Question
One angle of a triangle is 71°71\degree71° . How many degrees must one of the other angles be for the Pythagorean Theorem to be true for the triangle?
Answers
Answered by
GPT-4o mini
The Pythagorean Theorem applies specifically to right triangles, which have one angle measuring \(90^\circ\). In a triangle, the sum of all angles is \(180^\circ\).
Given one angle of the triangle is \(71^\circ\), we can denote the other two angles as \(A\) and \(B\). For the triangle to have a right angle, one of these angles must be \(90^\circ\).
Assuming \(A\) is our right angle:
\[
71^\circ + A + B = 180^\circ
\]
Substituting \(A = 90^\circ\):
\[
71^\circ + 90^\circ + B = 180^\circ
\]
This simplifies to:
\[
161^\circ + B = 180^\circ
\]
Thus,
\[
B = 180^\circ - 161^\circ = 19^\circ
\]
The angles of the triangle would then be \(71^\circ\), \(90^\circ\), and \(19^\circ\).
Therefore, one of the other angles must be \(90^\circ\) for the Pythagorean Theorem to hold true in the triangle.
Given one angle of the triangle is \(71^\circ\), we can denote the other two angles as \(A\) and \(B\). For the triangle to have a right angle, one of these angles must be \(90^\circ\).
Assuming \(A\) is our right angle:
\[
71^\circ + A + B = 180^\circ
\]
Substituting \(A = 90^\circ\):
\[
71^\circ + 90^\circ + B = 180^\circ
\]
This simplifies to:
\[
161^\circ + B = 180^\circ
\]
Thus,
\[
B = 180^\circ - 161^\circ = 19^\circ
\]
The angles of the triangle would then be \(71^\circ\), \(90^\circ\), and \(19^\circ\).
Therefore, one of the other angles must be \(90^\circ\) for the Pythagorean Theorem to hold true in the triangle.
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