Common Core Geometry Unit 1 Starting Points
Unit 2: Triangles, Proof, and Similarity
Part 1: Triangles, Constructions, and Proof
Essential Questions:
o What are the criteria for triangles to be similar? How does this relate to
transformations?
o 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?
o 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?
o How does the perpendicular bisector of a segment relate to isosceles triangles?
How can constructions be used to verify this?
o What are the properties of isosceles triangles and how can they be used to
solve problems?
o What are the criteria for triangles to be congruent? How does this relate to
transformations?
o How does congruency relate to similarity?
o How can similarity and congruence be used to solve problems and/or prove
statements about or properties of triangles?
Curriculum Standards:
Understand similarity in terms of similarity transformations.
G.SRT.A.3 Use properties of similarity transformations to establish the AA criterion for
two triangles to be similar.
Prove geometric theorems. (This standard will be embedded throughout this unit)
G.CO.C.10 Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to; 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.
Prove theorems involving similarity.
G.SRT.B.4 Prove theorems about triangles. Theorems include: a line parallel to one
side of a triangle divides the other two proportionally, and conversely; the Pythagorean
Theorem proved using triangle similarity.
Make geometric constructions.
G.CO.C.12 Make formal geometric constructions with a variety of tools and methods
(compass and straightedge, string, reflective devices, paper folding, dynamic geometric
software, etc.). bisecting a segment; bisecting an angle; constructing perpendicular
lines, including the perpendicular bisector of a line segment.
This document represents one sample starting points for the unit. It is not all-inclusive and is only
one planning tool. Please refer to the wiki for more information and resources.
Prove geometric theorems.
G.CO.C.9 Prove theorems about lines and angles. Theorems include: points on a
perpendicular bisector of a line segment are exactly those equidistant from the segment’s
endpoints.
Use coordinates to prove simple geometric theorems algebraically.
G.GPE.B.6 Find the point on a directed line segment between two given points that
partitions the segment in a given ratio.
Understand congruence in terms of rigid motions.
G.CO.B.7 Use the definition of congruence in terms of rigid motions to show that two
triangles are congruent if and only if corresponding pairs of sides and corresponding pairs
of angles are congruent.
Understand congruence in terms of rigid motions.
G.CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
Prove theorems involving similarity.
G.SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures.
Approximate Length: 30 days
Standard(s)
Days
Notes
G.CO.C.10
1-2
Big Ideas:
The interior angles of triangles sum to 180
degrees.
In any triangle, the sum of two sides must be
greater than the third.
Theorems to
Prove/Develop
Angle Sum
theorem
Resources:
 Lesson: Triangle Sum
 Proof: Sketchpad Tutorial
G.SRT.A.3
G.SRT.B.5
5-8
Big Ideas:
Triangles that have two congruent angles are
always similar.
AA Similarity
Proportions can be used to find missing values
for similar figures.
Similar triangles can be used to solve real-world
problems.
This document represents one sample starting points for the unit. It is not all-inclusive and is only
one planning tool. Please refer to the wiki for more information and resources.
G.SRT.B.4
G.GPE.B.6
G.CO.C.10
G.SRT.B.5
6-8
Resources:
 Lesson: Angle Angle Similarity
Discovery
Big Ideas:
A line parallel to one side of a triangle will
divide the other two proportionally, creating
similar triangles.
Side Splitter
theorem
Triangle
Midsegment
theorem
A directed segment can be divided
proportionally on the coordinate plane, and this
can be used to create similar right triangles.
The midsegment of a triangle is half of the
length of the side to which it is parallel. This
information can be used to solve problems.
Parallel line theorems can be used to prove
similarity in triangle problems.
Similar triangles can be used to solve real-world
problems.
Resources:
 Lesson: Line Parallel to Side of a
Triangle Investigation
 Lesson: Intro to Partitioning a Segment
 Lesson: Partitioning a Segment
 Proof: Sketchpad Tutorial
Assessment Items:
 Illustrative Mathematics: Joining two
midpoints of sides of a triangle
G.CO.C.12
G.CO.C.10
2-3
Big Ideas:
Different points of concurrency occur when the
angles (incenter) or sides (circumcenter) of a
triangle are bisected.
There is also a point of concurrency where the
medians intersect (centroid).
These points of concurrency can help you to
solve different real-world problems.
Resources:
 Lesson: Geometric Constructions
 Lesson: Investigating the Centroid
This document represents one sample starting points for the unit. It is not all-inclusive and is only
one planning tool. Please refer to the wiki for more information and resources.
 Lesson Seed: Investigating Incenters and
Circumcenters
 Construction: Patty Paper Tutorials
Assessment Items:
 PARCC: Geometric Construction
Connection
 Illustrative Mathematics: Bisecting an
Angle
G.CO.B.7
G.CO.B.8
G.SRT.B.5
6-9
Big Ideas:
Congruence is a special case of similarity.
Rigid motion can be used to develop
congruence criteria for triangles.
SSS, ASA, SAS,
AAS congruence
criteria
CPCTC
Congruence theorems can be used to prove
statements and solve real-world problems.
Resources:
 Task: Bulletin Board Congruence
 Lesson: verifying Triangle Congruence
 Lesson: Criteria for Triangle
Congruence
Assessment Items:
 Illustrative Mathematics: Are the
Triangles Congruent?
 Illustrative Mathematics: Reflections
and Equilateral Triangles
 Illustrative Mathematics: Reflections
and Equilateral Triangles II
This document represents one sample starting points for the unit. It is not all-inclusive and is only
one planning tool. Please refer to the wiki for more information and resources.
G.CO.C.9
G.CO.C.10
4-6
Big Ideas:
Any point on the perpendicular line segment is
equidistant from the segment’s endpoints. This
can be used to develop theorems regarding
isosceles triangles.
Perpendicular
Bisector theorem
Base Angles
theorem
The properties of isosceles triangles can be
proven and used to solve real-world problems.
Resources:
 Task: Sailing with Congruence
 Task: Pennant Pride
 Task: High Scorer
 Proof: Patty Paper Tutorials
 Proof: Proof Blocks
 Proof: Sketchpad Tutorial
Assessment Items:
 Illustrative Mathematics: Reflections
and Isosceles Triangles
 Illustrative Mathematics: Points
Equidistant from Two Points in a Plane
 Illustrative Mathematics: Angle
Bisection and Midpoints of Line
Segments
G.SRT.B.5
G.CO.C.11
3-4
Big Ideas:
Congruence and similarity can be used to solve
real-world problems, which can extend to
include parallelograms.
Properties of
parallelograms
Theorems involving parallelograms can be
proven formally as an extension of triangle
proofs.
Assessment Items:
 Illustrative Mathematics: Extensions,
Bisections, and Dissections in a
Rectangle
This document represents one sample starting points for the unit. It is not all-inclusive and is only
one planning tool. Please refer to the wiki for more information and resources.
Part 2: Right Triangle Trigonometry
Essential Questions:
● 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?
● 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)
Curriculum Standards:
Define trigonometric ratios and solve problems involving right triangles.
G.SRT.B.6 Understand that by similarity, side ratios in right triangles are properties of
the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
Define trigonometric ratios and solve problems involving right triangles.
G.SRT.C.7 Explain and use the relationship between the sine and cosine of
complementary angles.
Define trigonometric ratios and solve problems involving right triangles.
G.SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangle
in applied problems.
Apply trigonometry to general triangles. (Geometry GT)
G.SRT.D.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems.
Apply trigonometry to general triangles. (Geometry GT)
G.SRT.D.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find
unknown measurements in right and non-right triangles (e.g., surveying problems,
resultant forces).
Apply trigonometry to general triangles. (Geometry GT)
G.SRT.D.9(+) Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing
an auxiliary line from a vertex perpendicular to the opposite side.
This document represents one sample starting points for the unit. It is not all-inclusive and is only
one planning tool. Please refer to the wiki for more information and resources.
Approximate Length: 12-15 days (18-20 for G/T)
Standards
Days
Notes
G.SRT.B.4
2-3
Big Ideas:
Altitudes to the hypotenuse of a right
triangle create similar triangles that can be
used to solve problems.
These similar triangles can be used to
prove the Pythagorean Theorem.
Theorems to
Prove/Develop
Pythagorean
theorem (using
similar triangles)
Resources:
 Lesson Seed: Pythagorean
Theorem Proofs
 Proof: Sketchpad Tutorial
Assessment Items:
 Illustrative Mathematics: Points
from Directions
G.SRT.B.6
G.SRT.C.7
G.SRT.C.8
12-13
Big Ideas:
The definitions of trigonometric ratios can
be developed through similarity.
Trigonometric ratios can be used in
conjunction with the Pythagorean theorem
to solve real-world problems involving
right triangles.
The sine of an angle is equal to the cosine
of that angle’s complement. This can be
used to solve problems.
Resources:
 Task: How Tall Is It?
 Lesson: Similar Right Triangles
and Trig Ratios
 Lesson Seed: Sine and Cosine of
Complementary Angles
 Lesson: Inverse Trig
 Exploration: Developing 45-45-90
Triangles
 Exploration: Developing 30-60-90
Triangles
This document represents one sample starting points for the unit. It is not all-inclusive and is only
one planning tool. Please refer to the wiki for more information and resources.
Additional G/T Standards
G.SRT.D.10 (+)
4-5
G.SRT.D.11 (+)
Big Ideas:
The lengths of the sides of a triangle
can be related to the sine or cosine of
its angles in order to solve for missing
values.
Law of Sines
Law of Cosines
Law of Sines can lead to ambiguity in
problems involving SSA.
The Laws of Sines and Cosines can be
used to solve real-world problems.
Resources:
 Illuminations: Deriving Law of
Sines and Law of Cosines
 Task: Campus Conundrum
 Lesson: Applying Laws of
Sines and Cosines
G.SRT.D.9(+)
1-2
Big Ideas:
The sine ratio can be used to find the
area of a triangle.
Resources:
 Task: Can We Build It?
 Lesson: Discovering a Formula
for the Area of a Triangle
 Proof: Sketchpad Tutorial
Assessment Items:
 Illustrative Mathematics;
Finding the Area of an
Equilateral Triangle
Howard County Public Schools Office of Secondary Mathematics Curricular Projects has
licensed this product under a Creative Commons Attribution-NonCommercial-NoDerivs
3.0 Unported License.
This document represents one sample starting points for the unit. It is not all-inclusive and is only
one planning tool. Please refer to the wiki for more information and resources.