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Prove that it is a right triangle. Select the correct answer from each drop-down menu.

____ is congruent to ____ because segment ____ was constructed so that ____. ____ is congruent to ____ because segment ____ was constructed so that ____. Since ____ is a right triangle, by the ____.

We are given that ____. Since ____ and ____, by the ____.

Also, by the ____.

Taking the square root of both sides of the equation gives ____. So, ____ is congruent to ____ by the definition of congruence. Applying the ____,

____. By CPCTC, ____. Therefore, ____ is a right angle, and ____ is a right triangle.

Answer :

Final answer:

To prove a right triangle using geometric principles, congruence of hypotenuse and an acute angle or congruence of sides and angles can be employed in accordance with certain geometric axioms and theorems.

Explanation:

To prove that a given figure is a right triangle, we can use several geometric principles and theorems.

For instance, if we are given two right triangles and we know that their hypotenuses are congruent and one of their acute angles is equal, then by the Corollary to the Pythagorean theorem, we can deduce that the triangles are congruent.

In the case of triangles OAC and OBC, assuming they are right triangles, we know that if side OC which is common to both triangles is congruent, and sides AC and BC are also congruent, then by the definition of congruence and the properties of isosceles triangles, two angles are congruent.

However, as stipulated by the axiom IV, 6, it's essential to have the third sides, OA and OB, congruent or not.

This detail becomes crucial when applying geometric theorems since congruence of sides and angles firmly establishes whether two triangles are congruent and, in the context of right triangles, whether a given triangle is indeed right-angled.

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