Two-column Proof

Annie Fetter, The Geometry Forum

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From: (Annie Fetter)
Subject:    2-column proof example
Date:       Wed, 25 Jan 1995 10:51:58 -0500

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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