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. 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. by substitution, c² = n². using the square - root property and the principle root, we can take the square root of both sides to get c = n. by, triangles 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. asa sss the pythagorean theorem the definition of right triangle

Explanation:

Step1: Recall right - triangle property

Since $\triangle DEF$ is a right - triangle, by the Pythagorean theorem, $a^{2}+b^{2}=n^{2}$.

Step2: Use congruence criterion

We have $c = n$, $a=a$, and $b = b$. By the SSS (Side - Side - Side) congruence criterion, $\triangle ABC\cong\triangle DEF$.

Step3: Identify corresponding angles

Since $\triangle ABC\cong\triangle DEF$ and $\angle F$ is a right - angle, by CPCTC (Corresponding Parts of Congruent Triangles are Congruent), $\angle C$ is also a right - angle. And by the definition of a right - triangle (a triangle with one right - angle), $\triangle ABC$ is a right - triangle.

Answer:

1. the Pythagorean theorem
2. SSS
3. C
4. the definition of right triangle