Theorem 1. (Exterior Angle Inequality) The measure of an exterior
... Proof: By Lemma 2, the angle sum of 4ABC ≤ 180◦ and the angle sum of 4ACD ≤ 180◦ . If both of these inequalities were equalities we would have m∠1 + m∠2 + m∠B + m∠D + m∠3 + m∠4 = 360, in which case the angle sum of 4ABD = m∠1 + m∠2 + m∠B + m∠D = 360 − m∠3 − m∠4 = 180◦ , contradicting our hypothesis. ...
... Proof: By Lemma 2, the angle sum of 4ABC ≤ 180◦ and the angle sum of 4ACD ≤ 180◦ . If both of these inequalities were equalities we would have m∠1 + m∠2 + m∠B + m∠D + m∠3 + m∠4 = 360, in which case the angle sum of 4ABD = m∠1 + m∠2 + m∠B + m∠D = 360 − m∠3 − m∠4 = 180◦ , contradicting our hypothesis. ...
Final exam key
... 3. (25 pts.) Prove (within neutral geometry) the hypotenuse–leg theorem: Two right triangles are congruent if the hypotenuse and one other side of one triangle are congruent (respectively) to the hypotenuse and a side of the other triangle. [See Ex. 4.4, p. 193. Note that it is not enough to move th ...
... 3. (25 pts.) Prove (within neutral geometry) the hypotenuse–leg theorem: Two right triangles are congruent if the hypotenuse and one other side of one triangle are congruent (respectively) to the hypotenuse and a side of the other triangle. [See Ex. 4.4, p. 193. Note that it is not enough to move th ...
