High School Geometry
Prove Geometric Theorems
Introduction
The cluster representing geometric
theorems consists of three different standards.
The first standard is HCG-CO.C.9. This standard
requires students to prove theorems about lines
and angles. The theorems need to include vertical
angles are congruent, when a transversal crosses
parallel lines, alternate interior angles are
congruent and corresponding angles are
congruent; points on a perpendicular bisector of a
line segment are exactly those equidistant from
the segment’s endpoints.
The second standard is HSG-CO.C.10 and
requires student to prove theorems about
triangles. The theorems include measures of
interior angles of a triangle sum to 180 degrees,
base angles of isosceles triangles are congruent,
the segment of joining midpoints of two sides of a
triangle is parallel to the third side and half the
length; the medians of a triangle meet at a point.
The third and last standard of the cluster
is HSG-CO.C.11. This standard has students prove
theorems about parallelograms. The theorems
included are opposite sides are congruent,
opposite angles are congruent, the diagonals of a
parallelogram bisect each other, and conversely,
rectangles are parallelograms with congruent
diagonals.
Students will be required to use
inquisitive learning to determine what each
theorem listed should say. Working in groups will
do this. Each section will include a corresponding
geometric figure in which students will work
together to discover what the theorem that is
being focused on should say. Prompts for the
discussion should encompass the following, but
can be individualized:
 Do you notice any patterns?
 Are there any similarities?
 Would you be able to prove that?
 Why do you think that is true?
 Do you know any previous theorems that
can support that thought?
Additional assignments and supports can
be found in the textbook Geometry authored by
Boswell, Larson, and Stiff. Further, the instructor
should allow for more or less time depending on
the needs of both the class and/or individual
HSG-CO.C.9: Prove theorems about lines and
angles. Theorems include: vertical angles are
congruent; when a transversal crosses
parallel lines, alternate interior angles are
congruent and corresponding angles are
congruent; points on a perpendicular bisector
of a line segment are exactly those equidistant
from the segment’s endpoints.
students.
HSG-CO.C.9
Students will need to be able to prove
theorems about lines and angles in order to
master this section. In order to determine what
theorems should be discussed, students will be led
by the instructor on guided discover to predict
what the theorems should say. There will be three
geometric shapes that will be given to the student
over the time period needed for this section:
crossing lines that form vertical angles,
perpendicular lines that have a transversal, and a
line segment that has a perpendicular.
The students should be allowed to use
rulers, compasses, protractors, or any other tool
that will help them to notice the patterns that are
existing within the section. Once the students have
noticed patterns, it will be up to write their own
theorems about the given geometric figures. It is
at this point that the teacher should give the
students the formal theorem and have the
student’s self evaluate how close their theorem
was to the formal theorem.
This image will help students to determine
that vertical angles are congruent
Formal theorems are as follows:
Vertical Angles are congruent:
If two angles are vertical angles, then they are
congruent.
Alternate Interior Angles:
If two parallel lines are cut by a transversal, then
the pairs of alternate interior angles are
congruent.
Corresponding Angles are Congruent:
If two parallel lines are cut by a transversal, then
the corresponding interior angles are congruent.
The following are assignments that can be used to
reinforce and assess how the students understand
the concepts at hand:
pp. 132-133 (11-25 odd, 33, 35)
pp. 138-140 (1-19 odd)
pp. 146-148 (1-29 odd)
This image will guide students in the
discovery of alternate interior angles,
corresponding angles, exterior angles,
etc.
Perpendicular Bisector
If a perpendicular line is drawn so that it bisects a
line segment, then the new point is equidistant
from the existing points of the line segment.
pp. 153-154 (1-27 odd, 28)
The odd questions were specifically chosen so that
students would be able to compare their answers
with those in the answers reference that the
textbook supplies. This technique allows the
students to self evaluate their master of the
standard and determine if more work needs to be
done.
Lastly, there should be a quiz once students have
finished this section to use as a benchmark
assessment of the standard.
HSG-CO.C.10
Similarly to the section on lines and
angles, students will be given a series of geometric
images to help prove theorems about triangles.
These images will include an equilateral triangle,
an isosceles triangle, and a generic triangle. Again,
students will need to determine, with guided
questions from the instructor, what the theorems
will be. They then will be required to compare the
theorems they developed with the formal
theorems.
HSG-CO.C.10: Prove theorems about
triangles. Theorems include: measures of
interior angles of a triangle sum to 180°;
base angles of isosceles triangles are
congruent; the segment joining midpoints
of two sides of a triangle is parallel to the
third side and half the length; the
medians of a triangle meet at a point.
A list of the needed theorems is as
follows:
Interior Angles sum to 180 degrees
If the polygon is a triangle, the interior angles will
sum to 180 degrees.
Base Angles of an Isosceles Triangle are
Congruent:
If a triangle is an Isosceles triangle, then the base
angles of the triangle are congruent.
Converse:
If the base angles of a triangle are congruent, then
the triangle is isosceles.
Triangle Midpoint Theorem
The segment joining the midpoints of two sides of
a triangle is parallel to the third side and half of
the length.
This triangle can be used for interior
angle theorem, medians theorem, and
midpoint theorem.
The following assignments can be given from the
text for support and assessment:
PP. 198-199 (10-15, 17-21 odd, 22-26, 27-39 odd)
PP. 206-207 (11-31 odd)
PP. 216-218 (1-27 odd)
PP. 223-224 (1-21 odd)
PP. 239-241 (1-33 odd)
This image is an isosceles triangle that
can be used for the base angle
theorems.
Dan Meyer’s Best Triangle Activity
Once the section is over, there should be a quiz on
the properties and theorems of triangles to
provide a benchmark assessment of the standard.
HSG-CO.C.11
The instruction of this standard will have
a great deal of discovery from the students. They
will most likely have worked with parallelograms,
but did not know they were. During this section,
students will be given a parallelogram and asked
to discover what theorems can be derived from
the parallelogram. Students should develop their
own theorems and compare them to the actual
theorem.
Prove theorems about parallelograms.
Theorems include: opposite sides are
congruent, opposite angles are congruent, the
diagonals of a parallelogram bisect each
other, and conversely, rectangles are
parallelograms with congruent diagonals.
The formal theorems are as follows:
Opposite sides are congruent:
If the quadrilateral is a parallelogram then the
opposite sides are congruent.
Opposite angles are congruent:
If the quadrilateral is a parallelogram then the
opposite angles are congruent.
Bisecting Diagonals
If the quadrilateral is a parallelogram then the
diagonals of the parallelogram bisect each other.
Rectangle Diagonals
If the parallelogram has diagonals that are
congruent, then the parallelogram is a rectangle.
The following assignments can be given from the
text for support and assessment:
PP. 333-335 (1-39 odd)
PP. 342 (1-23 odd)
PP. 351-352 (1-41 odd)
Dan Meyer’s Best Square activity
Once this section is complete, there should be a
quiz to use as a benchmark assessment. Further,
this is the end of the cluster of standards.
Therefore, there should be a summative
assessment for the overall unit of geometric
proofs.
This parallelogram can be used to discover
all theorems about parallelograms. It
should be noted that students should
attempt to see if there are any “special”
parallelograms as well.