Segments and Angles

Key Concepts

• Write proofs using geometric theorems
• Know about the theorems: congruence of angles and congruence of segments
• Prove angles are congruent

Proof 

A proof is a logical argument that shows a statement is true. There are several formats for proofs. 

Two-column proof 

A two-column proof has numbered statements and corresponding reasons that show an argument in a logical order. 

In a two-column proof, each statement in the left-hand column is either given information or the result of applying a known property or fact to statements already made. Each reason in the right-hand column is the explanation for the corresponding statement. 

Theorem 

The reasons used in a proof can include definitions, properties, postulates, and theorems.  

A theorem is a statement that can be proven. Once you have proven a theorem, you can use the theorem as a reason in other proofs. 

THEOREM 1

Congruence of Segments 

Segment congruence is reflexive, symmetric, and transitive. 

Reflexive For any segment AB, AB- ≅ AB-

Symmetric If  AB- ≅ CD , then CD- ≅ AB-. 

Transitive If AB- ≅ CD and CD- ≅ EF, then AB- ≅ EF-.  

THEOREM 2

Congruence of Angles 

Angle congruence is reflexive, symmetric, and transitive. 

Reflexive For any angle A, ∠A ≅ ∠A

Symmetric If ∠A ≅ ∠B, then ∠B ≅ ∠A. 

Transitive If ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C.  

Solve Examples 

Write a two-column proof  

Write a two-column proof for the following: 

Given: m∠1 = m∠3 

Prove: m∠EBA = m∠DBC 

Solution: 

Name the property shown 

Name the property illustrated by the statement. 

• If ∠R ≅ ∠T and ∠T ≅ ∠P, then ∠R ≅ ∠P.  
• If NK ≅ BD, then BD ≅ NK. 

Solution: 

• Transitive property of angle congruence 
• Symmetric property of segment congruence 

Use properties of equality 

Prove this property of midpoints: If you know that M is the midpoint of AB, prove that AB is two times AM, and AM is one half of AB. 

Given: M is the mid point of AB-. 

Prove: a. AB = 2 AM   

   b. AM = 1/2 AB   

Solution: 

Solve a multi-step problem 

SHOPPING MALL Walking down a hallway at the mall, you notice the music store is halfway between the food court and the shoe store. The shoe store is halfway between the music store and the bookstore. Prove that the distance between the entrances of the food court and music store is the same as the distance between the entrances of the shoe store and bookstore. 

Solution: 

Step 1: Draw and label a diagram. 

Step 2: Draw separate diagrams to show mathematical relationships. 

Step 3: State what is given and what is to be proved for the situation. Then write a proof. 

Given: B is the mid point of AB-

C is the mid point of BD-

Prove: AB = CD 

Questions to solve 

Question 1:

Write a two-column proof for the situation: 

Given: AC = AB + AB  

Prove: AB = BC 

Solution:  

Question 2:

Name the property illustrated by the statement. 

1. CD ≅ CD
2. If ∠Q ≅ ∠V then ∠V ≅ ∠Q. 

Solution: 

1. Reflexive property of segment congruence 
2. Symmetric property of angle congruence  

Question 3:

Solve for x using the given information. Explain your steps. 

Given:

QR− ≅ PQ−

RS− ≅ PQ−

Solution: 

According to transitive property of segment congruence, QR = RS. 

2x + 5 = 10 – 3x  

5x + 5 = 10  

5x = 5  

x = 1 

Key Concepts Covered  

1. Writing a two-column proof is a formal way of organizing your reasons to show a statement is true. 
2. Before writing a proof, organize your reasoning by copying or drawing a diagram for the situation described. Then identify the GIVEN and PROVE statements. 
3. Congruence of Segments: Segment congruence is reflexive, symmetric, and transitive. 
4. Congruence of Angles: Angle congruence is reflexive, symmetric, and transitive.

Exercise

• What is a theorem? How is it different from a postulate?
• You can use theorems as reasons in a two-column proof. What other types of statements can you use as reasons in a two-column proof? Give examples.

Name the property illustrated by the following statements:

• If 5≅3 then 3≅5.
• If a≅b and b≅c then a c≅a.
• VWX≅VWX
• Sketch a diagram that represents the given information.
• CRYSTALS: The shape of a crystal can be represented by intersecting lines and planes. Suppose a crystal is cubic, which means it can be represented by six planes that intersect at right angles.
• BEACH VACATION: You are on vacation at the beach. Along the boardwalk, the bike rentals are halfway between your cottage and the kite shop. The snack shop is halfway between your cottage and the bike rentals. The arcade is halfway between the bike rentals and the kite shop.

Solve for x using the given information. Explain your steps.