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[For more discussions about two-column proofs, search the archives of the newsgroup geometry-pre-college for the words two column proof and browse the threads returned.]
Much as it pains me to do this :-) here's an example of a two column proof. The problem was one of our problems of the week last year. ********************************************************************** Problem of the Week for January 31 to February 4 (1994) Two engineers are scouting out an area where they plan to build a bridge. They are standing on the edge of the river, directly across from where the other end of the bridge will be. Their problem is this: They don't know how far it is across the river. They puzzle about this for a minute, and then Jill says to Jimmy, "I know how to find out how far it is." D Jill and Jimmy walk along the edge of the ----------------*-------- river to point B. Jimmy stays there. Then Jill walks to C, making sure that the distance river from B to C is equal to the distance from A to B. Then she turns 90 degrees, and walks ----*-----*-----*--------- away from the river until Jimmy, who is C| B A standing at B, is directly in line with D, | the point where they want the other end of | bridge. This is point E. Jill claims that | the distance from C to E is the same as the * distance from A to D. Is she right? Why E or why not? ********************************************************************** Given: BD = 3DAB, <BCE & <DAB right angles. Want to prove: CE=3DDA? (~ is the congruent symbol in this case) Statement | Reason -------------------------|---------------------- S 1. BC = 3DBA | 1. Given <BCE, <AD rt. angles | 2. Construct DE; DBE | 2. Given are collinear | A 3. <BCE ~ <BAD | 3. All right angles are congruent A 4. <CBE ~ <ABD | 4. Vertical angles are congruent 5. Triangle ECB = 3D~ | 5. ASA triangle DAB | 6. CE ~ DA | 6. CPCTE So that's a two-column proof. Just so people can see that not all students are obsessed with this idea, I've attached a couple of solutions that students sent in last year. As the students note, we aren't actually told that <DAB is a right angle, but it's assumed, and there isn't usually a spot in a two-column proof for the reason "assumed," though I suppose that could be step 1b. - Annie ********************************************************************** Jill is correct if the bridge is perpendicular to the river, forming a right angle at point A and if the river is straight. The distance between C and B is equal to the distance between A and B because of the given information. Angle DAB is congruent to angle BCE also because of the given information and the assumption that the bridge forms a right angle with the edge of the river. Because D, B, and E & C, B, and A form lines, angles CBE and DBA are vertical angles (Two angles are vertical angles if their sides form two pairs of opposite rays.) These two angles are congruent because of the Vertical Angle Theorem that states that vertical angles are congruent. Points B, C. and E form a triangle and points A, B, and D also form a triangle. Triangles ECB and DAB are congruent because two angles and the included side of those angles are congruent (All triangles with an ASA correspondence are congruent.) This means that CE=3DDA because corresponding parts of congruent triangles are equal and Jill is correct. Tammy Manski Grade 10 Shaler Area High School ********************************************************************** At the start of the problem I realized that I could prove she was right by using an ASA correspondence. To do come at this conclusion I did the following: First I knew that the 2 right angles were congruent because all right angles are congruent. The distances between BC and BA were equal because that was given. The third pair of angles was congruent because they were vertical angles and vertical angles are congruent. After proving those pairs congruent I proved the two triangles congruent by Angle Side Angle. Finally, I knew that DA equaled CE because corresponding parts of congruent triangles are congruent. My conclusion was that Jill's method for finding the distance across the river worked. Bipin Mujumdar Grade 10 Shaler Area High School ********************************************************************** |

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