Wednesday, December 3, 2014

Section 7-3: Triangle Congruence Proofs (Day 72)

Section 7-3 of the U of Chicago text discusses triangle congruence proofs. Finally, this is what most geometry students and teachers think of when they hear about "proofs."

Every year, Theoni Pappas, a Berkeley mathematician, publishes her Mathematical Calendar.

I've already purchased my 2015 calendar -- the 35th edition. But on the current 2014 calendar, the question for today happens to be a geometry question -- furthermore, one that's very relevant to the Section 7-3 of the text:

If M is a midpoint of segment AB and angles A and B are congruent, then triangle ACM is congruent to triangle BDM by:

1) SSS
2) SAS
3) ASA
4) hypotenuse leg theorem
5) none of the above

Now the diagram accompanying this problem is nearly identical to the one that appears in Example 2 in the U of Chicago text, except that the calendar labels one segment AB where the U of Chicago text labels it EF. In the text, M is the midpoint of both CD and EF, and so SAS is used. But on the calendar, the proof would be different:

Statements                   Reasons
1. M midpoint AB,       1. Given
angle A = angle B
2. AM = MB                 2. Definition of midpoint
3. angle AMC = BMD  3. Vertical Angles Theorem
4. tri. ACM = tri. BDM 4. ASA Congruence Theorem (steps 1, 2, 3)

So the correct answer is choice 3) -- and notice that today's date is the 3rd. Pappas likes to make the answer to each day's question the date. This means, of course, that multiple choice questions such as this one can only occur during the first week of each month -- otherwise there would have to be a question with 31 choices on the last day of the month. Most questions are, therefore, free response rather than multiple choice.

If I ever get to teach full-time, I'm considering having a daily warm-up consisting of a single question to which the answer -- just as on the Mathematical Calendar -- is just the date. This should alert students that what I'm looking for is the work and not just the answer -- so don't try to copy only the answer from another student.

Once again, I'm still working on my scanner, so let me choose the problems from the U of Chicago text that I would give as exercises. Notice that the U of Chicago provides the proof of a theorem that serves as a converse to the Isosceles Triangle Theorem. This proof is based on AAS, which itself requires a Parallel Postulate. But as I mentioned earlier, we could use ASA instead if line AD were the perpendicular bisector of BC rather than an angle bisector. Also, we could use ASA if we used that quick-and-dirty Pappus proof of showing that a triangle is congruent to itself. (And that's Pappus, the ancient Greek mathematician, not Pappas, the modern California mathematician. Of course, Theoni is probably of Greek descent -- maybe she's a descendant of the great Alexandrian after all.)

1. Finish this statement of the converse of the Isosceles Triangle Theorem: If two angles of a triangle are congruent, then...

2. Fill in the missing reasons in this proof of the converse of the ITT:
Given: angle B = angle C
Prove: AB = AC
(Draw ray AD, the bisector of angle A.)
Statements                  Reasons
1. angle B = angle C   1. Given
2. angle BAD = CAD  2. Definition of angle bisector
3. AD = AD                 3.
4. tri. ABD = tri. ACD 4. AAS Congruence Theorem (steps 1, 2, 3)
5. AB = AC                  5.

3. Multiple Choice. If M is a midpoint of segment AB and angles A and B are congruent, then triangle ACM is congruent to triangle BDM by:
1) SSS
2) SAS
3) ASA
4) hypotenuse leg theorem
5) none of the above

4.a.With measures angle A = angle C = 61, angle B = 58, which sides of triangle ABC are congruent?
   b. What is your justification for your answer to part a?

5. Given: ABC is isosceles with vertex angle B. D is the midpoint of AC.
    Prove: triangles ABD and CBD are congruent

6. Given: AB = CD, BC = AD
    Prove: triangles ABC and CDA are congruent

Review Exercises:

7. Same as #6, except all five segments (the four mentioned plus AC) are congruent. Make and justify at least two conclusions from this given information.

8. Tony and Trisha each made a triangle out of straws with lengths 3 cm, 4 cm, and 6 cm. True/false?
a. The two triangles must be congruent.
b. The two triangles must have the same orientation.

9. The sum of the measures of the acute angles in a right triangle is _____.

10.a. Draw a figure to test this conjecture: If the midpoints of the sides of a rhombus are connected in order, the resulting figure is a square.
     b. Do you think that the conjecture is true?

Bonus Exercise:
a. What would be an SSSS condition for two quadrilaterals to be congruent?
b. Show by drawing a counterexample that there is no SSSS Congruence Theorem for quadrilaterals.

Note on the Bonus Exercise, for teachers:
The easiest counterexample would be two rhombuses -- one a square, the other not.

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