Question

given: △abc with a² + b² = c² and right △def constructed with legs a and b and hypotenuse n. prove: △abc is a right triangle. complete the missing parts of the paragraph proof. we are given a² + b² = c² for △abc and right △def constructed with legs a and b and hypotenuse n. since △def is a right triangle, we know that a² + b² = n² because of the square - root property and the principle root. using the substitution, c² = n². by taking the square root of both sides to get c = n. by cpctc, △abc and △def are congruent. since it is given that ∠f is a right angle, then ∠ is also a right angle by cpctc. therefore, △abc is a right triangle by cpctc.

Explanation:

Step1: Apply Pythagorean theorem to △DEF

Since △DEF is a right - triangle with legs \(a\) and \(b\) and hypotenuse \(n\), by the Pythagorean theorem, \(a^{2}+b^{2}=n^{2}\).

Step2: Use the given equation for △ABC

We are given \(a^{2}+b^{2}=c^{2}\) for △ABC. Substituting \(a^{2}+b^{2}=n^{2}\) (from △DEF) into \(a^{2}+b^{2}=c^{2}\), we get \(c^{2}=n^{2}\).

Step3: Take square roots

Taking the square root of both sides of \(c^{2}=n^{2}\), we have \(c = n\) (we consider the non - negative square root since \(c\) and \(n\) represent side lengths).

Step4: Prove congruence and conclude

Since in △ABC and △DEF, \(a=a\), \(b = b\), and \(c=n\), by the Side - Side - Side (SSS) congruence criterion, △ABC and △DEF are congruent. Since ∠F is a right angle, then ∠C is also a right angle by CPCTC (Corresponding Parts of Congruent Triangles are Congruent). Therefore, △ABC is a right triangle.

Answer:

We are given \(a^{2}+b^{2}=c^{2}\) for △ABC and △DEF is a right - triangle with legs \(a\) and \(b\) and hypotenuse \(n\). Since △DEF is a right - triangle, we know that \(a^{2}+b^{2}=n^{2}\) because of the Pythagorean theorem. Using substitution, \(c^{2}=n^{2}\). By the square root property and the principle root, we can take the square root of both sides to get \(c = n\). By SSS, △ABC and △DEF are congruent. Since ∠F is a right angle, then ∠C is also a right angle by CPCTC. Therefore, △ABC is a right triangle.