In the diagram below, BC = QR = x, AC = PR = y, and triangle PQR is a right

Question

triangle. Use the drop-down menus to complete the proof that if the equation
x2 + y2 = z2 is true of the side lengths in triangle ABC, then triangle ABC must be a
right triangle.

P

y

Z

y

X

F

x2 + y2 = z2

X

Triangle PQR is a right triangle

R

In the diagram below, BC = QR = x, AC = PR = y, and triangle PQR is a right
triangle. Use the drop-down menus to complete the proof that if the equation
x2 + y2 = z2 is true of the side lengths in triangle ABC, then triangle ABC must be a
right triangle.

P

y

Z

y

X

F

x2 + y2 = z2

X

Triangle PQR is a right triangle

R

Click the arrows to choose an answer from each menu.

Triangle PQR is given to be a right triangle, so the Pythagorean theorem can be used to
. It is given that x2 + y2 = 22 in triangle ABC, so
. This information shows that the triangles are congruent
because their three sides are congruent. Therefore, to show that triangle PQR is a right triangle,
we can use corresponding parts of the congruent triangles to show that

determine that

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4

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Answer 1

To complete the proof that if the equation \( x^2 + y^2 = z^2 \) is true of the side lengths in triangle ABC, then triangle ABC must be a right triangle, we can follow this structure:

1. Triangle PQR is given to be a right triangle, so the Pythagorean theorem can be used to determine that \( x^2 + y^2 = z^2 \).

2. It is given that \( x^2 + y^2 = z^2 \) in triangle ABC, so triangle ABC satisfies the conditions of the Pythagorean theorem.

3. This information shows that the triangles are congruent because their three sides are congruent. Therefore, to show that triangle PQR is a right triangle, we can use corresponding parts of the congruent triangles to show that triangle ABC must also be a right triangle.

So, the filled proof statement would look like this:

• Triangle PQR is given to be a right triangle, so the Pythagorean theorem can be used to determine that \( x^2 + y^2 = z^2 \).
• It is given that \( x^2 + y^2 = z^2 \) in triangle ABC, so triangle ABC satisfies the conditions of the Pythagorean theorem.
• This information shows that the triangles are congruent because their three sides are congruent. Therefore, to show that triangle PQR is a right triangle, we can use corresponding parts of the congruent triangles to show that triangle ABC must also be a right triangle.

Make sure to load your proof with any specific conditions required by your instructor or the guidelines of your coursework.