Geometry Problems of the Day (Geometry Regents, June 2013)

Now that I'm caught up with the current New York State Regents exams, I'm revisiting some older ones.

More Regents problems.

Geometry Regents, June 2013

Part IV: A correct answer will receive 6 credits. Partial credit is possible.


38. In the diagram of MAH below, MH ≅ AH and medians AB and MT are drawn.

Prove: ∠MBA ≅ ∠ATM

Answer:

To prove that ∠MBA and ∠ATM are congruent, you may first wamt to prove that traingles MBA and ATM are congruent. However, you don't have enough information. I remember this problem first appearing, and I remember several math teachers trying to reason it out. When they got to around 11 steps, which they felt certain couldn't be the best approach, another teacher pointed out a different method.

A better approach is to show that triangle HTM ≅ triangle HBA.

We know that this is an isosceles triangle, which gives us a pair of sides. We have medians which give us a second side. And angle H is congruent to itself with the reflexive property.

Statement	Reason
1. MH ≅ AH, AB and MT are medians	Given
2. B and T are midpoints	Definition of median
3. BH ≅ TH	Division Property of Equality
4. ∠H ≅ ∠H	Reflexive Property
4. Triangle HTM ≅ triangle HBA	SAS
5. ∠HBA ≅ ∠HTM	CPCTC
6. ∠MBA ≅ ∠ATM	Angles supplementary to congruent angles are congruent.

End of Exam.

More to come. Comments and questions welcome.

More Regents problems.

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