Question

ab is congruent to de because segment de was constructed so that de = ab. bc is congruent to ef because segment ef was constructed so that ef = bc. since △def is a right triangle, de² + ef² = df² by the. we are given that ab² + bc² = ac². since de = ab and ef = bc, de² + ef² = ac² by the. also, df² = ac² by the. taking the square root of both sides of the equation gives df = ac. so, ac is congruent to df by the definition of congruence. applying the, △abc ≅ △def. by cpctc, ∠b ≅ ∠e. therefore ∠b is a right angle and △abc is a right triangle.

Explanation:

Step1: Pythagorean Theorem

In a right - triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse. For right - triangle \(\triangle DEF\), \(DE^{2}+EF^{2}=DF^{2}\) by the Pythagorean Theorem.

Step2: Substitution Property

Since \(DE = AB\) and \(EF = BC\), we substitute \(AB\) for \(DE\) and \(BC\) for \(EF\) in the equation \(DE^{2}+EF^{2}\). So, \(DE^{2}+EF^{2}=AC^{2}\) (because \(AB^{2}+BC^{2}=AC^{2}\)) by the Substitution Property.

Step3: Transitive Property

We know that \(DE^{2}+EF^{2}=DF^{2}\) and \(DE^{2}+EF^{2}=AC^{2}\). By the Transitive Property (if \(a = b\) and \(a = c\), then \(b = c\)), \(DF^{2}=AC^{2}\).

Step4: SSS (Side - Side - Side) Congruence Criterion

We have \(AB = DE\), \(BC = EF\), and \(AC = DF\). By the SSS (Side - Side - Side) Congruence Criterion, \(\triangle ABC\cong\triangle DEF\).

Answer:

First blank: Pythagorean Theorem; Second blank: Substitution Property; Third blank: Transitive Property; Fourth blank: SSS (Side - Side - Side) Congruence Criterion.