Proving Congruent Triangles: SSS & SAS Ch 4 Lesson 3

Two Triangles Are Congruent Have to prove 6 parts of the triangle are congruent –Congruent Side (3 sides) –Congruent Angles (3 angles)

Postulate 19: Side-Side-Side, SSS If three sides of one triangle are congruent to three sides of another triangle, the two triangles are congruent. Side AB ≅ DE Side BC ≅ EF Side CA ≅ FD  Δ ABC ≅ Δ DEF

Postulate 19: Side-Side-Side, SSS If three sides of one triangle are congruent to three sides of another triangle, the two triangles are congruent. Side AB ≅ DE Side BC ≅ EF Side CA ≅ FD  Δ ABC ≅ Δ DEF

Postulate 19: Side-Side-Side, SSS If three sides of one triangle are congruent to three sides of another triangle, the two triangles are congruent. Side AB ≅ DE Side BC ≅ EF Side CA ≅ FD  Δ ABC ≅ Δ DEF

Postulate 19: Side-Side-Side, SSS If three sides of one triangle are congruent to three sides of another triangle, the two triangles are congruent. Side AB ≅ DE Side BC ≅ EF Side AC ≅ DF  Δ ABC ≅ Δ DEF

Postulate 19: Side-Side-Side, SSS If three sides of one triangle are congruent to three sides of another triangle, the two triangles are congruent. Side AB ≅ DE Side BC ≅ EF Side CA ≅ FD  Δ ABC ≅ Δ DEF

Postulate 20: Side-Angle-Side, SAS If two sides and the angle between those two sides are congruent to the two sides and the angle between the two sides of another angle  the two triangles are congruent. Side AB ≅ DE Angle <A ≅ <D Side AC ≅ DF

Postulate 20: Side-Angle-Side, SAS If two sides and the angle between those two sides are congruent to the two sides and the angle between the two sides of another angle  the two triangles are congruent. Side AB ≅ DE Angle <A ≅ <D Side AC ≅ DF

Postulate 20: Side-Angle-Side, SAS If two sides and the angle between those two sides are congruent to the two sides and the angle between the two sides of another angle  the two triangles are congruent. Side AB ≅ DE Angle <A ≅ <D Side AC ≅ DF

Postulate 20: Side-Angle-Side, SAS If two sides and the angle between those two sides are congruent to the two sides and the angle between the two sides of another angle  the two triangles are congruent. Side AB ≅ DE Angle <A ≅ <D Side AC ≅ DF  Δ ABC ≅ Δ DEF

Postulate 20: Side-Angle-Side, SAS If two sides and the angle between those two sides are congruent to the two sides and the angle between the two sides of another angle  the two triangles are congruent. Side AB ≅ DE Angle <A ≅ <D Side AC ≅ DF  Δ ABC ≅ Δ DEF

Example #1 Prove that Δ ABC ≅ Δ DEF Given BE ≅ EC and AE ≅ ED Set up two column proof

Prove that Δ ABC ≅ Δ DEF Statement BE ≅ EC and AE ≅ ED <1 ≅ <2 Δ ABC ≅ Δ DEF Reason Given Vertical Angles SAS congruent postulate

Prove that Δ ABC ≅ Δ DEF Statement BE ≅ EC and AE ≅ ED <1 ≅ <2 Δ ABC ≅ Δ DEF Reason Given Vertical Angles SAS congruent postulate

Prove that Δ ABC ≅ Δ DEF Statement BE ≅ EC and AE ≅ ED <1 ≅ <2 Δ ABC ≅ Δ DEF Reason Given Vertical Angles SAS congruent postulate

Prove that Δ ABC ≅ Δ DEF Statement BE ≅ EC and AE ≅ ED <1 ≅ <2 Δ ABC ≅ Δ DEF Reason Given Vertical Angles SAS congruent postulate

Example #2 Decide whether the two triangles are congruent Given QP ≅ PS Given QR ≅ SR Set up two column proof

Decide whether the two triangles are congruent Statement QP ≅ PS QR ≅ SR PR ≅ PR ΔQPR ≅ ΔSPR Reason Given Reflexive property SSS congruent postulate

Decide whether the two triangles are congruent Statement QP ≅ PS QR ≅ SR PR ≅ PR ΔQPR ≅ ΔSPR Reason Given Reflexive property SSS congruent postulate

Decide whether the two triangles are congruent Statement QP ≅ PS QR ≅ SR PR ≅ PR ΔQPR ≅ ΔSPR Reason Given Reflexive property SSS congruent postulate

Decide whether the two triangles are congruent Statement QP ≅ PS QR ≅ SR PR ≅ PR ΔQPR ≅ ΔSPR Reason Given Reflexive property SSS congruent postulate

Example #3 Given DR ⊥ AG and AR ≅ RG Prove ΔARD ≅ ΔGRD Set up two column proof

Prove ΔARD ≅ ΔGRD Statement DR ⊥ AG <DRA=90°&<DRG =90° <DRA ≅ <DRG AR ≅ RG DR ≅ DR ΔARD ≅ ΔGRD Reason Given Def of ⊥ lines Right Angle congruent th. Given Reflexive Prop. SAS congruent Pos.

Prove ΔARD ≅ ΔGRD Statement DR ⊥ AG <DRA=90°&<DRG =90° <DRA ≅ <DRG AR ≅ RG DR ≅ DR ΔARD ≅ ΔGRD Reason Given Def of ⊥ lines Right Angle congruent th. Given Reflexive Prop. SAS congruent Pos.

Prove ΔARD ≅ ΔGRD Statement DR ⊥ AG <DRA=90°&<DRG =90° <DRA ≅ <DRG AR ≅ RG DR ≅ DR ΔARD ≅ ΔGRD Reason Given Def of ⊥ lines Right Angle congruent th. Given Reflexive Prop. SAS congruent Pos.

Prove ΔARD ≅ ΔGRD Statement DR ⊥ AG <DRA=90°&<DRG =90° <DRA ≅ <DRG AR ≅ RG DR ≅ DR ΔARD ≅ ΔGRD Reason Given Def of ⊥ lines Right Angle congruent th. Given Reflexive Prop. SAS congruent Pos.

Prove ΔARD ≅ ΔGRD Statement DR ⊥ AG <DRA=90°&<DRG =90° <DRA ≅ <DRG AR ≅ RG DR ≅ DR ΔARD ≅ ΔGRD Reason Given Def of ⊥ lines Right Angle congruent th. Given Reflexive Prop. SAS congruent Pos.

Prove ΔARD ≅ ΔGRD Statement DR ⊥ AG <DRA=90°&<DRG =90° <DRA ≅ <DRG AR ≅ RG DR ≅ DR ΔARD ≅ ΔGRD Reason Given Def of ⊥ lines Right Angle congruent th. Given Reflexive Prop. SAS congruent Pos.

Prove ΔARD ≅ ΔGRD Statement DR ⊥ AG <DRA=90°&<DRG =90° <DRA ≅ <DRG AR ≅ RG DR ≅ DR ΔARD ≅ ΔGRD Reason Given Def of ⊥ lines Right Angle congruent th. Given Reflexive Prop. SAS congruent Pos.

More examples Decide whether the triangles are congruent and state the theorem or postulate.

SAS

SSS