Geometry proof

This problem is adapted from problem #15 in section 3.4 of

Rhoad, R., Milsaukas, G., & Whipple, R. Geometry for Enjoyment and Challenge (New Edition). McDougal Littell, 1991. Page 137.

All references to theorems are taken taken this text.

Problem:

Given the figure

and the statements

$\overline {AB} \cong \overline{AC}$

$\overline {BD} \cong \overline{CE}$,

Prove: $\angle ABC \cong \angle ACB$

Comment: This could be proved in a single step using Theorem 21 (page 149) "If two sides of a triangle are congruent, the angles opposite the sides are congruent." However, this theorem will not be proved until a later chapter, so we must arrive at our proof by different means.

Historical note: This problem as actually a disguised version of the "pons asinorum" (or "Bridge of Asses") proof from Euclid's Elements. In that problem only the $\triangle ABC$ is given, and the the student must prove that if it is isosceles then its base angles are congruent. The proof given is surprisingly complex (though simpler proofs are known); hence it was said to separate the students who could understand it from the "asses" who could not.

Solution:

	Statement	Justification
1.	$\overline {AB} \cong \overline{AC}$	Given
2.	$\overline {BD} \cong \overline{CE}$	Given
3.	$\overline {AD} \cong \overline{AE}$	Addition Property (1, 2)
4.	$\angle A \cong \angle A$	Reflexive Property
5.	$\triangle ADC \cong \triangle AEB$	SAS (3, 4, 1)
6.	$\overline{DC} \cong \overline{EB}$	CPCTC (5)
7.	$\angle D \cong \angle E$	CPCTC (5)
8.	$\triangle BDC \cong \triangle CEB$	SAS (2, 7, 6)
9.	$\angle DBC \cong \angle ECB$	CPCTC (8)
10.	$\angle DBC$ is supp. to $\angle CBA$	Def. of supp. $\angle$'s
11.	$\angle ECB$ is supp. to $\angle BCA$	Def. of supp. $\angle$'s
12.	$\angle DBC$ is supp. to $\angle BCA$	Substitution (9, 11)
13.	$\angle CBA \cong \angle BCA$	Transitive property (10, 12)

Q. E. D.