Based on the marks on the diagram, we know that:
(single tick marks).
(double tick marks).
(single arc marks).
(double arc marks).
Lesson Summary
As we move forward toward more formal reasoning, we have reviewed the basic postulates and expressed them more formally. We saw that most geometric situations involve diagrams. In diagrams we can assume some facts, and we cannot assume others.
Points to Consider
In upcoming lessons you will organize your reasoning pattern into the two-column proof. This is a traditional pattern that still works very well today. It gives us a clear, direct format, and uses the basic rules of logic that we saw in earlier lessons. We will prove many important geometric relationships called theorems throughout the rest of this geometry course.
Review Questions
Use the diagram to answer questions 1-8.
Name a right angle.
Name two perpendicular lines (not segments).
Given that is true? Explain your answer completely.
Given that is a rectangle? Explain your answer informally. (Note: This is a new question. Do not assume that the given from a previous question is included in this question.)
Fill in the blanks: . Why?
. Why?
Fill in the blanks:
Given that , prove
Given that , prove:
What geometric objects does the real-world model suggest?
Model: two railroad tracks
Model: a floor and a ceiling
Model: two lines on a piece of graph paper
Model: referee’s arms when signaling a touchdown
Model: capital letter
Model: the spine of a book where the front and back covers join
Review Answers
and
Yes
Yes. It’s given that (so ). Since and then must be equal to , and this would make a rectangle.
Parallel lines
Parallel planes
Parallel or perpendicular lines
Parallel lines or segments
Perpendicular segments
Intersecting planes
Two-Column Proof
Learning Objectives
Draw a diagram to help set up a two-column proof.
Identify the given information and statement to be proved in a two-column proof.
Write a two-column proof.
Introduction
You have done some informal proofs in earlier sections. Now we raise the level of formality higher. In this section you will learn to write formal two-column proofs. You’ll need to draw a diagram, identify the given and prove, and write a logical chain of statements. Each statement will have a reason, such as a definition, postulate, or previously proven theorem, that justifies it.
Given, Prove, and Diagram
Example 1
Write a two-column proof for the following:
If , , , and are points on a line, in the given order, and , then .
Comments: The if part of the statement contains the given. The then part is the section that you must prove. A diagram should show the given facts.
We start with the given, prove, and a diagram.
Given: , and are points on a line in the order given. .
Prove: .
points on the line;
Now it’s time to start with the given. Then we use logical reasoning to reach the statement we want to prove. Often (not always) the proof starts with the given information.
In the two column format, Statements go on the left side, and Reasons for each statement on the right. Reasons are generally definitions, postulates, and previously proved statements (called theorems).
Statement Reason
1. Given
2. , and are collinear in that order
Given
3.
Reflexive
4. and
Segment Addition Postulate
5.
Addition Property of Equality
6.
Substitution
is what we were given to prove, and we’ve done it.
Example 2
Write a two-column proof of the following:
Given: bisects
Prove:
Statement Reason
1. bisects Given
2. Definition of Angle Bisector
3. Angle Addition Postulate
4. Angle Addition Postulate
5. Substitution
6. Given
7. Substitution
8. Subtract from both sides
(Reminder: Angle measures are all real numbers, so properties of equality apply.)
9. Definition of congruent angles
This is the end of the proof. The last statement is the requirement made in the proof above. This is the signal that the proof is completed.
Lesson Summary
In this section you have seen two examples illustrating the format of two-column proofs. The format of two-column proofs is the same regardless of the specific details. Geometry originated many centuries ago using this same kind of deductive reasoning proof.
Points to Consider
You will see and write many two-column proofs in future lessons. The framework will stay the same, but the details will be different. Some of the statements that we prove are important enough that they are identified by their names. You will learn about many theorems and use them in proofs and problem solving.
Review Questions
Use the diagram below to answer questions 1-10.
Which of the following can be assumed to be true from the diagram? Answer yes or no.
bisects
is a square
is a rectangle
Use the diagram below to answer questions 11-14.
Given: bisects is the midpoint of , and .
How many segments have two of the given points as endpoints? What is the value of each of the following?
Write a two-column proof for the following:
Given: bisects
Prove:
Review Answers
No
No
Yes
No
Yes
No
No
No
No
No
Statement Reason
1. bisects Given
2. Definition of angle bisector
3. Angle Addition Postulate
4. Given
5. Definition of perpendicular segments
6. Substitution
7. Algebra (Distributive Property)
8. Multiplication Property of Equality
Segment and Angle Congruence Theorems
Learning Objectives
Understand basic congruence properties.
Prove theorems about congruence.
Introduction
In an earlier lesson you reviewed many of the basic properties of equality. Properties of equality are about numbers. Angles and segments are not numbers, but their measures are numbers. Congruence of angles and segments is defined in terms of these numbers. To prove congruence properties, we immediately turn congruence statements into number statements, and use the properties of equality.
Equality Properties
Reminder: Here are some of the basic properties of equality. These are postulates—no proof needed. For each of these there is a corresponding property of congruence for segments, and one for angles. These are theorems—we’ll prove them.
Properties of Equality for real numbers , and .
Reflexive
Symmetric If then
Transitive If and , then
These properties are convertibles; we can convert them quickly and easily into congruence theorems.
Note that diagrams are needed to prove the congruence theorems. They are about angles and segments...ALL angles and segments, wherever and whenever they are found. No special setting (diagram) is needed.
/> Segment Congruence Properties
In this section we’ll prove a series of segment theorems.
Reflexive:
Statement Reason
1. Reflexive Property of Equality
2. Definition of congruent segments
Symmetric: If , then
Given:
Prove:
Statement Reason
1. Given
2. Definition of congruent segments
3. Symmetric Property of Equality
4. Definition of congruent segments
Transitive: If and , then
Given:
Prove:
Statement Reason
1. Given
2. Definition of congruent segments
3. Transitive property of equality
4. Definition of congruent segments
Angle Congruence Properties
Watch for proofs of the Angle Congruence Properties in the Lesson Exercises.
Reflexive:
Symmetric: If , then
Transitive: If and , then
Lesson Summary
In this lesson we looked at old information in a new light. We saw that the properties of equality—reflexive, symmetric, transitive—convert easily into theorems about congruent segments and angles. In the next section we’ll move ahead into new ground. There we’ll get to use all the tools in our geometry toolbox to solve problems and to create new theorems.
Points to Consider
We are about to transition from introductory concepts that are necessary but not too “geometric” to the real heart of geometry. We needed a certain amount of foundation material before we could begin to get into more unfamiliar, challenging concepts and relationships. We have the definitions and postulates, and analogs of the equality properties, as the foundation. From here on out, we will be able to experience geometry on a richer and deeper level.
Review Questions
Prove the Segment Congruence Properties, in questions 1-3.
Reflexive:
Symmetric: If , then
Transitive: If and , then
Is the following statement true? If it’s not, give a counterexample. If it is, prove it.
If and , then
Give a reason for each statement in the proof below.
If and are collinear, and , then .
Given: , and are collinear, and
Prove:
Is the following statement true? Explain your answer. (A formal two-column proof is not required.)
Let and be points in a single plane. If is in the interior of , and is in the interior of , then is in the interior of .
Note that this is a bit like a Transitive Property for a ray in the interior of an angle.
Review Answers
Statement Reason
A. Reflexive Property of Equality
B. Definition of Congruent Angles
Given:
Prove:
Statement Reason
A. Given
B. Definition of Congruent Angles
C. Symmetric Property of Equality
D. Definition of Congruent Angles
Given: and
Prove:
Statement Reason
A. and Given
B. and Definition of Congruent Angles
C. Transitive Property of Equality
D. Definition of Congruent Angles
Yes Given:
and
Prove:
Statement Reason
A. and Given
B. Definition of Congruent Angles
C. Addition Property of Equality
D. Substitution
Statement Reason
, and are collinear A._____ Given
B._____ Given
C._____ Definition of Congruent Segments
D._____ Addition Property of Equality
E._____ Commutative Property of Equality
F._____ Definition of Collinear Points
G._____ Definition of Collinear Points
H._____ Substitution Property of Equality
I._____ Definition of Congruent Segments
True. Since is in the interior of Since is in the interior of , then . So
is in the interior of by the angle addition property.
Proofs About Angle Pairs
Learning Objectives
State theorems about special pairs of angles.
Understand proofs of the theorems about special pairs of angles.
Apply the theorems in problem solving.
Introduction
So far most of the things we have proven have been fairly straightforward. Now we have the tools to prove some more in-depth theorems that may not be so obvious. We’ll start with theorems about special pairs of angles. They are:
right angles
supplementary angles
complementary angles
vertical angles
Right Angle Theorem
If two angles are right angles, then the angles are congruent.
Given: and are right angles.
Prove:
Statement Reason
1. and are right angles.
Given
2.
Definition of right angle
3.
Substitution
4.
Definition of congruent angles
Supplements of the Same Angle Theorem
If two angles are both supplementary to the same angle (or congruent angles) then the angles are congruent.
Comments: As an example, we know that if is supplementary to a angle, then . If is also supplementary to a angle, then too, and
Given: and are supplementary angles. and are supplementary angles.
Prove:
Statement Reason
1. and are supplementary angles. Given
2. and are supplementary angles. Given
3. Definition of Supplementary Angles
4. Substitution
5. Addition Property of Equality
6. Definition of Congruent Angles
Example 1
Given that , what other angles must be congruent?
Answer:
by the Right Angle Theorem, because they’re both right angles.
by the Supplements of the Same Angle Theorem and the Linear Pair Postulate: and are a linear pair, which makes them supplementary. and are also a linear pair, which makes them supplementary too. Then by Supplements of the Same Angle Theorem, because they’re supplementary to congruent angles and .
Complements of the Same Angle Theorem
If two angles are both complementary to the same angle (or congruent angles) then the angles are congruent.
Comments: Only one word is different in this theorem compared to the Supplements of the Same Angle Theorem. Here we have angles that are complementary, rather than supplementary, to the same angle.
The proof of the Complements of the Same Angle Theorem is in the Lesson Exercises, and it is very similar to the proof above.
Vertical Angles Theorem
Vertical Angles Theorem: Vertical angles are congruent.
Vertical angles are common in geometry problems, and in real life wherever lines intersect: cables, fence lines, highways, roof beams, etc. A theorem about them will be useful. The vertical angle theorem is one of the world’s briefest theorems. Its proof draws on the new theorems just proved earlier in this section.
Given: Lines and intersect.
Prove: , and
Statement Reason
1. Lines and intersect. Given
2. and , and are linear pairs. Definition of Linear Pairs
3. and are supplementary, and and are supplementary. Linear Pair Postulate
4. Supplements of the Same Angle Theorem
This shows that . The same proof can be used to show that .
Example 2
Given:
Each of the following pairs of angles are congruent. Give a reason.
and answer: Vertical Angles T
heorem
and answer: Complements of Congruent Angles Theorem
and answer: Vertical Angles Theorem
and answer: Vertical Angles Theorem
and answer: Vertical Angles Theorem and Transitive Property
and answer: Vertical Angles Theorem and Transitive Property
and answer: Complements of Congruent Angles Theorem
Example 3
Given:
Prove:
Statement Reason
1. Given
2. Vertical Angles Theorem
3. Transitive Property of Congruence
Lesson Summary
In this lesson we proved theorems about angle pairs.
Right angles are congruent.
Supplements of the same, or congruent, angles are congruent.
Complements of the same, or congruent, angles are congruent.
Vertical angles are congruent.
We saw how these theorems can be applied in simple or complex figures.
Points to Consider
No matter how complicated or abstract the model of a real-world situation may seem, in the final analysis it can often be expressed in terms of simple lines, segments, and angles. We’ll be able to use the theorems of this section when we encounter complicated relationships in future figures.
CK-12 Geometry Page 8
