Proving Triangles Congruent (Intro to Proofs)

Proving Triangles Congruent

by Angela Preston

​Intro to Proofs

​What is a Proof?

​A proof is made up of given statements that prove a mathematical concept is true.

​

In order for a proof to be proven true, it has to include multiple steps. These steps are made up of statements and reasons.

Let's try a few proofs...​

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​Example #1: 

​Example #1: 

​Step 1:

​Mark the given information

​Example #1: 

​Step 1:

​Mark the given information

​Example #1: 

​Step 2:

​Mark the reflexive sides or vertical angles

​Example #1: 

​Step 2:

​Mark the reflexive sides or vertical angles

​Example #1: 

​Step 3:

​Choose the correct triangle congruence shortcut

​Example #1: 

​Step 3:

​Choose the correct triangle congruence shortcut

​Example #1: 

​Step 4:

​List the parts in order according to the congruence statement

​Example #1: 

​Step 4:

​List the parts in order according to the congruence statement

​Example #1: 

​Step 5:

​List the reasons for each statement.

​Example #1: 

​Step 5:

​List the reasons for each statement.

Is there more?

​Example #1: 

​Let's Review the "Algebra Properties"

Multiple Select

Choose all statements that show symmetric statements

• $$\ if\ 4=8\div2,\ then\ 8\div2=4$$

• $$\angle CAT\cong\angle CAT$$

• $$if\ m\angle1\ +m\angle2\ =\ 90\circ\ ,\ then\ 90\circ\ =\ m\angle1\ +m\angle2\ \$$

Multiple Select

Which of these are examples of transitive property?

• if M = N then N = M

• R+T =180 , J +K = 180, so R+T= J+K

• x = y and p = y so p = x

• J=K, K=R so J=R

Fill in the Blank

What would you type next to make the statement reflexive?

$$\angle A\cong\angle\ \$$  

​Example 2:

​1) Mark the given information

​2) Mark the reflexive sides/vertical angles

​3) Choose a shortcut

​4) List the parts in order

​5) List the reasons

​Example 3:

​1) Mark the given information

​2) Mark the reflexive sides/vertical angles

​3) Choose a shortcut

​4) List the parts in order

​5) List the reasons

​CPCTC

​CPCTC stands for "Corresponding Parts of Congruent Triangles are Congruent"

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​This is a common reason used in a Proof.

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​It usually comes after one of the Triangle Congruence Shortcuts.

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​Once a pair of triangles have been proven congruent by SSS, SAS, ASA, AAS, or HL, the additional parts are also congruent by CPCTC!

​Example:

​What is reason #5?

Let's Practice...​

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Multiple Choice

What is statement #1?

• BC≅DC
• AC≅EC
• BC≅DC, AC≅EC
• ∆BCA≅∆DCE

Multiple Choice

What is reason #3?

• SAS
• ASA
• SSS
• AAS

Multiple Choice

What does CPCTC stand for?

• Corresponding Parts of Congruent Triangles are Congruent
• Congruent Parts of Congruent Triangles are Corresponding
• Colorful Pieces of Corresponding Triangles are Congruent
• Congruent Pieces of Congruent Triangles are Congruent

Multiple Choice

AB = AB

• Reflexive POE
• Transitive POE
• Symmetric POE
• Distributive POE

Multiple Choice

What is the justification?

• Symmetric Prop.
• Angle addition postulate
• Reflexive Prop
• Transitive Prop.

Multiple Choice

What is reason #1?

• Reflexive Property
• Definition of Bisect
• Prove
• Given

Multiple Choice

What is reason #2?

• Definition of Midpoint
• Definition of Perpendicular
• Definition of Bisect
• Definition of Right Triangle

Multiple Choice

What is statement #3?

• FH≅FH

• GF≅GF

• IH≅IH

• GF≅FI

Multiple Choice

What is statement #4?

• ∆HFG≅∆IHF

• ∆FHG≅∆IHF

• ∆GFH≅∆IFH

• ∆HFG≅∆IFH

Multiple Choice

What is reason #4?

• SAS

• AAS

• SSS

• HL

Multiple Choice

What is reason #2?

• Vertical angles are congruent.

• Consecutive angles are congruent.

• Alternate interior angles are congruent.

• Definition of Bisect