Triangles (SSS,SAS,ASA)

Geometric Proofs

What is known?

•  $\overline{AB}\ \cong\ \overline{CB}\$  

•  $\overline{BD}\$  bisects  $\angle ABC$  

•  $\Delta ABD\ \cong\ \Delta CBD$  

How do we know that  $\Delta ABD\ \cong\ \Delta CBD$  

•  $\overline{AB}\ \cong\ \overline{CB}$  and  $\overline{BD}$  bisects  $\angle ABC$  are GIVEN

•  $\angle ABD\ \cong\ \angle CBD$   : For what reason?

• Reflexive property next, what sides?

Multiple Choice

By what theorem are the two triangles congruent?

• Side-Angle-Side

• Angle-Side-Angle

• Side-Side-Side

• Angle-Angle-Angle

• Side-Side-Angle

What is known?

• Q is the midpoint of  $\overline{SN}$  

•  $\angle1\ \cong\ \angle2$  

•  $\Delta SQR\ \cong\ \Delta NQM$  

How do we know that  $\Delta SQR\ \cong\ \Delta NQM$  

• Q is the midpoint of  $\overline{SN}$  and  $\angle1\ \cong\ \angle2$  are GIVEN

•  $\overline{SQ}\ \cong\ \overline{NQ}$  ... 

Multiple Choice

Knowing that Q is the midpoint of  $\overline{SN}$  . How do we know that  $\overline{SQ}\ \cong\ \overline{NQ}$  ?

• Definition of a midpoint

• Definition of a perpendicular bisector

• Definition of an angle bisector

• Reflexive property

Multiple Choice

For what reason are  $\angle MQN\ \cong\ \angle RQS$  

• Vertical Angles

• Corresponding Angles

• Alternate Interior Angles

• Angle-Angle-Angle

Multiple Choice

By what theorem are triangles  $\Delta MNQ\ \ and\ \ \Delta RSQ$  congruent?

• Side-Angle-Side

• Angle-Side-Angle

• Angle-Angle-Side

• Side-Side-Angle

• Side-Side-Side

Fill in the Blanks

Based on the image, what reason can be given for the statement: 

O is the midpoint of  $\overline{MP}$  and  $\overline{NQ}$  

Fill in the Blanks

Given that reason 1 is given, what would the best reason for statement 2?

Fill in the Blanks

Given the two reasons we determined earlier, what would the third reason be?

Fill in the Blanks

What would the final reasoning be? Why are the two triangles congruent?

Multiple Select

Recap: What are the theorems for congruence in triangles?

• Side-Angle-Side

• Angle-Side-Angle

• Side-Side-Side

• Side-Side-Angle

• Angle-Angle-Angle

Multiple Choice

What theorem makes $\Delta ABC\ \cong\ \Delta DEF$ ?

• Angle-Side-Angle

• Side-Angle-Side

• Side-Side-Side

• Side-Side-Angle

Multiple Choice

What theorem makes  $\Delta ABC\ \cong\ \Delta DEF$  ?

• Angle-Side-Angle

• Side-Angle-Side

• Side-Side-Side

• Hypotenuse-Leg

Multiple Choice

What theorem makes $\Delta ABC\ \cong\ \Delta DEF$ ?

• Angle-Side-Angle

• Side-Angle-Side

• Side-Side-Side

• Side-Side-Angle