Use this plan to write a paragraph proof of the Converse of...

Use this plan to write a paragraph proof of the Converse of the Pythagorean Theorem. Given: $\triangle A B C$ with $a^{2}+b^{2}=c^{2}$ Prove: $\triangle A B C$ is a right triangle. Plan: Draw $\triangle P Q R$ with $\angle R$ as the right angle, leg lengths of $a$ and $b,$ and a hypotenuse of length $x .$ By the Pythagorean Theorem, $a^{2}+b^{2}=x^{2} .$ Use substitution to compare $x$ and $c .$ Show that $\triangle A B C \cong \triangle P Q R$ and thus $\angle C$ is a right angle.

Use this plan to write a paragraph proof of the Converse of the Pythagorean Theorem.
Given: $\triangle A B C$ with $a^{2}+b^{2}=c^{2}$
Prove: $\triangle A B C$ is a right triangle.
Plan: Draw $\triangle P Q R$ with $\angle R$ as the right angle, leg lengths of $a$ and $b,$ and a hypotenuse of length $x .$ By the Pythagorean Theorem, $a^{2}+b^{2}=x^{2} .$ Use substitution to compare $x$ and $c .$ Show that $\triangle A B C \cong \triangle P Q R$ and thus $\angle C$ is a right angle.

Key Concepts

Substitution Method

The substitution method in proofs involves replacing one expression with another equivalent one in order to simplify or transform an equation or inequality. Here, it is used to replace the expression for the squared hypotenuse obtained from the auxiliary right triangle with the corresponding side length in the given triangle, thereby setting up a comparison that supports the congruence argument.

Auxiliary Construction

Auxiliary construction is a technique used in proofs where an additional geometric figure is introduced to help reveal hidden relationships or properties. In this context, constructing an auxiliary right triangle with specified leg lengths allows one to utilize the Pythagorean Theorem and establish a basis for comparing the given triangle’s side lengths, ultimately aiding in proving that the triangle is right-angled.

Triangle Congruence

Triangle congruence involves demonstrating that two triangles are identical in shape and size, which often requires proving that corresponding sides and angles are equal. In the context of proving the converse of the Pythagorean Theorem, establishing congruence between a constructed right triangle and the given triangle confirms that the angle in question is indeed a right angle.

Pythagorean Theorem

The Pythagorean Theorem states that in any right triangle, the sum of the squares of the lengths of the two legs is equal to the square of the length of the hypotenuse. This relationship is fundamental in Euclidean geometry and is frequently employed to establish relationships among the sides of right triangles.

Converse of the Pythagorean Theorem

The Converse of the Pythagorean Theorem asserts that if in a triangle the sum of the squares of two sides is equal to the square of the third side, then the triangle is a right triangle. This logical inversion of the original theorem serves as a criterion to verify the presence of a right angle in a triangle based solely on its side lengths.

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