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2.4: Algebraic and Congruence Properties

Difficulty Level: At Grade Created by: CK-12
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Learning Objectives

  • Understand basic properties of equality and congruence.
  • Solve equations and justify each step.
  • Fill in the blanks of a 2-column proof.

Review Queue

Solve the following problems.

1. Solve .

2. If two angles are a linear pair, they are supplementary.

If two angles are supplementary, their sum is .

What can you conclude? By which law?

3. Draw a picture with the following:

Know What? Three identical triplets are sitting next to each other. The oldest is Sara and she always tells the truth. The next oldest is Sue and she always lies. Sally is the youngest of the three. She sometimes lies and sometimes tells the truth.

Scott came over one day and didn't know who was who, so he asked each sister who was sitting in the middle. Who is who?

Properties of Equality

Recall from Chapter 1 that the = sign and the word “equality” are used with numbers. The basic properties of equality were introduced to you in Algebra I. Here they are again:

Reflexive Property of Equality

Symmetric Property of Equality and

Transitive Property of Equality and , then

Substitution Property of Equality — If and , then

Addition Property of Equality — If , then or

Subtraction Property of Equality — If , then or

Multiplication Property of Equality — If , then or

Division Property of Equality — If , then or

Distributive Property

Properties of Congruence

Recall that if and only if and if and only if . The Properties of Equality work for and .

Just like the properties of equality, there are properties of congruence. These properties hold for figures and shapes.

For Line Segments For Angles

Reflexive Property

of Congruence

Symmetric Property

of Congruence

If , then If , then

Transitive Property

of Congruence

If and ,


If and ,


Using Properties of Equality with Equations

When you solve equations in algebra you use properties of equality. You might not write out the property for each step, but you should know that there is an equality property that justifies that step. We will abbreviate “Property of Equality” “” and “Property of Congruence” “.”

Example 1: Solve and write the property for each step (also called “to justify each step”).


Example 2: , and . Are points , and collinear?

Solution: Set up an equation using the Segment Addition Postulate.

Because the two sides are not equal, and are not collinear.

Example 3: If and , prove that is an acute angle.

Solution: We will use a 2-column format, with statements in one column and their reasons next to it, just like Example 1.

Two-Column Proof

Examples 1 and 3 are examples of two-column proofs. They both have the left side, the statements, and on the right side are the reason for these statements. Here we will continue with more proofs and some helpful tips for completing one.

Example 4: Write a two-column proof for the following:

If , and are points on a line, in the given order, and , then .

Solution: When the statement is given in this way, the “if” part is the given and the “then” part is what we are trying to prove.

Always start with drawing a picture of what you are given.

Plot the points in the order on a line.

Add the given, .

Draw the 2-column proof and start with the given information.

Statement Reason
1. , and are collinear, in that order. Given
2. Given
3. Reflexive
4. Addition


Segment Addition Postulate
6. Substitution or Transitive

Once we reach what we wanted to prove, we are done.

When completing a proof, these keep things in mind:

  • Number each step.
  • Start with the given information.
  • Statements with the same reason can (or cannot) be combined into one step. It is up to you. For example, steps 1 and 2 above could have been one step. And, in step 5, the two statements could have been written separately.
  • Draw a picture and mark it with the given information.
  • You must have a reason for EVERY statement.
  • The order of the statements in the proof is not fixed. For example, steps 3, 4, and 5 could have been interchanged and it would still make sense.
  • Reasons will be definitions, postulates, properties and previously proven theorems. “Given” is only used as a reason if the information in the statement column was told in the problem.

Example 5: Write a two-column proof.

Given: bisects ;


Solution: First, put the appropriate markings on the picture. Recall, that bisect means “to cut in half.” Therefore, .

Statement Reason
1. bisects Given
2. Definition of an Angle Bisector
3. If angles are , then their measures are equal.


Angle Addition Postulate
5. Substitution
6. Substitution
7. Subtraction
8. If measures are equal, the angles are .

Use symbols and abbreviations for words within proofs. For example, was used in place of the word congruent above. You could also use for the word angle.

Know What? Revisited Analyzing the picture and what we know the sister on the left cannot be Sara because she lied (if we take what the sister in the middle said as truth). So, let’s assume that the sister in the middle is telling the truth, she is Sally. However, we know this is impossible, because that would have to mean that the sister on the right is lying and Sarah does not lie. From this, that means that the sister on the right is Sara and she is telling the truth, the sister in the middle is Sue. So, the first sister is Sally. The order is: Sally, Sue, Sara.

Review Questions

  • Questions 1-8 are similar to Examples 1 and 3.
  • Questions 9-14 use the Properties of Equality.
  • Questions 15-17 are similar to Example 2.
  • Questions 18 and 19 are similar to Examples 8 and 9.
  • Questions 20-34 are review.

For questions 1-8, solve each equation and justify each step.

For questions 9-14, use the given property or properties of equality to fill in the blank. , and are real numbers.

  1. Symmetric: If , then ______________.
  2. Distributive: If , then ______________.
  3. Transitive: If and , then ______________.
  4. Symmetric: If , then ______________.
  5. Transitive: If and , then ______________.
  6. Substitution: If and , then ______________.
  7. Given points , and and , and . Determine if and are collinear.
  8. Given points and and , and . Are the three points collinear? Is the midpoint?
  9. If and , explain how must be an obtuse angle.

Fill in the blanks in the proofs below.

  1. Given: Prove:
Statement Reason
1. Given


3. Addition
  1. Given: is the midpoint of . is the midpoint Prove:
Statement Reason
1. Given
2. Definition of a midpoint

Use the diagram to answer questions 20-25.

  1. Name a right angle.
  2. Name two perpendicular lines.
  3. Given that , is true? Explain your answer.
  4. Is a right angle? Why or why not?
  5. Fill in the blanks:

  1. Fill in the blanks:

Use the diagram to answer questions 26-31.

Which of the following must be true from the diagram?

Take each question separately, they do not build upon each other.

  1. is a square
  2. bisects

Use the diagram to answer questions 32-34.

Given: bisects , is the midpoint of and .

What is the value of each of the following?

Review Queue Answers

  1. If 2 angles are a linear pair, then their sum is . Law of Syllogism.

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