Given: and Given that , does it follow that

In Example 1, we could just as easily have used CPCTC to prove that two angles are congruent. If we had been asked to prove that , then the final statement would have read ∠T ⬵ ∠V Reminder CPCTC means “corresponding parts of congruent triangles are congruent.” 6. 6. CPCTC ∠T ⬵ ∠V We can take the proof in Example 1 a step further by proving triangles congruent and then using CPCTC to reach another conclusion, such as parallel or perpendicular lines. In Example 1, suppose we had been asked to prove that bisects . Then steps 1–6 would have remained as is, and a seventh step would have read TV WZ 7. bisects 7. If a line segment is divided into two parts, then it has been bisected ⬵ TV WZ In our study of triangles, we will establish three types of conclusions: 1. Proving triangles congruent, such as 2. Proving corresponding parts of congruent triangles congruent, like Note that two have to be proved before CPCTC can be used.

3. Establishing a further relationship, like bisects

Note that we must establish that two are and also apply CPCTC before this goal can be reached. ⬵ 䉭s TV WZ ⬵ 䉭s TZ ⬵ VZ 䉭TWZ ⬵ 䉭VWZ STRATEGY FOR PROOF 왘 Proofs that Involve Congruent Triangles Little is said in this book about a “plan for proof,” but every geometry student and teacher must have a plan before a proof can be completed. Though we generally do not write the “plan,” we demonstrate the technique in Example 2. EXAMPLE 2 GIVEN: See Figure 3.16. PROVE: PLAN FOR PROOF: By showing that , we can show that by CPCTC. Then 1 and 2 are congruent alternate interior angles for and , which must be parallel. WX ZY ∠s ∠1 ⬵ ∠2 䉭ZWX ⬵ 䉭XYZ ZY 7 WX ZY ⬵ WX ZW ⬵ YX W X Y Z 2 1 Figure 3.16 쮿 PROOF Statements Reasons

1. ;

2. 3. 4. 5. ZY 7 WX ∠1 ⬵ ∠2 䉭ZWX ⬵ 䉭XYZ ZX ⬵ ZX ZY ⬵ WX ZW ⬵ YX 1. Given 2. Identity 3. SSS 4. CPCTC 5. If two lines are cut by a transversal so that the alt. int. are , these lines are 7 ⬵ ∠s Exs. 4–6 SUGGESTIONS FOR PROVING TRIANGLES CONGRUENT Because many proofs depend upon establishing congruent triangles, we offer the following suggestions. W X Y Z 2 1 Figure 3.17 Leg Leg Hypotenuse Figure 3.18 Figure 3.19 Suggestions for a proof that involves congruent triangles:

1. Mark the figures systematically, using:

a A square in the opening of each right angle b The same number of dashes on congruent sides c The same number of arcs on congruent angles 2. Trace the triangles to be proved congruent in different colors. 3. If the triangles overlap, draw them separately. NOTE: In Figure 3.17, consider like markings. STRATEGY FOR PROOF 왘 Drawings Used to Prove Triangles Congruent Exs. 7–9 RIGHT TRIANGLES In a right triangle, the side opposite the right angle is the hypotenuse of the triangle, and the sides of the right angle are the legs of the triangle. These parts of a right trian- gle are illustrated in Figure 3.18. Another method for proving triangles congruent is the HL method, which applies exclusively to right triangles. In HL, H refers to hypotenuse and L refers to leg. The proof of this method will be delayed until Section 5.4. HL METHOD FOR PROVING TRIANGLES CONGRUENT 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. THEOREM 3.2.1 The relationship described in Theorem 3.2.1 HL is illustrated in Figure 3.19. In Example 3, the construction based upon HL leads to a unique right triangle. EXAMPLE 3 GIVEN: and in Figure 3.20a; note that . See page 141. CONSTRUCT: The right triangle with hypotenuse of length equal to AB and one leg of length equal to CA Solution Figure 3.20b: Construct perpendicular to at point C. Figure 3.20c: Now mark off the length of on . Í CQ CA Í EF Í CQ AB 7 CA CA AB The following theorem can be applied only when a triangle is a right triangle contains a right angle. Proof of the theorem is delayed until Section 5.4. Figure 3.20 Finally, with point A as center, mark off a length equal to that of as shown. is the desired right . 䉭 䉭ABC AB b Q F E C c C A B Q F E Geometry in the Real World In the manufacturing process, the parts of many machines must be congruent. The two sides of the hinge shown are congruent. A B E C D 1 2 Figure 3.21 EXAMPLE 4 Cite the reason why the right triangles and in Figure 3.21 are congruent if: a and b and C is the midpoint of c and d and bisects Solution a HL b AAS c ASA d SAS 쮿 BD EC AB ⬵ EC ∠1 ⬵ ∠2 BC ⬵ CD BD ∠A ⬵ ∠E AC ⬵ ED AB ⬵ EC 䉭ECD 䉭ABC Exs. 10–11 The square of the length c of the hypotenuse of a right triangle equals the sum of squares of the lengths a and b of the legs of the right triangle; that is, . c 2 = a 2 + b 2 PYTHAGOREAN THEOREM a A B C A 쮿