠3 = 3x – 50 and m∠7 = 2x, what is the value of x?
If m∠1 = 2x + 50 and m∠8 = 3x - 20, what is the value of x?
Find the value of each of the variables.
a° b° d° c°
h°
4
e° 120° g° f °
Proving Lines Parallel Converse of the Alternate Interior Angles Theorem: If two lines and a transversal form Alternate Interior angles that are congruent, then the lines are parallel.
Converse of the Same-Side Interior Angles Theorem: If two lines and a transversal form Same-Side Interior angles that are supplementary, then the lines are parallel.
Converse of the Alternate Exterior Angles Theorem: If two lines and a transversal form Alternate Exterior angles that are congruent, then the lines are parallel.
Converse of the Same-Side Exterior Angles Theorem: If two lines and a transversal form Same-Side Exterior angles that are supplementary, then the lines are parallel.
Converse of the Corresponding Angles Theorem: If two lines and a transversal form corresponding angles that are congruent, then the lines are parallel.
5
t
n 1 5 2 6 3 7 4 8
9
13 10 14 11 15 12 16
l m
Use the given information to determine which lines, if any, are parallel. Justify each conclusion with a theorem or postulate. ∠14 is supplementary to ∠15
6
∠4 is supplementary to ∠16
∠4 ≅ ∠2
∠1 ≅ ∠6
∠9 ≅ ∠2
∠7 ≅ ∠12
∠4 ≅ ∠13
∠1 ≅ ∠11
Parallel and Perpendicular Lines In a plane, if two lines are parallel to the same line, then they are parallel to each other.
In a plane, if two lines are perpendicular to the same line, then they are parallel to each other.
In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other.
7
Parallel Lines and Triangles Triangle Sum Theorem: The sum of the measures of the angles of a triangle is 180.
Exterior Angle of a Polygon: An angle formed by a side and an extension of an adjacent side. Remote Interior Angles: The two nonadjacent interior angles corresponding to each exterior angle of a triangle.
Triangle Exterior Angle Theorem: The measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles.
Find the value of each variable.
32°
75°
8
42°
x° y°
z°
Equations of Lines in the Coordinate Plane Slope:
Slope-Intercept Form:
Point-Slope Form:
Find an equation if you are given the slope and a point:
Find an equation if you are given two points:
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Slopes of Parallel and Perpendicular Lines Slopes of Parallel Lines If two nonvertical lines are parallel, then their slopes are equal. If the slopes of two distinct nonvertical lines are equal, then the lines are parallel. Any two vertical lines or any two horizontal lines are parallel. Slopes of Perpendicular Lines If two nonvertical lines are perpendicular, then their slopes opposite reciprocals. If the slopes of two lines are opposite reciprocals of each other, then the lines are perpendicular. Any horizontal line and vertical line are perpendicular.
1
Write the equation of the line that is parallel to y = x – 3 2 and goes through the point (6 , -3).
Write the equation of the line that is perpendicular to 1 y = x – 3 and goes through the point (6 , -3). 2
10
Constructing Parallel and Perpendicular Lines Constructing Parallel Lines Given: line l and point N not on l Construct: line m through N with m ǁ l
N
l
Step 1: Label two points H and J on l. Draw ⃡HN. Step 2: At N, construct ∠1 congruent to ∠NHJ. Label the new line m.
mǁl 11
Perpendicular at a Point on a line Given: point P on line l Construct: ⃡CP with ⃡CP | l
l
P
Step 1: Construct two points on l that are equidistant from P. Label the points A and B. Step 2: Open the compass wider so the opening is 1
greater than AB. With the compass tip on 2 A, draw an arc above point P. Step 3: Without changing the compass setting, place the compass point on point B. Draw an arc that intersects the arc from step 2. Label the point of intersection C. ⃡ . Step 4: Draw CP ⃡𝐂𝐏 | l 12
Perpendicular from a Point to a Line Given: line l and point R not on l Construct: ⃡RG with ⃡RG | l
R
l
Step 1: Open your compass to size greater than the distance form R to l. With the compass on point R, draw an arc that intersects l at two points. Label the points E and F. Step 2: Place the compass point on E and make an arc. Step 3: Keep the same compass setting. With the compass tip on F, draw an arc that intersects the arc from step 2. Label the point of intersection G. Step 4: Draw ⃡RG. ⃡𝐑𝐆 | l
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