Question

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 ____. 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 ____.
a) Hypotenuse-Leg (HL) Congruence
b) Corresponding Angles Postulate
c) Side-Angle-Side (SAS) Congruence
d) Definition of a Right Triangle

Answer 1

In a right triangle, the Pythagorean theorem is applied to relate the lengths of the legs and the hypotenuse. By using substitution and taking the square root, we can show that the hypotenuse is equal to the hypotenuse of another right triangle. Using the HL congruence and CPCTC, we can conclude that the triangles are congruent and angle A is a right angle. Therefore, triangle ABC is a right triangle.

Step-by-step explanation:

Since △DEF is a right triangle, we know that a² + b² = n² because of the Pythagorean theorem. 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 the Hypotenuse-Leg (HL) Congruence, triangles ABC and DEF are congruent. Since it is given that ∠F is a right angle, then ∠A is also a right angle by CPCTC (Corresponding Parts of Congruent Triangles are Congruent). Therefore, △ABC is a right triangle by the Definition of a Right Triangle.