Lines and Angles
Lines Relationship
Parallel Planes
Thursday, April 9, 2020
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Lines and Angles: Parallel Lines, Skew Lines, Perpendicular Lines, and Planes, Exercises of Analytical Geometry
The concepts of parallel lines, skew lines, and perpendicular lines, as well as parallel planes. It includes definitions, examples, and illustrations. The document also discusses the Parallel and Perpendicular Postulate and its implications for identifying parallel and perpendicular lines and planes.
Typology: Exercises
2021/2022
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Lines and Angles
Lines Relationship
Parallel Planes
Thursday, April 9, 2020
Lines Relationship
๏ฏ Parallel Lines
๏ฏ Skew Lines
๏ฏ Perpendicular Lines
Two lines are parallel lines if they do not intersect
and are coplanar.
Two lines are skew lines if they do not intersect and
are not coplanar.
Two lines are perpendicular lines if they intersect
and form right angles.
Parallel Planes
Example
Identify parallel planes
A B C D E F G H
Plane ADE and plane BCH
Plane ABG and plane CDE
Two planes are parallel lines if they do not intersect.
Parallel and Perpendicular Postulate
Parallel Line 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.
Illustration
n P
There is exactly one line through P parallel to line n
- >
Skew, Parallel Lines, Planes, Perpendicular d. (^) Plane ( s ) parallel to plane EFG and containing point A c. Line ( s )^ perpendicular to^ CD^ and containing point^ A a. Line ( s ) parallel to CD and containing point A b. Line ( s ) skew to CD and containing point A Think of each segment in the figure as part of a line. Which line ( s ) or plane ( s ) in the figure appear to fit the description? AB , HG , and EF all appear parallel to CD , but only AB contains point A. Both AG and AH is skew to CD and contain point A. BC , AD , DE , and FC all appear perpendicular to CD , but only AD contains point A. Plane ABC appears parallel to plane EFG and contains point A.
Skew, Parallel Lines, Planes, Perpendicular
Try it yourself! Guided Practice
1. (^) Look at the diagram in Example 1. Name the lines through point H that appear skew to CD. ANSWER AH , EH
Transversal Line
๏ฏ Transversal Line is a line that intersect two or
more coplanar lines at different points
l 1 l 2 l 3 l 1 l 2 l 3
Example: Name the transversal Line
l
Transversal Line : 1 Transversal Line :
l
l
l
Angles Formed by Transversal Line
๏ฏ When a transversal line intersect two lines, 8 angles
are formed as shown in the figure below.
1 2 3 4 (^5 ) (^7 )
l 1 l 2 l 3
CORRESPONDING ANGLES: ALTERNATE EXTERIOR ANGLES: SAME SIDE/ CONSECUTIVE INTERIOR ANGLES: ALTERNATE INTERIOR ANGLES: ๏1 and ๏5, ๏1 and ๏8, ๏3 and ๏6, ๏3 and ๏5, ๏2 and ๏6, ๏3 and ๏7, ๏4 and^ ๏^8 ๏2 and ๏ 7 ๏4 and ๏ 5 ๏4 and ๏ 6
Angles Formed by Transversal Line
Identify each pair of angles as alternate interior, alternate
exterior, consecutive interior, corresponding or vertical.
Try it yourself!
3. (^) 4. 5. Corresponding angles Alternate exterior angles Alternate interior angles
Properties of Parallel Lines
Lesson 3 โ 3
Proofs with Parallel Lines
Theorems on Angles Formed by a Transversal
Thursday, April 9, 2020
Theorems on Angles Formed by a Transversal
Find the measure of the angle
1. ) If m ๏ 1 = 125^0 , find m ๏ 2 =_____ 2. ) If m ๏ 8 =135^0 , find m ๏ 2 =_____ 3. ) If m ๏ 5 = 87 0 , find m ๏ **4 =_____
- ) If m** ๏ 1 = 118^0 , find m ๏ **8 = _____
- ) If m** ๏ 5 = 78^0 , find m ๏ 2 =_____ 6. ) If m ๏ 2 = 126^0 , find m ๏ 4 =____ 7. ) If m ๏ 9 =5x + 8 and m ๏ 11 = 12x + 2 8. ) If m ๏ 9 =2x + 7 and m ๏ 14 = 3x - 13 9. ) If m ๏ 11= 14x - 56 and m ๏ 13 = 6x 10. ) If m ๏ 5=5x + 25 and m ๏ 2= 3x - 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 For #โs 7-10, Solve for x and the angles 1250 1350 870 1180 102 0 540 x = 10 m ๏ 9 = 58^0 m ๏ 11 = 122^0 x = 20 m ๏ 9 = 47^0 m ๏ 11 = 47^0 x = 7 m ๏ 11 = 42^0 m ๏ 13 = 42^0 x = 20 m ๏ 6 = 125^0 m ๏ 2 = 55^0
Example
Theorems on Angles Formed by a Transversal
Try it yourself!
1. If m ๏1 = 105ยฐ, find m ๏ 4 , m ๏ 5 , and m ๏ 8. Tell
which postulate or theorem you use in each case.
2. If m ๏3 = 68ยฐ and m ๏8 = (2 x + 4)ยฐ, what is the
value of x? Show your steps.
m ๏4 = 105ยฐ
m ๏5 = 105ยฐ
m ๏8 = 105ยฐ
Vertical Angles Congruence Corresponding Angles Alternate Exterior Angles
x = 54
Prove Lines Parallel
Postulate 16: Corresponding Angles Converse
If two lines are cut by a transversal so the
corresponding angles are congruent, then the
lines are parallel.
Illustration
1 2