Proofs, Parallel Lines, and Angle Pairs Review

Corresponding Angle Theorem

Same Side Interior Angle Theorem

Given

Substitution Property

$$\angle3\cong\angle5$$  

$$m\angle4+m\angle5=180\degree$$  

$$\angle1\cong\angle5$$  

$$\angle3\cong\angle1$$  

$$\angle1\ and\ \angle3\ are\ vertical\ angles$$  

$$\angle4\ and\ \angle3\ are\ linear\ pair$$  

Transitive Property

$$m\angle4+m\angle3=180\degree$$  

Linear Pair Postulate

Vertical Angle Theorem

Diagram

$$p\parallel q$$  

$$m\angle4+m\angle8=180\degree$$  

$$\angle2\ and\ \angle3\ are\ linear\ pair$$  

Transitive Property

$$m\angle2+m\angle3=180\degree$$  

$$\angle2\cong\angle8$$  

Substitution Property

Diagram

$$\angle4\cong\angle8$$  

Corresponding Angle Converse Theorem

$$\angle2\ and\ \angle4\ are\ vertical\ angles$$  

$$p\parallel q$$  

Vertical Angle Theorem

Given

$$\angle2\cong\angle4$$  

Alternate Exterior Angle Converse Theorem

Linear Pair Postulate

Transitive Property

$$\angle2\ and\ \angle3\ are\ linear\ pair$$  

Substitution Property

$$\angle1\cong\angle6$$  

Linear Pair Postulate

$$\angle2\cong\angle6$$  

Corresponding Angle Theorem

$$p\parallel q$$  

$$\angle3\cong\angle4$$  

Vertical Angle Theorem

Alternate Interior Angle Theorem

$$m\angle2+m\angle3=180\degree$$  

$$\angle1\ and\ \angle3\ are\ vertical\ angles$$  

Given

$$m\angle3+m\angle6=180\degree$$  

Diagram

Transitive Property

Given

Alternate Interior Angle Converse Theorem

$$\angle3\cong\angle4$$  

$$m\angle2+m\angle3=180\degree$$  

$$\angle1\ and\ \angle3\ are\ vertical\ angles$$  

Congruent Supplement Theorem

$$\angle2\ and\ \angle3\ are\ linear\ pair$$  

$$p\parallel q$$  

$$\angle4\cong\angle6$$  

$$m\angle3+m\angle6=180\degree$$  

Vertical Angle Theorem

Corresponding Angle Converse Theorem

$$\angle2\cong\angle6$$  

Diagram

Linear Pair Postulate