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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

Uploaded on 09/27/2022

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Lines and Angles
Lines Relationship
Parallel Planes
Thursday, April 9, 2020
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Download Lines and Angles: Parallel Lines, Skew Lines, Perpendicular Lines, and Planes and more Exercises Analytical Geometry in PDF only on Docsity!

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: ๏ƒ1 and ๏ƒ5, ๏ƒ1 and ๏ƒ8, ๏ƒ3 and ๏ƒ6, ๏ƒ3 and ๏ƒ5, ๏ƒ2 and ๏ƒ6, ๏ƒ3 and ๏ƒ7, ๏ƒ4 and^ ๏ƒ^8 ๏ƒ2 and ๏ƒ 7 ๏ƒ4 and ๏ƒ 5 ๏ƒ4 and ๏ƒ 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 ๏ƒ 1 = 125^0 , find m ๏ƒ 2 =_____ 2. ) If m ๏ƒ 8 =135^0 , find m ๏ƒ 2 =_____ 3. ) If m ๏ƒ 5 = 87 0 , find m ๏ƒ **4 =_____

  1. ) If m** ๏ƒ 1 = 118^0 , find m ๏ƒ **8 = _____
  2. ) If m** ๏ƒ 5 = 78^0 , find m ๏ƒ 2 =_____ 6. ) If m ๏ƒ 2 = 126^0 , find m ๏ƒ 4 =____ 7. ) If m ๏ƒ 9 =5x + 8 and m ๏ƒ 11 = 12x + 2 8. ) If m ๏ƒ 9 =2x + 7 and m ๏ƒ 14 = 3x - 13 9. ) If m ๏ƒ 11= 14x - 56 and m ๏ƒ 13 = 6x 10. ) If m ๏ƒ 5=5x + 25 and m ๏ƒ 2= 3x - 25 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 For #โ€™s 7-10, Solve for x and the angles 1250 1350 870 1180 102 0 540 x = 10 m ๏ƒ 9 = 58^0 m ๏ƒ 11 = 122^0 x = 20 m ๏ƒ 9 = 47^0 m ๏ƒ 11 = 47^0 x = 7 m ๏ƒ 11 = 42^0 m ๏ƒ 13 = 42^0 x = 20 m ๏ƒ 6 = 125^0 m ๏ƒ 2 = 55^0
Example

Theorems on Angles Formed by a Transversal

Try it yourself!

1. If m ๏ƒ1 = 105ยฐ, find m ๏ƒ 4 , m ๏ƒ 5 , and m ๏ƒ 8. Tell
which postulate or theorem you use in each case.
2. If m ๏ƒ3 = 68ยฐ and m ๏ƒ8 = (2 x + 4)ยฐ, what is the
value of x? Show your steps.
m ๏ƒ4 = 105ยฐ
m ๏ƒ5 = 105ยฐ
m ๏ƒ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

In the figure, ๏ƒ 1 ๏€ ๏ƒ2 then line a // line b

Prove Lines Parallel

Theorem 3.4: Alternate Interior Angles Converse

If two lines are cut by a transversal so the

alternate interior angles are congruent, then the

lines are parallel.

Illustration

In the figure, ๏ƒ 1 ๏€ ๏ƒ2 then line a // line b