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CHAPTER
13
Geometry as a Mathematical System
Content Summary Having experienced all the concepts of a standard geometry course, students are ready to examine the framework of the geometry knowledge they have built. Students now review and deepen their understanding of those concepts by proving some of the most important conjectures in the context of a logical system, starting with the premises of geometry.
Premises and Theorems A complete deductive system must begin with some assumptions that are clearly stated and, ideally, so obvious that they need no defense. Chapter 13 begins by laying out its assumptions: properties of arithmetic and equality, postulates of geometry, and a definition of congruence for angles and line segments. These basic assumptions are called premises. Everything else builds on these premises. Next, students develop proofs of their conjectures concerning triangles, quadrilaterals, circles, similarity, and coordinate geometry. Once a conjecture has been proved, it is called a theorem. Each step of a proof must be supported by a premise or a previously proved theorem.
Developing a Proof Developing a proof is more art than science. Mathematicians don’t sit down and write a proof from beginning to end, so encourage your student not to expect to do so. Proofs require thought and creativity. Generally, the student will start by writing down what’s given and what’s to be shown—the beginning and end of the proof. Then, perhaps using diagrams, the student will restate these first and last statements in several ways, looking for an idea of how to get from one to the other logically. You might remind your student about the reasoning strategies that can help in planning a proof: ● ● ● ● ● ●
Draw a labeled diagram and mark what you know Represent a situation algebraically Apply previous conjectures and definitions Break a problem into parts Add an auxiliary line Think backward
Questions you may ask your student include “What can you conclude from the given statements?” and “What’s needed to prove the last statement?” Seeing connections, the student can develop a plan, perhaps expressed using flowcharts. There are several ways to express the plan, and your student may use more than one. Then he or she can write the proof, being careful to check the reasoning. A good way to be careful about details is to write a two-column proof, with statements in the first column and reasons in the second. Your student might find gaps in his or her reasoning and have to go back to the planning stage. If there doesn’t seem to be any way to prove the statement, you might suggest indirect reasoning, in which the student proves that the negation of the theorem is false. It then follows that the theorem must be true.
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Chapter 13 • Geometry as a Mathematical System (continued) Summary Problem Draw a diagram that gives a family tree for all triangle theorems that appear in the exercises for Lesson 13.3. Include postulates and theorems, but not definitions or properties. Questions you might ask in your role as student to your student: ● ●
Can you build on the tree on page 707? What do the arrows represent in words?
Sample Answers A family tree shows how theorems support each other. A theorem may rely on several theorems, each of which relies on other theorems, and so on, all the way up to the postulates of geometry. The top of the family tree should be all postulates, and all the arrows should flow down from there. The diagrams can get quite complex. Don’t worry too much about neatness or completeness; the goal is to see how the structure can be built while reviewing the theorems. Here is the complete tree. SAS Postulate
ASA Postulate
Parallel Postulate
CA Postulate
Linear Pair Postulate
Angle Addition Postulate
SSS Postulate
VA Theorem AIA Theorem Perpendicular Bisector Theorem
Isosceles Triangle Theorem
Triangle Sum Theorem Third Angle Theorem SAA Congruence Theorem
Converse of Isosceles Triangle Theorem
Angle Bisector Theorem
Converse of Angle Bisector Theorem
Angle Bisector Concurrency Theorem
Angle Bisectors to Congruent Sides Theorem
Altitudes to Congruent Sides Theorem
Medians to Congruent Sides Theorem
Perpendicular Bisector Concurrency Theorem
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Converse of Perpendicular Bisector Theorem
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Chapter 13 • Review Exercises Name
Period
Date
1. (Lesson 13.1) Name the property that supports each statement:
CD and CD EF , then AB EF . a. If AB CD , then AB CD. b. If AB 2. (Lessons 13.2, 13.3) In Lesson 13.2, Example B, the Triangle Sum
Theorem is proved with a flowchart proof. Rewrite this proof using a two-column proof. Given: 1, 2, and 3 are the three angles of ABC Show: m1 + m2 + m3 180° 3. (Lessons 13.2, 13.4) Answer the following questions for the statement,
“The diagonals of an isosceles trapezoid are congruent.” a. Task 1: Identify what is given and what you must show. b. Task 2: Draw and label a diagram to illustrate the given information. c. Task 3: Restate what is given and what you must show in terms of
your diagram. 4. (Lesson 13.6) Write a proof for the Parallel Secants Congruent Arcs
Theorem: Parallel lines intercept congruent arcs on a circle. 5. (Lesson 13.7) Write a proof for the Corresponding Altitudes Theorem: If
two triangles are similar, then corresponding altitudes are proportional to the corresponding sides.
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SOLUTIONS TO CHAPTER 13 REVIEW EXERCISES
1. a. Transitive Property b. Definition of Congruence 2. K 4
C
2 5
1
A
3
B
Statement
Reason
1, 2, and 3 of ABC
Given
AB Construct KC
Parallel Postulate
1 4; 3 5
Alternate Interior Angles Theorem
m1 m4; m3 m5
Definition of Congruence
m4 m2 mKCB
Angle Addition Postulate
KCB and 5 are supplementary
Linear Pair Postulate
mKCB m5 180°
Definition of Supplementary
m4 m2 + m5 180°
Substitution Property of Equality
m1 m2 m3 180°
Substitution Property of Equality
3. a. Given: Isosceles trapezoid Show: Diagonals are congruent b. A D
B
C
DC ; AD BC c. Given: AB Show: AC DB 4.
B C A D
DC Given: AB Show: AD BC
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Statement
Reason
DC AB
Given
Construct AC
Line Postulate
DCA BAC
Alternate Interior Angles Theorem
mDCA mBAC
Definition of Congruence
1 mBAC 2
Multiplication Property of Equality
1 mDCA 2
1mDCA mAD 2
Inscribed Angle Theorem
1mBAC mBC 2
Inscribed Angle Theorem
mA A D mBC
Substitution Property of Equality
BC AD
Definition of Congruence
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J C K
B
L
R
P
D
and JR Given: CBD JKL; Altitudes CP C P CB Show: JR JK
Statement
Reason
and JR CBD JKL; Altitudes CP
Given
BD ; JR KL CP
Definition of Altitude
CPB and JRK are right angles
Definition of Perpendicular
CPB JRK
Right Angles are Congruent Theorem
CBD JKL
Corresponding Angles of Similar Triangles are Congruent
CBP JKR
AA Similarity Postulate
C P C B JR JK
Corresponding Sides of Similar Triangles are Proportional
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