Download 8.2

Chapter 8: Informal Geometry - Answers
Section 8.2: Lines, Planes, and Angles [Oh My!! ]
Relations among Lines and Planes

Parallel Planes: If the intersection of two distinct planes is empty, the planes are said to be parallel.

Angle: An angle is the union of two rays that have the same endpoint.
Properties of a Line and a Plane:
1. A line and a plane may intersect in exactly one point.
2. The entire line may lie in the plane.
3. The line may be parallel to the plane.

Sides : the rays that form an angle

Vertex: the common endpoint of two rays that form an angle
Naming an Angle:
1. Use the ∠ with the vertex only
2. Use the ∠ with three points, where the point in the middle is the vertex
3. Use the ∠ with a Greek letter, like 𝜃 or , , , and so on
Measuring Angles

Degrees: Babylonians created over 2500 years ago; divide a circle into 6 equal parts and then divide
those parts into 60 equal parts each; thus a circle has 360 degrees

Right angle: an angle measuring 90 degrees
Classifications of Angles:
1. A right angle has a measure of 90o ; In diagrams, a right angle is indicated by a box at the vertex
2. A straight angle has a measure of 180 degrees
3. An acute angle has a measure of between 0 degrees and 90 degrees
4. An obtuse angle has a measure of between 90 degrees and 180 degrees
5. A reflex angle has a measure of between 180 degrees and 360 degrees
Perpendicular Lines

Perpendicular: Lines, rays, and line segments that form right angles; written: 𝑟 ⊥ 𝑠

Can a line and a plane be perpendicular? Yes, as long as the angle formed by the point of intersection
(the vertex) with any line containing that point is a right angle.

What would be true about every or any line in the plane if another line was perpendicular to the plane?
Every and any line in the plane would be perpendicular to the plane containing the line.

What about two planes – Can they be perpendicular? Two planes are perpendicular if and only if one
plain contains a line perpendicular to the other plane.
Other Relations among Angles

Interior: area between the two rays and end point forming an angle

Exterior: area outside the two rays and end point forming an angle
Relationships between Two Angles:
1. Two angles are supplementary if the sum of their measures is 180 degrees
2. Two angles are complementary if the sum of their measures is 90 degrees
3. Two coplanar (in the same plane) angles that have a common vertex and a common side are called
adjacent angles if their interiors are disjoint (have NO points in common)

Give an example of two adjacent angles:

Give an example of two angles with a common vertex that are NOT adjacent:

Vertical angles: the pairs of nonadjacent angles formed by the intersection of two lines are called
verticle angles

Congruent angles: vertical angles have the same measure; angles with the same measure are
congruent

Measures: measures are equal; figures are congruent
Transversals

Transversal: any line that intersects a pair of lines in exactly two points

Alternate interior angles: angles on opposite sides of the transversal and between the parallel lines

Alternate exterior angles: angles on opposite sides of the transversal and exterior to the parallel lines

Corresponding angles: angles on the same side of the transversal and one is interior and one is
exterior to the pair of parallel lines

If I give you one of the angles can you give me the others? Give a reason why for each pair as you go
(vertical, alternate interior, and so on).
6 5
1 4
2 3
7 8
146 34
34 146
146 34
34 146
o ∡1 = 34𝑜
AWV: 1  5 : vertical angles; 1  3 : alternate interior
angles; 1  7 : corresponding angles; 4 = 180 - 1 = 180 - 34 = 146 :
supplementary angles; 4  6 : vertical angles; 6  8 : alternate exterior angles; 2 
8 : vertical angles;
113.5 66.5
66.5 113.5
113.5 66.5
66.5
113.5
𝑜
AWV: 6  4 : vertical angles; 6  8 : alternate
o ∡6 = 113.5
exterior angles; 6  2 : corresponding angles; 1 = 180 - 6 = 180 - 113.5 = 66.5 :
supplementary angles; 1  5 : vertical angles; 1  3 : alternate interior angles; 3 
7 : vertical angles;
x + 103 77-x
77-x x + 103
x + 103 77-x
𝑜
o ∡4 = 𝑥 + 103
77-x x + 103
AWV: 4  6 : vertical angles; 6  8 : alternate
exterior angles; 6  2 : corresponding angles; 1 = 180 - 6 = 180 - 113.5 = 66.5 :
supplementary angles; 1  5 : vertical angles; 1  3 : alternate interior angles; 3 
7 : vertical angles;
Properties of Angles and Transversals:
1. If lines l and m in a plane are intersected by a transversal in such a way that alternate interior angles
are of equal measure, then l is parallel to m
2. Alternate interior angles formed by two parallel lines and a transversal are of equal measure