Download Geometric Relationship Sample Tasks with Solutions

Sample Task Alignment
Unit: Geometric Relationships
Content Strands
G.G.1 Know and apply that if a line is perpendicular to each of two intersecting lines at their point
of intersection, then the line is perpendicular to the plane determined by them.
G.G.1a
Jim is a carpenter and would like to install a flagpole in his front yard. Carpenters use a tool called a level,
shown in the figure below, to determine if objects are level (horizontal) or plumb (vertical). Describe how
Jim can use a level to ensure that the flagpole appears vertical from any direction.
Jim, being a carpenter who knows how to use a level, will properly use said level against alleged vertical
flag pole from multiple “sides” of the pole (north and east).
G.G.1b
Study the drawing below of a pyramid whose base is quadrilateral
ABCD . John claims that line
segment EF is the altitude of the pyramid. Explain what John must do to prove that he is correct.
John must show that
EF  AC or EF  DB .
G.G.2
Know and apply that through a given point there passes one and only one
plane perpendicular to a given line
G.G.2a
Examine the diagram of a pencil below:
Explain how the pencil illustrates the fact that if two lines are perpendicular to the same line, then they
must be parallel.
Two consecutive lines going down the length of the pencil are perpendicular to the line formed by the
bottom of the pencil. The two lines are parallel.
Explain how the pencil illustrates the fact that if two lines are parallel and the first line is perpendicular to a
third line, then the second line must be perpendicular to the third line.
Same lines as above. The consecutives are parallel to each other. One is perpendicular to bottom. Viola
second is also perpendicular.
G.G.3
Know and apply that through a given point there passes one and only one line
perpendicular to a given plane
G.G.3a
Examine the diagram of a right triangular prism.
Describe how a plane and the prism could intersect so that the intersection is:
a line parallel to one of the triangular bases (The plane touches only an edge of a triangle)
a line perpendicular to the triangular bases (A vertical plane touching one of the vertical edges)
a triangle (Plane parallel to triangle base, cutting through prism)
a rectangle (Plane parallel to a rectangular side, cutting through the prism)
a trapezoid (A plane intersecting both triangle bases)
G.G.4
Know and apply that two lines perpendicular to the same plane are coplanar
G.G.4a
The figure below is a right hexagonal prism.
On a copy of the figure sketch a symmetry plane. Then write a description of the symmetry plane
that uses the word parallel.
On a copy of the figure sketch another symmetry plane. Then write a description that uses the
word perpendicular.
Plane 1 – parallel to edges of
prism, perpendicular to vertical
faces it intersects and bases
Visualize the same symmetry
plane is drawn in white ink.
G.G.4b The figure below in three dimensional space, where AB is perpendicular to
BC and DC is
perpendicular to BC , illustrates that two lines perpendicular to the same line are not necessarily parallel.
Must two lines perpendicular to the same plane be parallel? Discuss this problem with a partner.
Yes they must be parallel because
two lines perpendicular to the
same plane are coplanar and two
lines perpendicular to the same
line in the same plane are
perpendicular.
G.G.5
Know and apply that two planes are perpendicular to each other if and only if one
plane contains a line perpendicular to the second plane
G.G.5a
Julie wishes to construct a wall in the basement of her apartment. Describe what she must do to ensure that
the wall is perpendicular to the floor of the basement. Explain why Julie’s procedure will work, using
appropriate mathematical terminology.
Julie must measure 3 feet up the wall and 4 feet on the floor out from the wall. The endpoints
must be 5 feet apart. By the Pythagorean theorem, the wall and floor should form a right angle
and therefore be perpendicular.
G.G.6 Know and apply that 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
G.G.6a
Justify the fact that if one edge of a triangular prism is perpendicular to its base then the prism is a right
triangular prism.
Since the lateral edge is perpendicular to the base, it is an altitude, so a right angle is formed, therefore
the prism is a right prism.
G.G.7 Know and apply that if a line is perpendicular to a plane, then every plane containing the
line is perpendicular to the given plane
G.G.7a
Examine the four figures below:
Each figure has how many symmetry planes?
Describe the location of all the symmetry planes for each figure.
#
Descriptions
G.G.8
Rectangular prism
3
Vertical left/right,
Vertical front/back,
horizontal
Pentagonal prism
6
5 Vertical- one from
each vertex to
midpoint of opposite
side ,
horizontal
Square puramid
4
2 intersecting
midpoints of base
edges,
2 intersecting
opposite vertices
Cone

Vertical planes
that intersect the
center of the base
Know and apply that if a plane intersects two parallel planes, then the intersection is
two parallel lines
G.G.8a
The figure below is a right hexagonal prism.
On a copy of the figure sketch a symmetry plane. Then write a description that uses the word
parallel.
On a copy of the figure sketch another symmetry plane. Then write a description that uses the
word perpendicular.
See G.G.4a
G.G.9
Know and apply that if two planes are perpendicular to the same line, they
are parallel
G.G.9a
The figure below shows a right hexagonal prism.
A plane that intersects a three-dimensional figure such that one half is the reflected image of the other half
is called a symmetry plane. On a copy of the figure sketch a symmetry plane. Then write a description of
the symmetry plane that uses the word parallel.
On a copy of the figure sketch another symmetry plane. Then write a description that uses the
word perpendicular.
See G.G.4a
G.G.9b
Draw a line on a piece of cardboard. Use additional pieces of cardboard to construct two planes that are
perpendicular to the line that you drew. Make a conjecture regarding those two planes and justify your
conjecture.
The planes are parallel. If two planes are perpendicular to the same line, then the planes are parallel.
G.PS.1a
Obtain several different size cylinders made of metal or cardboard. Using stiff paper, construct a cone with
the same base and height as each cylinder. Fill the cone with rice, then pour the rice into the cylinder.
Repeat until the cylinder is filled. Record your data.
What is the relationship between the volume of the cylinder and the volume of the corresponding
cone?
Collect the class data for this experiment.
Use the data to write a formula for the volume of a cone with radius r and height h.
3 cones worth of rice fit evenly into one cylinder with the same base and height.
1
Vcone  hr 2
3
Does not match any of the content strands but goes with the topic.
G.PS.2c, G.PS.3a, G.PS.7b, G.RP.3a, G.RP.8a, G.CM.7a
Consider the following conjecture: The intersection of two distinct planes can be a point. Find a “real
world” example that supports the conjecture or provides a counterexample to the conjecture. Share your
example with a partner and use your knowledge of geometry in three dimensional space to justify it.
Not true. A wall and a ceiling represent planes. Their intersection is a line.
G.R.7
Use mathematics to show and understand social phenomena (e.g., determine if
conclusions from another person’s argument have a logical foundation)
G.R.7a
Melvin claims that two lines in space that do not intersect must always be parallel. To support his
conjecture he refers to the line of intersection of a wall and the ceiling and the line of intersection of the
same wall and the floor. Discuss the validity of his conjecture and his justification.
He is wrong. The road under an overpass is not parallel to the overpass and does not intersect it. They
are skew.
G.G.35 Determine if two lines cut by a transversal are parallel, based on the measure of given pairs
of angles formed by the transversal and the lines
G.G.35a
Investigate the two drawings using dynamic geometry software. Write as many conjectures as you can for
each drawing.






When the same side
interior angles are not
supplementary, the lines
are not parallel.
Vertical angles are
congruent.
Adjacent angles that form
a line are supplementary.
When the same side
interior angles are
supplementary, the lines
are parallel.
Vertical angles are
congruent.
Adjacent angles that form
a line are supplementary.
G.G.35b
In the following figure, certain angle measures are given. Use the information provided to show that AB
is parallel to
CD and that EF is not parallel to CD .
CDA and A are supplementary, therefore,
AB // DC .
DCF and CFE are not supplementary,
therefore, EF is not // to CD