Question

prove that $\triangle abc$ is a right triangle. select the correct answer from each drop-down menu.

$\overline{ab}$ is congruent to $\overline{de}$ because segment $de$ was constructed so that $de = ab$. $\overline{bc}$ is congruent to $\overline{ef}$ because segment $ef$ was constructed so that $ef = bc$. since $\triangle def$ is a right triangle, $de^2 + ef^2 = df^2$ by the $\boldsymbol{\
abla}$. we are given that $ab^2 + bc^2 = ac^2$. since $de = ab$ and $ef = bc$, $de^2 + ef^2 = ac^2$ by the $\boldsymbol{\
abla}$. also, $df^2 = ac^2$ by the $\boldsymbol{\
abla}$. taking the square root of both sides of the equation gives $df = ac$. so, $\overline{ac}$ is congruent to $\overline{df}$ by the definition of congruence. applying the $\boldsymbol{\
abla}$, $\triangle abc \cong \triangle def$. by cpctc, $\angle b \cong \angle e$. therefore $\angle b$ is a right angle and $\triangle abc$ is a right triangle.

Explanation:

Step1: Identify the first drop - down (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 \( \triangle DEF \), \( DE \) and \( EF \) are the legs and \( DF \) is the hypotenuse. So \( DE^{2}+EF^{2} = DF^{2} \) by the Pythagorean Theorem.

Step2: Identify the second drop - down (Substitution Property)

We know that \( DE = AB \) and \( EF = BC \). So we can substitute \( DE \) with \( AB \) and \( EF \) with \( BC \) in the equation \( DE^{2}+EF^{2} \). Since \( AB^{2}+BC^{2}=AC^{2} \), by the Substitution Property, \( DE^{2}+EF^{2}=AC^{2} \).

Step3: Identify the third drop - down (Transitive Property of Equality)

We have \( DE^{2}+EF^{2}=DF^{2} \) (from Pythagorean Theorem) and \( DE^{2}+EF^{2}=AC^{2} \) (from substitution). By the Transitive Property of Equality (if \( a = b \) and \( b = c \), then \( a = c \)), \( DF^{2}=AC^{2} \).

Step4: Identify the fourth drop - down (SSS Congruence Criterion)

We know that \( AB = DE \), \( BC = EF \), and we just proved \( AC = DF \). So by the SSS (Side - Side - Side) Congruence Criterion, if three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent. So \( \triangle ABC\cong\triangle DEF \) by SSS.

Answer:

1. First drop - down: Pythagorean Theorem
2. Second drop - down: Substitution Property
3. Third drop - down: Transitive Property of Equality
4. Fourth drop - down: SSS (Side - Side - Side) Congruence Criterion