Written to the Common Core State Standards for Mathematics, Discovering Geometry creates learning trajectories that match the progression of mathematics content from middle school to high school.

Discovering Geometry encourages students to inquire about concepts and ideas and provides the opportunity to experiment, hypothesize, measure, analyze, test, talk, write, explain, and justify their ideas as they explore principles of geometry.

Get Point-of-Use Access to the StandardsTo provide a more useful and authentic correlation to the CCSS, Discovering Geometry lessons differentiate the mathematical standards.

Applied refers to standards that were covered in previous courses and lessons. Developed identifies the standards that are most closely aligned to the learning objectives within the lessons. Introduced standards are those for which conceptual groundwork is being laid and will be covered in greater depth in future lessons.

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NEWEdition

These standards are addressed in Chapter 4, Lesson 5. They are easily found in the Teacher Edition wraparound and through a menu in the Teacher’s eBook.

The four-step LISA instructional model ensures lessons provide the required rigor and coherence that is called for in the CCSS.

Launch – connects lesson learning objectives to prior learning and engages students in the mathematics to be covered

Investigate – develops understanding and skills through engaging, scaffolded tasks

Summarize – helps students abstract math concepts and make important mathematical connections

Apply – demonstrates mastery of lesson objectives by using math procedures and conceptual understanding to solve new problems

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Digital eBooks and Resources Support LearningDiscovering Geometry student and teacher eBooks are compatible with most Internet-enabled devices and are accessible through Flourish, the Kendall Hunt digital learning network. They contain all the content of the print textbooks with additional features that make teaching easier and enhance student learning.

Users also get access to CAS, Kendall Hunt’s cohesive assessment system. It is accessible from any Web browser and can be used to manage assessment content, create and assign online tests, provide feedback, and analyze student performance data.

INVESTIGATION

204 C H A P T E R 4 Discovering and Proving Triangle Properties

LESSON

4.1

“Teaching is the art of

assisting discovery.”

ALBERT VAN DOREN

Triangle Sum Conjecture

Triangles have certain

properties that make

them useful in all kinds of

structures, from bridges to

high-rise buildings. One

such property of triangles

is their rigidity. Another

application of triangles is a

procedure used in surveying

called triangulation. This

procedure allows surveyors

to locate points or positions

on a map by measuring

angles and distances and

creating a network of triangles. Triangulation is based on an important property of

plane geometry that you will discover in this lesson.

The Triangle Sum

There is an endless variety of triangles that you can draw

with different shapes and angle measures. Do their angle

measures have anything in common? Start by drawing

different kinds of triangles. Make sure your group has at

least one acute and one obtuse triangle.

Step 1 Measure the three angles of each triangle as

accurately as possible with your protractor.

Step 2 Find the sum of the measures of the three angles

in each triangle. Compare results with others

in your group. Does

everyone get about the

same result? What is it?

Step 3 Check the sum another

way. Write the letters a,

b, and c in the interiors

of the three angles of

one of the triangles,

and carefully cut out

the triangle.

a

b

c

YOU WILL NEED

• a protractor

• a straightedge

• scissors

• patty paper

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Advanced

Support

ELL

INVESTIGATION

204 C H A P T E R 4 Discovering and Proving Triangle Properties

LESSON

4.1

“Teaching is the art of assisting discovery.”ALBERT VAN DOREN

Triangle Sum ConjectureTriangles have certain properties that make them useful in all kinds of structures, from bridges to high-rise buildings. One such property of triangles is their rigidity. Another application of triangles is a procedure used in surveying called triangulation. This procedure allows surveyors to locate points or positions on a map by measuring angles and distances and creating a network of triangles. Triangulation is based on an important property of

plane geometry that you will discover in this lesson.

The Triangle SumThere is an endless variety of triangles that you can draw

with different shapes and angle measures. Do their angle

measures have anything in common? Start by drawing

different kinds of triangles. Make sure your group has at

least one acute and one obtuse triangle.Step 1 Measure the three angles of each triangle as accurately as possible with your protractor.

Step 2 Find the sum of the measures of the three angles in each triangle. Compare results with others in your group. Does everyone get about the same result? What is it?Step 3 Check the sum another way. Write the letters a, b, and c in the interiors of the three angles of one of the triangles, and carefully cut out the triangle.

ab

c

YOU WILL NEED• a protractor• a straightedge• scissors• patty paper

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204 C H A P T E R 4 Discovering and Proving Triangle Properties

LESSON

4.1

Pose the question from the One Step. Students may be surprised to find that the angle measures of both triangles are about the same. Remind students that real-world problems often make hidden assumptions, so students need to be skeptical.

Demonstrate the investiga-tion as a follow-along activity by cutting out the triangles, manipulating, and measuring the pieces. Introduce paragraph proof by having students brain-storm all the statements that are true about the diagram. Complete the proof together as a class.

Demonstrate how to find the sum of the angles in a trian-gle and make sure students understand the definition and location of interior angles. Using the visuals in Developing Proof will help explain the necessity of including an auxiliary line.

In this lesson students discover the Triangle Sum conjecture. They begin to formalize proof by using deductive strategies to write a paragraph proof. COMMON CORE STATE STANDARDSApplied

G.CO.128.G.5

DevelopedG.CO.10 IntroducedG.CO.4

Objectives •Discover and explain the Triangle

Sum conjecture.•Use deductive reasoning to create a paragraph proof. • Learn the strategy of adding an auxiliary line to help with a proof. Vocabulary

auxiliary lineparagraph proof

Materials• construction tools•protractors

• scissors

LaunchConstruct a large right triangle. Measure the angles and find their sum.Bisect the right angle to create two

new triangles. Find the sum of the angles in each triangle.

What do you notice? How do these sums compare to each of the triangles? Answers will vary.

InvestigateIn this investigation, students find the sum of the measures of the angles of a triangle using two different methods.

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Discovering Geometry provides hands-on Investigations that provide opportunities for students to work in cooperative groups, share their ideas verbally and in writing, and discover ways to connect geometry concepts to the world around them.

Teacher eBook

Dynamic Explorations provide students and teachers with interactive models that illustrate key mathematical concepts.

Teacher eBook