HCPSS Curriculum Framework
Common Core Geometry, Unit 2
Triangles, Proof, and Trigonometry
Overview (Big Ideas):
Triangles
Students will use their knowledge of similarity and congruence to build an understanding of
similar and congruent triangles. They will use similarity transformations to establish the AA and
other Triangle Similarity Theorems and rigid motions to establish the SSS, SAS, and ASA
Triangle Congruence Theorems. Students will understand that congruence is a special case of
similarity, where the ratio of side lengths is 1:1. They will use multiple formats to prove various
theorems about triangles throughout the unit, including those pertaining to congruence, midsegments, and medians.
Right Triangle Trigonometry
Students will use their knowledge of similarity of right triangles (and other triangles, in
Geometry GT) to establish an understanding of the trigonometric ratios of angles in these
triangles. They will understand the interrelationships between the trigonometric functions. They
will use these ratios, and the Pythagorean theorem to solve right triangles, given various initial
information.
In Geometry GT, students will further their exploration or triangles using the Law of Sines and
the Law of Cosines. They will both solve triangles that are not right triangles and they will
develop and use a general formula for the area of triangles which uses trigonometry.
Enduring Understandings:
1. A diagram is a sophisticated mathematical device for thinking and communicating. A
diagram is a built geometric artifact, with both a history- a narrative of successive
construction- and a purpose. A diagram is not a picture. It needs to be interpreted:
learning how to read a diagram can be like learning a new language.
2. Empirical verification is an important part of the process of proving, but it can never, by
itself, constitute a proof. Geometry uses a wide variety of kinds of proofs.
3. The processes of proving include a variety of activities, such as developing conjectures,
considering the general case, exploring with examples, looking for structural similarities
across cases, and searching for counterexamples.
4. Making sense of others’ arguments and determining their validity are proof-related
activities.
5. A proof can have many different valid representational forms, including narrative,
picture, diagram, two-column presentation, or algebraic form.
6. Underlying any geometric theorem is an invariance- something that does not change
while something else does.
7. Similar geometric figures have angles that are congruent and segments that are
proportional in length.
8. Similar geometric figures can be created by transformations. All transformations create
similar geometric figures. Dilations, in particular create figures that are similar, but may
not be congruent.
9. Congruence is also similarity. It is just a more specifically defined similarity where the
ratio of lengths is 1:1.
Sources:
Ellis, A. B., Bieda, K., & Knuth, E. (2012). Developing Essential Understanding of Proof and
Proving. Reston, VA: The National Council of Teachers of Mathematics, Inc.
Pimm, D., Sinclair, N., & Skelin, M. (2012). Developing Essential Understanding of Geometry.
Reston, VA: The National Council of Teachers of Mathematics, Inc.
Essential Questions:
Triangles
● What are the criteria for triangles to be similar? How does this relate to transformations?
● How can a line drawn parallel to one of the sides of a triangle be used to solve problems
involving triangles? How can the median be used?
● How does the perpendicular bisector of a segment relate to isosceles triangles? How can
constructions be used to verify this?
● What are the properties of isosceles triangles and how can they be used to solve
problems?
● How can the side of a triangle be partitioned into segments of a given ratio? How can
this information be used to solve problems involving similar triangles?
● What are the criteria for triangles to be congruent? How does this relate to
transformations?
● How does congruency relate to similarity?
● How can similarity and congruence be used to solve problems and/or prove statements
about or properties of triangles?
Right Triangle Trigonometry
● How are the values of the trigonometric functions of right triangles connected to the
similarity of the right triangles?
● How are the sine and cosine related to each other?
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What relationships exist between the trigonometric functions and the Pythagorean
theorem?
How can the trigonometric ratios and the Pythagorean theorem be used to solve a right
triangle?
+ How can triangles that are not right triangles be solved? (GT Geometry)
+ How can the Law of Sines and the Law of Cosines be established or proven? (GT
Geometry)
+ How can the area of a triangle be found using the Laws of Sines and Cosines, and by
using the general formula for area (A=1/2 a b sin(C)? (GT Geometry)
Common Misconceptions:
Triangles
● Students struggle to see congruence as a type of similarity.
● Students have difficulty when setting up proportions for similar triangles, particularly
when triangles overlap in any way or share common sides or parts of sides. Students
have difficulty seeing both triangles.
● Students have trouble visualizing the various congruence and similarity theorems.
● Triangle proofs tend to be longer and more difficult for students.
Right Triangle Trigonometry
● Students see the answers to the sine, cosine or tangent of an angle as a “magical” number
that helps to solve triangles and not as a ratio of sides of all right triangles with the given
acute angle.
● Using memory devices (like SOHCAHTOA) help students to remember which ratio of
two sides each trigonometric function uses. However, care must be taken that these do
not mute the conceptual understanding of the ratios of similar triangles.
● +Students see Law of Sines and Cosines as universally applicable to all given
information when solving a triangle, when in fact the type of information given (various
combinations of sides and angles) dictates which law can be used. (Geometry GT)
● +Students believe that there is only one possible answer to a solved triangle. The
ambiguous case, when ASS information is given, leads to the possibility of two solutions.