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PRACTICE: Proofs Name: ____________________ What is wrong with these Isosceles Triangle Theorem proofs? 1. 2. Connect A to a point P on BC. Obviously The base angles must be AP=AP. It is given that AB=AC. When two equal, because if they sides are equal, the third sides must be weren’t, the triangle would equal, so BP = PC. We have SSS, so not be isosceles. ΔAPB ≅ ΔAPC, and the corresponding parts are equal. So ∠ABC=∠ACB. What is wrong with these Converse of the Isosceles Triangle Theorem proofs? 3. 4. Draw the line from A to the midpoint M of Draw the line from A to the BC. Obviously, AM=AM. BM=MC since M midpoint M of BC. Obviously, is the midpoint. ∠ABC = ∠ACB is given. So AM=AM. BM=MC since M is the midpoint. AB = AC since ΔAMB ≅ ΔAMC, and the corresponding the triangle is isosceles. So parts are equal, so AB=AC. we have SSS, and the corresponding parts are equal, so AB=AC. 5. Drop a perpendicular from A to BC. Say it meets BC at point H. Obviously, AH=AH, and BH = HC. ∠AHC = ∠AHB = 90°. Therefore ΔAHB ≅ ΔAHC by SAS, and the corresponding parts are equal, so AB=AC. 6. Drop a perpendicular from A to BC. Say it meets BC at point H. When two angles are equal, so are the third angles. So ΔAHB ≅ ΔAHC by AAA, and the corresponding parts are equal, so AB=AC. PRACTICE: Proofs 7. Given: ∠ A and ∠ O are right angles and Segment AO bisects Segment RN. Prove: ΔRAD ≅ ΔNOD. 8. Given: Ray MT bisects ∠ M and MA = MH. Prove: ΔMAT is congruent to ΔMHT 9. Given: ∠ 3 = ∠ 4. BH = HC. Prove: Δ BHI ≅ Δ CHR. Name: ____________________ PRACTICE: Proofs 10. What is the area of the square? 12. How do you explain the apparent contradiction? Name: ____________________ 11. The pieces of the square can be rearranged to make the figure on the right. What is its area? This puzzle shows that just because something looks true, it is not necessarily so! This is why proofs need to be carefully constructed to be completely logically tight. In the example above, we assumed the pieces fit together as shown, without proving it, which led us to an impossible situation. PRACTICE: Proofs Name: ____________________ What is wrong with these Isosceles Triangle Theorem proofs? 1. 2. Connect A to a point P on BC. Obviously The base angles must be AP=AP. It is given that AB=AC. When two equal, because if they sides are equal, the third sides must be weren’t, the triangle would equal, so BP = PC. We have SSS, so not be isosceles. ΔAPB ≅ ΔAPC, and the corresponding parts are equal. So ∠ABC=∠ACB. What is wrong with these Converse of the Isosceles Triangle Theorem proofs? 3. 4. Draw the line from A to the midpoint M of Draw the line from A to the BC. Obviously, AM=AM. BM=MC since M midpoint M of BC. Obviously, is the midpoint. ∠ABC = ∠ACB is given. So AM=AM. BM=MC since M is the midpoint. AB = AC since ΔAMB ≅ ΔAMC, and the corresponding the triangle is isosceles. So parts are equal, so AB=AC. we have SSS, and the corresponding parts are equal, so AB=AC. 5. Drop a perpendicular from A to BC. Say it meets BC at point H. Obviously, AH=AH, and BH = HC. ∠AHC = ∠AHB = 90°. Therefore ΔAHB ≅ ΔAHC by SAS, and the corresponding parts are equal, so AB=AC. 6. Drop a perpendicular from A to BC. Say it meets BC at point H. When two angles are equal, so are the third angles. So ΔAHB ≅ ΔAHC by AAA, and the corresponding parts are equal, so AB=AC. PRACTICE: Proofs 7. Given: ∠ A and ∠ O are right angles and Segment AO bisects Segment RN. Prove: ΔRAD ≅ ΔNOD. 8. Given: Ray MT bisects ∠ M and MA = MH. Prove: ΔMAT is congruent to ΔMHT 9. Given: ∠ 3 = ∠ 4. BH = HC. Prove: Δ BHI ≅ Δ CHR. Name: ____________________ PRACTICE: Proofs 10. What is the area of the square? 12. How do you explain the apparent contradiction? Name: ____________________ 11. The pieces of the square can be rearranged to make the figure on the right. What is its area? This puzzle shows that just because something looks true, it is not necessarily so! This is why proofs need to be carefully constructed to be completely logically tight. In the example above, we assumed the pieces fit together as shown, without proving it, which led us to an impossible situation.
