3.2 Corresponding Parts of Congruent Triangles CPCTC: Corresponding parts of congruent triangles are congruent. Using CPCTC to prove parts of congruent triangles are congruent. Ex 1 p. 138 ▬ ▬ ▬ ▬ ▬ W T Z V Given: WZ bisects TWV WT  WV Prove: TZ  VZ PROOF Statements Reasons ___ WZ bisects TWV 1. Given TWZ  VWZ 2. The bisector of an angle __ ___ separates it into two  ’s WT  WV 3. Given WZ  WZ 4. Identity Δ TWZΔ VWZ 5. SAS TZ TZ 6. CPCTC 11/9/2018 Section 3.2 Nack

Three Types of Conclusions Involving Triangles Proving triangles congruent. Proving corresponding pairs of congruent triangles. Note that the two triangles have to be proven congruent before you use CPCTC. Establishing a further relationship. Note that the two triangles have to be proven congruent and also apply CPCTC first. 11/9/2018 Section 3.2 Nack

Example 2 p. 139 __ __ Given: ZW  YX Z Y ZY  WX 2 Prove: ZY || WX 1 __ __ Given: ZW  YX ZY  WX Prove: ZY || WX Plan for Proof: Show that ΔZWX  ΔXYZ ; then we can say that 1  2 by CPCTC. Since these are alt. int angles for ZY and WX, these lines must be parallel. Statements Reasons ___ __ __ __ ZW  YX;ZY WX 1. Given ZX  ZX 2. Identity ΔZWX  ΔXYZ 3. SSS 1  2 4. CPCTC ZY || WX 5. If two lines are cut by a transversal so that the alt. Int. s are , these lines are parallel. Z Y 2 1 W X 11/9/2018 Section 3.2 Nack

Suggestions for Proving Triangles Congruent Mark the figures systematically, using: A square in the opening of each right angle. The same number of dashes on congruent sides. The same number of arcs on congruent angles. Trace the triangles to be proved congruent in different colors. If the triangles overlap, draw them separately. 11/9/2018 Section 3.2 Nack

Right Triangles and the HL method for proving Right Triangles Congruent. Theorem 3.2.1: If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of a second right triangle then the triangles are congruent. (HL) This theorem is illustrated in the construction on p. 141, Example 3, Fig. 3.20 Ex. 4 p. 141 Pythagorean Theorem: The square of the length (c) of the hypotenuse of a right triangle equals the sum of the squares of the lengths (a and b) of the legs of the right triangle. a² + b² = c² Ex. 5 p. 142 Hypotenuse Leg 11/9/2018 Section 3.2 Nack