Question

Prove that if triangle ABC and triangle DEF are triangles with equal defects and a pair of congruent sides, their associated Saccheri quadrilaterals are congruent. (HINT: Recall that the angle sum for a triangle is equal to the sum of the measures of the summit angles of its associated Saccheri quadrilateral and that the two summit angles of a Saccheri quadrilateral are congruent.)

Answer 1

According to theorem 7.5

Π ABB'A' ≅ Π DEE'D'

therefore by transitivity of equivalence it is proven that triangle ABC and triangle DEF are triangles with equal defects and a pair of congruent sides

Explanation:

To prove that triangle ABC and triangle DEF are triangles with equal defects and a pair of congruent sides :

Assume: б(Δ ABC ) = б(Δ DEF ) and also AB ≅ DE

let Π ABB'A' and DEE'D' be taken as the saccheri quadrilaterals that corresponds to Δ ABC and Δ DEF respectively

Following the Lemma above; б(Π ABB'A' ) = б( Π DEE'D' ) given that

AB = summit of ABB'A' and DE = summit of DEE'D' also AB ≅ DE

According to theorem 7.5

Π ABB'A' ≅ Π DEE'D'

therefore by transitivity of equivalence it is proven that triangle ABC and triangle DEF are triangles with equal defects and a pair of congruent sides