New York State Common Core Math Geometry, Module 3, Lesson 5
Worksheets for Geometry, Module 3, Lesson 5
- Students describe properties of points, lines, and planes in three-dimensional space.
The following three-dimensional right rectangular prism has dimensions 3 × 4 × 5. Determine the length of 𝐴𝐶′. Show
a full solution.
- Two points 𝑃 and 𝑄 determine a distance
𝑃𝑄, a line segment 𝑃𝑄, a ray 𝑃𝑄, a vector
𝑃𝑄, and a line 𝑃𝑄.
- Three non-collinear points 𝐴, 𝐵, and 𝐶
determine a plane 𝐴𝐵𝐶 and, in that plane,
determine a triangle 𝐴𝐵𝐶.
- Two lines either meet in a single point, or
they do not meet. Lines that do not meet
and lie in a plane are called parallel. Skew
lines are lines that do not meet and are not
- Given a line ℓ and a point not on ℓ, there is
a unique line through the point that is
parallel to ℓ.
- Given a line ℓ and a plane 𝑃, then ℓ lies in
𝑃, ℓ meets 𝑃 in a single point, or ℓ does
not meet 𝑃, in which case we say ℓ is
parallel to 𝑃. (Note: This implies that if
two points lie in a plane, then the line
determined by the two points is also in the
- Two planes either meet in a line, or they do
not meet, in which case we say the planes
- Two rays with the same vertex form an
angle. The angle lies in a plane and can be
measured by degree.
- Two lines are perpendicular if they meet,
and any of the angles formed between the
lines is a right angle. Two segments or rays
are perpendicular if the lines containing
them are perpendicular lines.
- A line ℓ is perpendicular to a plane 𝑃 if
they meet in a single point, and the plane
contains two lines that are perpendicular
to ℓ, in which case every line in 𝑃 that
meets ℓ is perpendicular to ℓ. A segment
or ray is perpendicular to a plane if the line
determined by the ray or segment is
perpendicular to the plane.
- Two planes perpendicular to the same line
- Two lines perpendicular to the same plane
- Any two line segments connecting parallel
planes have the same length if they are
each perpendicular to one (and hence
both) of the planes.
- The distance between a point and a plane
is the length of the perpendicular segment
from the point to the plane. The distance
is defined to be zero if the point is on the
plane. The distance between two planes is
the distance from a point in one plane to
SEGMENT: The segment between points 𝐴 and 𝐵 is the set consisting of 𝐴, 𝐵, and all points on ⃡𝐴𝐵 between 𝐴 and 𝐵.
The segment is denoted by 𝐴𝐵, and the points 𝐴 and 𝐵 are called the endpoints.
LINE PERPENDICULAR TO A PLANE:
A line 𝐿 intersecting a plane 𝐸 at a point 𝑃 is said to be perpendicular to the plane 𝐸 if
𝐿 is perpendicular to every line that (1) lies in 𝐸 and (2) passes through the point 𝑃. A segment is said to be
perpendicular to a plane if the line that contains the segment is perpendicular to the plane.
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