4.4 How do we prove corresponding parts of congruent triangles congruent? Do Now: If the following two triangles are congruent, name the six congruent parts. A C B D F E ≈ ** Remember that figures are usually labeled in ABC order! ** Angles Sides angle A ≈ angle D segment AC ≈ segment DE angle C ≈ angle E segment CB ≈ segment EF angle B ≈ angle F segment AB ≈ segment DF If triangles are congruent, then we can say that their corresponding parts are congruent! Proving Corresponding Parts of Congruent Triangles are Congruent 1.) Given: AB and CD intersect at E AE is congruent to EB EA and EB are right angles Prove: a.) triangle ACE is congruent to triangle BDE b.) angle C is congruent to angle D Statements Reasons 1. AB and CD intersect at E 1. Given 2. angle AEC is congruent to angle DEB 2. Vertical angles are congruent. 3. AE is congruent to EB 3. Given 4. angle A and angle B are right angles 4. Given 5. All right angles are congruent. 5. All right angles are congruent. 6. triangle ACE is congruent to triangle BDE 6. ASA Congruence Postulate. 7. angle C is congruent to angle D 7. Corresponding parts of congruent triangles are congruent. 2.) Given: AB is congruent to AD AC bisects angle BAD Prove: BC is congruent to CD Statements Reasons 1. AB is congruent to AD 1. Given 2. AC bisects angle BAD 2. Given 3. angle BAC is congruent to angle DAC 3. An angle bisector divides an angle into two congruent angles. 4. AC is congruent to AC 4. Reflexive Property. 5. triangle BAC is congruent to triangle DAC 5. S.A.S. is congruent to S.A.S. 6. BC is congruent to CD 6. Corresponding parts of congruent triangles are congruent. To prove Corresponding Parts (segments or angles) of Congruent Triangles are Congruent (also known as CPCTC), first prove two triangles that contain these parts are congruent. 1.) Given: XW is the perpendicular bisector of YZ. Prove: angle Z is congruent to angle Y. 2.) Given: RS is perpendicular to SW, XW is perpendicular to SW, SV is congruent to TW, RS is congruent to XW Prove: angle R is congruent to angle X