4.1 – 4.3 Triangle Congruency Geometry

SWBAT: Recognize congruent triangles Prove that triangles are congruent using the ASA Congruence Postulate, SSS Congruence Postulate, SAS Congruence Postulate and the AAS Congruence Theorem Use congruence postulates and theorems in real-life problems.

Congruent Means that corresponding parts are congruent, Matching sides and angles will be congruent

B A C Y X Z

Naming ORDER MATTERS!!!!

Example 1 If two triangles are congruent… Name all congruent angles Name all congruent sides X S T Y Z

Reminder… If two angles of one triangle are congruent to two angles of another triangle then the 3rd angles are congruent

Keep in mind You can flip, turn or slide congruent triangles and they will maintain congruency!!

SSS If sides of one triangle are congruent to sides of a second triangle then the two triangles are congruent.

Example 1 Given STU with S(0, 5), T(0,0), and U(-2, 0) and XYZ with X (4, 8), Y (4, 5), and Z (6, 3), determine if

Included angles

SAS If two sides and the included angle of one triangle are congruent to two sides and included angle of another triangle then the two triangles are congruent.

Example 2 Write a proof Given: X is the midpoint of BD X is the midpoint of AC - Prove: D A X C B

Postulate 21: Angle-Side-Angle (ASA) Congruence Postulate If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent.

If asked to show that a pair of corresponding parts of two triangles are congruent you often prove the triangles are congruent the by definition of congruent triangles the parts are congruent (CPCTC)

Example 3 Given: Prove:

Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of a second triangle, then the triangles are congruent.

Theorem 4.5: Angle-Angle-Side (AAS) Congruence Theorem Given: A  D, C  F, BC  EF Prove: ∆ABC  ∆DEF

Ex. 1 Developing Proof A. In addition to the angles and segments that are marked, EGF JGH by the Vertical Angles Theorem. Two pairs of corresponding angles and one pair of corresponding sides are congruent. You can use the AAS Congruence Theorem to prove that ∆EFG  ∆JHG.

Ex. 1 Developing Proof Is it possible to prove the triangles are congruent? If so, state the postulate or theorem you would use. Explain your reasoning.

Ex. 1 Developing Proof B. In addition to the congruent segments that are marked, MP  NP. Two pairs of corresponding sides are congruent. This is not enough information to prove the triangles are congruent.

Ex. 2 Proving Triangles are Congruent Given: AD ║EC, BD  BC Prove: ∆ABD  ∆EBC Plan for proof: Notice that ABD and EBC are congruent. You are given that BD  BC . Use the fact that AD ║EC to identify a pair of congruent angles.

Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: 1.

Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: 1. Given

Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: Given

Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: Given Alternate Interior Angles

Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: Given Alternate Interior Angles Vertical Angles Theorem

Proof: Statements: BD  BC AD ║ EC D  C ABD  EBC ∆ABD  ∆EBC Reasons: Given Alternate Interior Angles Vertical Angles Theorem ASA Congruence Theorem

Note: You can often use more than one method to prove a statement. In Example 2, you can use the parallel segments to show that D  C and A  E. Then you can use the AAS Congruence Theorem to prove that the triangles are congruent.