Topic 3 Congruent Triangles Unit 2 Topic 4

Information Two triangles are congruent if they have the same size and shape. You can turn, flip and/or slide one so it fits exactly on the other. Congruent angles can be marked with symbols, such as an arc, or a dot. Congruent sides can be marked with small line segments, called hatch marks.

Information If ABC and DEF are congruent, then the corresponding angles and corresponding sides are equal. The symbol for congruence is . Since ABC and DEF are congruent, we write .

Information By definition, if two triangles are congruent, then all corresponding angles are equal and all corresponding sides are equal. However, to prove two triangles are congruent, you do not need to know that all corresponding sides and all corresponding angles are equal. There are three conditions that can be used to prove two triangles are congruent.

Information Triangle Congruence Conditions SSS Congruence Condition If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent. B Q ….. A C P R

Information Triangle Congruence Conditions SAS Congruence Condition If two sides and the contained angle of one triangle are equal to two sides and the contained angle of another triangle, then the triangles are congruent B Q ….. A C P R

Information Triangle Congruence Conditions ASA Congruence Condition If two angles and the contained side of one triangle are equal to two angles and the contained side of another triangle, then the triangles are congruent B Q ….. A C P R

Example 1 Try this on your own first!!!! Stating Congruence and Corresponding Angles and Sides. For each pair of triangles below, state the congruence theorem that proves they are congruent. Then, state the corresponding angles and sides for a). x A B C x X Y Z a) c) b) d) A B C 3.6 cm 3.5 cm 2.1 cm X Y Z 3.6 cm 2.1 cm 3.5 cm x A B C X Y Z x A B C 40˚ 50˚ 2.2 cm 40˚ 50˚ 2.2 cm X Z Y

Example 1a: Solution A B C 3.6 cm 3.5 cm 2.1 cm We can see that all three sides are equal. Therefore, the triangles are congruent by SSS congruence. Corresponding sides: Corresponding angles: AB = XY A = X BC = YZ B = Y AC = XZ C = Z X Y Z 3.6 cm 2.1 cm 3.5 cm

Example 1b: Solution x A B C b) Two angles and the contained side of one triangle is equal to two angles and the contained side of another triangle. Therefore, the triangles are congruent by the ASA congruence theorem. X Y Z x Corresponding sides: Corresponding angles: AB = XY A = X BC = YZ B = Y AC = XZ C = Z

Example 1c: Solution x A B C c) Two sides and the contained angle of one triangle is equal to two sides and the contained angle of another triangle. Therefore, the triangles are congruent by the SAS congruence theorem. x X Y Z Corresponding sides: Corresponding angles: AB = XY A = X BC = YZ B = Y AC = XZ C = Z

Example 1d: Solution d) Since angle B and angle Y are each 90°, we can add to the diagram. A B C 40˚ 50˚ 2.2 cm Two angles and the contained side of one triangle is equal to two angles and the contained side of another triangle. Therefore, the triangles are congruent by the ASA congruence theorem. 90° 40˚ 50˚ 2.2 cm X Z Y Corresponding sides: Corresponding angles: AC = XY A = X AB = XY B = Y BC = YZ C = Z 90°

More Information Using algebra is one way to present a proof. Another way to present a proof, often used in geometry, is called a two-column proof. A two-column proof is a presentation of a logical argument involving deductive reasoning in which the statements of the argument are in one column and the justification for the statements are written in another column. To prove two triangles are congruent you need to provide a logical argument that establishes one of the three congruence conditions: SSS, SAS or ASA. Your proof consists of a set of statements and accompanying reasons. There are many reasons that you can use to justify the statements in your proof.

More Information You can end a proof with Q.E.D., a Latin phrase that means "which had to be demonstrated”.

Example 2a Given: A = D and AB = DB Prove: ∆ABC ∆DBE Try this on your own first!!!! Proving Two Triangles Are Congruent Given: A = D and AB = DB Prove: ∆ABC ∆DBE Statement Reason

Example 2a Given: A = D and AB = DB Prove: ∆ABC ∆DBE Proving Two Triangles Are Congruent Given: A = D and AB = DB Prove: ∆ABC ∆DBE Statement Reason Given Opposite angles are equal ASA Congruency condition A A S

Example 2b Try this on your own first!!!! Proving Two Triangles Are Congruent Statement Reason

Example 2b: Solution Proving Two Triangles Are Congruent Statement Reason Given Common side SAS Congruency condition A S S

Example 3a Given: point E is the midpoint AC, Completing a Proof Using Definitions Given: point E is the midpoint AC, point E is the midpoint of BD Prove: AB = CD Statement Reason

Example 3a: Solution Statement Reason BE=DE CE=AE By definition of a midpoint Opposite angles are equal SAS Congruency condition C D A E B S A S

Example 3b Given: TP is perpendicular to AC, represented as TP ⊥ AC Try this on your own first!!!! Completing a Proof Using Definitions Given: TP is perpendicular to AC, represented as TP ⊥ AC TP bisects ATC Prove: AT = CT Helpful Hint To bisect is to divide in exactly half.   Statement Reason T P A C

Example 3b: Solution Given: TP is perpendicular to AC, represented as TP ⊥ AC TP bisects ATC Prove: AT = CT TP is perpendicular to AC meaning they meet at a right angle. Statement Reason TPA=TPC TP=TP ATP=CTP AT=CT By definition of a perpendicular Common side By definition of a bisect of ATC ASA Congruency By congruency T P A C A S A

Example 4 Given: and Prove: Try this on your own first!!!! Completing a Proof with Parallel Lines Given: and Prove: T U V W X Statement Reason

Example 4 Given: and Prove: Completing a Proof with Parallel Lines Statement Reason VTU=VXW TU=XW VUT=VWX Alternate interior angles are equal Given ASA Congruency S T U V W X A A

Example 5 Given: BC=ED,  OBA=  OEF, and  OCB=  ODE. Try this on your own first!!!! Example 5 Completing a Proof Using Supplementary Angles Given: BC=ED,  OBA=  OEF, and  OCB=  ODE. Prove:  BOC =  EOD. Statement Reason F O E D C B A  

Example 5: Solution Given: BC=ED,  OBA=  OEF, and  OCB=  ODE. Completing a Proof Using Supplementary Angles Given: BC=ED,  OBA=  OEF, and  OCB=  ODE. Prove:  BOC =  EOD. Statement Reason ABO=FEO OBC=OED BC=ED BCO=EDO BOC=EOD Given Supplementary of the given angles ASA Congruency By Congruency F O E D C B A   A A S

Need to Know: Congruent triangles have the same size and shape. To prove that triangles are congruent, it must be shown that the corresponding sides and corresponding angles in the triangles are equal. To do this, the following Triangle Congruence Conditions are used: SSS Congruence Theorem SAS Congruence Theorem ASA Congruence Theorem You’re ready! Try the homework from this section.