Download Lesson 2: Points, Lines and Planes

GEOMETRY
Lesson 2
NYS COMMON CORE MATHEMATICS CURRICULUM
Name________________________
M3
Period: _____Date______________
Lesson 2: Points, Lines and Planes
Learning Target: I can describe the properties of points, lines, and planes
Opening Exercise: Fill in the blanks. Draw any missing pictures in the 2nd column.
Element Description
A point is a position in space, and has no _____________________ (actual size). A point is
usually named with a capital _________________. In the coordinate plane, a point is named
by an ordered pair, (𝑥, 𝑦).
A Line has no thickness but its length extends in one dimension and goes on forever in both
⃡ .
directions. A line is named by a single lowercase letter, l, or by any two points on the line, 𝐴𝐵
A _________ is a part of a line that starts at an endpoint and extends forever in one direction.
A ray is named by the endpoint followed by a point on the ray, such as 𝐴𝐵
Opposite rays are 2 rays that lie on the same line, with a common endpoint and no other
points in common. Opposite rays form a straight ________ and/or a straight ________ (180°).
A Line segment is part of a line and consists of two points and all the points between. A
̅̅̅̅.
segment is named by its two ___________________, such as 𝐶𝐷
A plane has no thickness but extends indefinitely in all directions.
A plane is named by a single __________ (plane V ) or by three non-___________________
points (plane 𝑅𝑆𝑇).
Collinear Points are points that lie on the same line, such as ____ and ____ in the example at
right.
Coplanar points are points that lie in the ____________ plane such as 𝐴, 𝐵, 𝑎𝑛𝑑 𝐶 in the
example at right.
Parallel lines two lines that do not ___________________ .
Intersecting lines are lines that intersect at a _________________ .
Skew lines two _____-____________ lines that do not ________________ .
Example
GEOMETRY
Lesson 2
NYS COMMON CORE MATHEMATICS CURRICULUM
Name________________________
M3
Period: _____Date______________
Example 1.
1. Name a plane.
______________________________
2. Name a segment.
______________________________
3. Name a line.
______________________________
4. Name three collinear points. ________________________
5. Name three non collinear points. _____________________
6. Name the intersection of a line and a segment not on the line.
_________________
Example 2.
1. Name a pair of opposite rays. _________________________________
2. Name the points that determine plane 𝑹. _______________________
3. Name the point at which line 𝒎 intersect plane 𝑹. ________________
4. Name two lines in plane 𝑹 that intersect line 𝒎. __________________
5. Name a line in plane 𝑹 that does not intersect line 𝒎. ______________
Theorems related to lines and planes
1. One Line Perpendicular to One Plane
Through a given point on a line passes only one only one plane
perpendicular to a given line.
Through any point on a plane passes one and only one line
perpendicular to that plane.
2. One Line Perpendicular to Two or More Planes
Two planes perpendicular to the same line are parallel. 𝑃𝑙𝑎𝑛𝑒 𝑙1 and
𝑃𝑙𝑎𝑛𝑒 𝑙 are perpendicular to ⃡𝐶𝐷, therefore 𝑃𝑙𝑎𝑛𝑒 𝑙1 ∥ 𝑃𝑙𝑎𝑛𝑒 𝑙
GEOMETRY
Lesson 2
NYS COMMON CORE MATHEMATICS CURRICULUM
Name________________________
M3
Period: _____Date______________
3. Two Lines Perpendicular to One Plane
⃡ and
Two lines perpendicular to a plane are parallel and coplanar (𝐴𝐵
⃡ are perpendicular to plane q, therefore 𝐴𝐵
⃡ ∥ 𝐶𝐷
⃡ and they are
𝐶𝐷
coplanar)
4. Two Perpendicular Planes
Two planes are perpendicular to each other if and only if one plane
contains a line perpendicular to the second plane.
5. Two Perpendicular Lines and Planes at a Point
If a line is perpendicular to a plane, then any line perpendicular to the
given line at its point of intersection with the given plane is in the
given plane.
6. One Line Perpendicular to One Plane: Revisited
If a line is perpendicular to a plane, then every plane containing the
line is perpendicular to the given plane.
7. Parallel Planes Intersecting with Another Plane
If a plane intersects two parallel planes, then the intersection is two
parallel lines.
Example 3. Indicate whether each statement is always true (A), sometimes true (S), or never true (N).
a. ________
If two lines are perpendicular to the same plane, the lines are parallel.
b. ________
Two planes can intersect in a point.
c.
Two lines parallel to the same plane are perpendicular to each other.
________
d. ________
If a line meets a plane in one point, then it must pass through the plane.
e. ________
Skew lines can lie in the same plane.
f.
If two lines are parallel to the same plane, the lines are parallel.
________
g. ________
If two planes are parallel to the same line, they are parallel to each other.
h. ________
If two lines do not intersect, they are parallel.
GEOMETRY
Lesson 2
NYS COMMON CORE MATHEMATICS CURRICULUM
Name________________________
M3
Period: _____Date______________
Lesson 2: Points, Lines and Planes
Classwork
1. Answer all the questions using the diagram on the right
a. Name the plane.
______________________
b. Name a segment.
______________________
c. Name a line.
______________________
d. Name three collinear points. ______________________
e. Name three non collinear points.
______________________
f. Name the intersection of a line and a segment not on the line.
g. Name a pair of opposite rays.
______________________
______________________
h. Name the points that determine plane R.
______________________
i.
Name the point at with line ⃡𝐽𝐾 intersects plane R. ______________________
j.
Name two lines in plane R that intersect point 𝑃.
______________________
k. Name a line in plane R that does not intersect point 𝐷.
2. If
is contained in plane P, and
1)
2)
3)
4)
______________________
is perpendicular to plane R, which statement is true?
is parallel to plane R.
Plane P is parallel to plane R.
is perpendicular to plane P.
Plane P is perpendicular to plane R.
3. As shown in the diagram below,
, , and
intersect at 𝐴.
is contained in plane R,
are contained in plane S, and
and
Which fact is sufficient to show that planes R and S are perpendicular?
1)
2)
3)
4)
4. Point 𝐴 is not contained in plane B. How many lines can be drawn through point 𝐴 that will be
perpendicular to plane B ?
a. One
b. Two
c. Zero
d. Infinity
GEOMETRY
Lesson 2
NYS COMMON CORE MATHEMATICS CURRICULUM
Name________________________
M3
Period: _____Date______________
5. In the diagram at right, point 𝐾 is in plane P. How many lines can be
drawn through 𝐾, perpendicular to plane P ?
a. One
b. Two
c. Zero
d. Infinity
6. In plane P, lines 𝑚 and 𝑛 intersect at point 𝐴. If line 𝑘 is perpendicular to line 𝑚 and line 𝑛 at point
𝐴, then line 𝑘 is
contained in plane P
1)
parallel to plane P
2)
perpendicular to plane P
3)
skew to plane P
4)
7. Lines a and b intersect at point P. Line c passes through P and is perpendicular to the plane
containing lines a and b. Which statement must be true?
1) Lines a, b, and c are coplanar.
2) Line a is perpendicular to line b.
3) Line c is perpendicular to both line a and
line b.
4) Line c is perpendicular to line a or line b,
but not both.
8. Lines
and
intersect at point E. Line m is perpendicular to lines
Which statement is always true?
1) Lines and are perpendicular.
2) Line m is parallel to the plane determined
by lines and .
3) Line m is perpendicular to the plane
determined by lines and .
4) Line m is coplanar with lines and .
and
at point E.
GEOMETRY
NYS COMMON CORE MATHEMATICS CURRICULUM
Name________________________
Lesson 2
M3
Period: _____Date______________