Course Name: Geometry/Math II
Unit 6
Unit Title: Congruence and Special Segments
Enduring understanding (Big Idea): Students will understand that a) you can determine if 2 figures are congruent by comparing
corresponding parts b) triangles can be proven congruent without having to compare all corresponding parts c) the angles and sides of
Isosceles and Equilateral triangles have special relationships.
Essential Questions:
1) How do you identify corresponding parts of congruent triangles?
2) How do you show that 2 triangles are congruent?
3) How can you tell whether a triangle is isosceles or equilateral?
4) How do you solve problems that involve measurements of triangles?
BY THE END OF THIS UNIT:
Students will know…
Students will be able to:
 That two figures are congruent if a series of rigid motions
 use the definition of congruence in terms of rigid
carries one onto the other.
motions to show that two triangles are congruent if and
only if corresponding pairs of sides and corresponding
 That two triangles are congruent if all corresponding pairs
pairs of angles are congruent.
of sides are congruent and all corresponding pairs of
 explain how the criteria for triangle congruence (ASA,
angles are congruent.
Vocabulary: Congruent polygons , Third Angle Theorem, congruent
SAS, and SSS) follow from the definition of
triangles, SSS, SAS, and ASA Postulates, AAS Theorem, included
congruence in terms of rigid motions.
angle hypotenuse, legs of a right, Triangle Angle-Sum Theorem,
 Use properties of midsegments to solve problems
Exterior angle, remote interior angle, auxiliary line, legs, base, vertex
 Use properties of perpendicular and angle bisectors to
angle, base angles, Isosceles Triangle Theorem and Converse,
solve problems
corollary, hypotenuse, legs, HL Theorem, midsegment of a triangle,
Triangle Midsegment Theorem, median of a triangle, Concurrency of
Medians Theorem, centroid of the triangle circumcenter, concurrent,
equidistant, incenter, median, midsegment and altitude of a triangle,
orthocenter, bisector
Unit Resources:
Mathematical Practices in Focus:
Learning Task: Intro to Triangle Congruence
1. Make sense of problems and persevere in solving
Performance Task: From CMS Wikispaces
them
Project: fettuccine noodles construction project
3. Construct viable arguments and critique the reasoning
Online interactive website: students can click and drag
of others
To form congruent triangles
8. Look for and express regularity in repeated reasoning
Unit Review Game: Jeopardy Style Game
Abbreviation Key:
Test Specification Weights for the Common Exams in Common CC – Common Core Additional Lessons found in the Pearson
online materials.
Core Math II:
CB- Concept Bytes found in between lessons in the Pearson
Category
textbook.
% Constructed % MultipleStandard
Percentage
ER – Enrichment worksheets found in teacher resources per
Response
Choice
(Geometry)
chapter.
G-CO
0%
16% to 19%
27% to 30%
Suggested Order/Pacing:
Congruent Figures: 4.1
Triangle Congruence by SSS and SAS: 4.2
Triangle Congruence by ASA and AAS: 4.3
CPCTC: 4.4 (Brief)
Isosceles and Equilateral Triangles: 4.5
Congruence in Right Triangles: 4.6
Midsegments of Triangles: 5.1
Bisectors in Triangles: 5.3 (Use 5.2 as introduction)
Medians and Altitudes: 5.4
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 1
CCSS-M Included: G.CO.7, G.CO.8, G.CO.10
Instructional Expectations
The expectation for both pathways, in Math II and Math I, is to
understand that rigid motions are at the foundation of the
definition of congruence. Students reason from the basic
properties of rigid motions (that they preserve distance and
angle), which are assumed without proof. Rigid motions and
their assumed properties can be used to establish the usual
triangle congruence criteria, which can then be used to prove
other theorems.
Course Name: Geometry/Math II
Unit 6
Unit Title: Congruence and Special Segments
CORE CONTENT
Cluster Title: Understand congruence in terms of rigid motion
Standard G-CO.7 Use the definition of congruence in terms of rigid motion to show that 2 triangles are congruent if and only if
corresponding pairs of sides and corresponding pairs of angles are congruent.
Concepts and Skills to Master:
 Recognize congruent figures and their corresponding parts
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
Proving angles congruent
Academic Vocabulary:
Congruent polygons , Third Angle Theorem
Suggested Instructional Strategies:
Starting Resources:
 Have students draw the given figures so that each is oriented the same way Geometry Textbook Correlation:
Section 4.1
 Have students “work backwards” in the proof, i.e. start with the desired
conclusion work backwards to identify information given and needed
NCDPI Unpacking:
What does this standard mean that a student will know and be able to do?
Use the definition of congruence, based on rigid motion, to show two triangles are
congruent if and only if their corresponding sides and corresponding angles are
congruent.
This standard connects with GSRT.3. Students should connect that two triangles are
congruent if and only if they are similar with a scale factor of one. Building on their
definition of similarity, this means that a rigid transformation will preserve the angle
measures and the sides will change (or not change) by a scale factor of one.
Additional note from DPI for Level II:
Using Interactive Geometry Software or graph paper graph the following triangle M(1,1), N(-4,2) and P(-3,5) and perform the following transformations. Verify that the
preimage and the image are congruent. Justify your answer.
a. (x,y)→(x+2, y-6)
b. (x,y)→(-x,y)
c. (x,y)→(-y,x)
Sample Assessment Tasks
Skill-based task:
Complete the following statements:
Given: ΔQXR
ΔNYC
a) line segment QX
line segment _?_
b) Y
_?_
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 2
Problem Task:
If each angle in one triangle is congruent to its
corresponding angle in another triangle, are
the two triangles congruent? Explain.
Course Name: Geometry/Math II
Unit 6
Unit Title: Congruence and Special Segments
CORE CONTENT
Cluster Title: Understand congruence in terms of rigid motion
Standard: G-CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in
terms of rigid motions.
Concepts and Skills to Master:
 Prove 2 triangles congruent using ASA, SAS, AAS, SSS, and HL Theorem
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
Parts of a right triangle and congruent corresponding parts
Academic Vocabulary:
congruent triangles, SSS, SAS, and ASA Postulates, AAS Theorem, included angle hypotenuse, legs of a right triangle
Suggested Instructional Strategies:
Starting Resources:
Geometry Textbook
 Point out that for SAS the included angle MUST be between the 2 sides used in the proof
Correlation:
 Likewise, for ASA the included side MUST be between the 2 angles used in the proof
 Point out that ASS is NOT method of proving triangles congruent nor is AAA. However, in a right Sections 4.2, 4.3, 4.6
triangle, what students see as ASS can often be HL.
 Use patty paper investigation to allow students to discover which combinations work: SSS,
SAS, ASA, AAS, SSA, HL
NCDPI Unpacking:
What does this standard mean that a student will know and be able to do?
Use the definition of congruence, based on rigid motion, to develop and explain the triangle congruence
criteria; ASA, SSS, SAS, AAS, and HL. Students should connect that these triangle congruence criteria are
special cases of the similarity criteria in GSRT.3. ASA and AAS are modified versions of the AA criteria for
similarity. Students should note that the “S” in ASA and AAS has to be present to include the scale factor of
one, which is necessary to show that it is a rigid transformation. Students should also investigate why SSA and
AAA are not useful for determining whether triangles are congruent.
Additional note from DPI for Level II:
Andy and Javier are designing triangular gardens for their yards. Andy and Javier want to determine if their
gardens that they build will be congruent by looking at the measures of the boards they will use for the
boarders, and the angles measures of the vertices. Andy and Javier use the following combinations to build
their gardens. Will these combinations create gardens that enclose the same area? If so, how do you know?
a. Each garden has length measurements of 12ft, 32ft and 28ft.
b. Both of the gardens have angle measure of 110°, 25° and 45°.
c. One side of the garden is 20ft another side is 30ft and the angle between those two boards is 40°.
d. One side of the garden is 20ft and the angles on each side of that board are 60° and 80°.
e. Two sides measure 16ft and 18ft and the non-included angle of the garden measures 30°.
Students can make sense of this problem by drawing diagrams of important features and relationships and
using concrete objects or pictures to help conceptualize and solve this problem.
Sample Assessment Tasks
Skill-based task:
Problem Task:
Draw and label three pairs of triangles to illustrate the Side-Side-Side, Angle-Side-Angle, and SideAngle-Side Postulates. One pair of triangles should share a common side. The figures should provide
enough information to prove that they are congruent. Write the congruence statements for each pair.
Teacher Created Argumentation Tasks (W1-MP3&6)
What types of conditions help you to decide which method of triangle congruency to use when writing a proof? After reading the
Prentice Hall text sections on writing proofs for proving triangles congruent, listening in class to class instructions and demonstrations,
and using the Prentice Hall CD Rom resources to read and listen along to information regarding writing proofs to prove triangle
congruency, write a letter to a friend that was absent multiple days in class that compares using ASA, AAS, SSS, SAS, and HL
Theorem and argue what method is best to use. Include what types of conditions help you decide which method to use. Be sure to
support your position with evidence from any of the texts.
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 3
Course Name: Geometry/Math II
Unit 6
Unit Title: Congruence and Special Segments
CORE CONTENT
Cluster Title: Prove Geometric Theorems
Standard G-CO.10 Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of 2 sides of a triangle is parallel to the third side and half the length; the medians of a
triangle meet at a point.
Using logic and deductive reasoning, algebraic and geometric properties, definitions, and proven theorems to draw conclusions.
(Encourage multiple ways of writing proofs, to include paragraph, flow charts, and two-column format).
These proof standards should be woven throughout the course. Students should be making arguments about content throughout their
geometry experience. The focus in the G-CO.9-11 is not the particular content items that they are proving. However, the focus is on the
idea that students are proving geometric properties. Pay close attention to the mathematical practices especially number three,
“Construct viable argument and critique the reasoning of others.”
Concepts and Skills to Master:
 Use parallel lines to prove a theorem about triangles
 Find measures of angles in triangles
 Prove right triangles congruent using the Hypotenuse-Leg Theorem
SUPPORTS FOR TEACHERS
Critical Background Knowledge:
 The sum of the angle measures of triangles is always the same i.e. 180 degrees
 Understanding the parts of a right triangle
Academic Vocabulary:
Triangle Angle-Sum Theorem, Exterior angle, remote interior angle, auxiliary line, legs, base, vertex angle, base angles, Isosceles
Triangle Theorem and Converse, corollary, hypotenuse, legs, HL Theorem, midsegment of a triangle, Triangle Midsegment Theorem,
median of a triangle, Concurrency of Medians Theorem, centroid of the triangle
Suggested Instructional Strategies:
Starting Resources:
Use a number of examples with exterior angles and show how the 2 remote angles add up to the
Geometry Textbook
exterior angle. Draw the correlation between triangle interior angles totaling 180° and an exterior angle Correlation:
and its linear pair inside the triangle totaling 180°
Sections 3.5, 4.1, 4.2, 4.3,
4.5, 4.6, 5.1, 5.4
NCDPI Unpacking:
What does this standard mean that a student will know and be able to do?
Video resource:
Prove theorems pertaining to triangles.
Introduction Congruent
 Prove the measures of interior angles of a triangle have a sum of 180º.
Triangles
 Prove base angles of isosceles triangles are congruent.
 Prove the segment joining midpoints of two sides of a triangle is parallel to the third side and half
the length.
 Prove the medians of a triangle meet at a point.
Using any method you choose, construct the medians of a triangle. Each median is divided up by the
centroid. Investigate the relationships of the distances of these segments. Can you create a deductive
argument to justify why these relationships are true? Can you prove why the medians all meet at one point
for all triangles? Extension: using coordinate geometry, how can you calculate the coordinate of the
centroid? Can you provide an algebraic argument for why this works for any triangle?
Using Interactive Geometry Software or tracing paper, investigate the relationships of sides and angles
when you connect the midpoints of the sides of a triangle. Using coordinates can you justify why the
segment that connects the midpoints of two of the sides is parallel to the opposite side. If you have not done
so already, can you generalize your argument and show that it works for all cases? Using coordinates justify
that the segment that connects the midpoints of two of the sides is half the length of the opposite side. If you
have not done so already, can you generalize your argument and show that it works for all cases?
Sample Assessment Tasks
Skill-based task:
Problem Task:
Pearson website Lesson 3.5 and 4.5 Enrichment and Solve it! 3.5,
Pearson website Activities, Games and Puzzles 3.5
Standards are listed in alphabetical /numerical order not suggested teaching order.
PLC’s must order the standards to form a reasonable unit for instructional purposes.
PAGE 4