Question

the diagram shows two triangles. in \\( \triangle t u v, u v=a, t v=b, t u=c \\), and \\( a^{2}+b^{2}=c^{2} \\). in \\( \triangle w x y, x y=a, w x=b \\), and \\( m \angle w x y=90^{circ} \\).

select all of the statements that are true.

\\( m \angle t u v+m \angle w x y=180^{circ} \\)
\\( \overline{t v} \\) is the hypotenuse of \\( \triangle t u v \\).
\\( \angle v \\) is a right angle.
\\( w y=c \\)

Explanation:

Step1: Analyze the first statement

Since \(a^{2}+b^{2}=c^{2}\) in \(\triangle TUV\), by the Pythagorean theorem, \(\angle V = 90^{\circ}\). And \(m\angle WXY=90^{\circ}\). So \(m\angle TUV + m\angle WXY=90^{\circ}+90^{\circ}=180^{\circ}\).

Step2: Analyze the second statement

In \(\triangle TUV\), because \(a^{2}+b^{2}=c^{2}\) (where \(a = UV\), \(b = TV\), \(c = TU\)), by the Pythagorean theorem, the side opposite the right - angle is the hypotenuse. The right - angle is \(\angle V\), and the side opposite \(\angle V\) is \(TU\). Wait, no. Wait, in a triangle, if \(a^{2}+b^{2}=c^{2}\), the side with length \(c\) is the hypotenuse. Here \(TU = c\), so \(TU\) is the hypotenuse. Wait, no, wait the formula \(a^{2}+b^{2}=c^{2}\) for a right - triangle, where \(c\) is the hypotenuse. In \(\triangle TUV\), \(UV=a\), \(TV = b\), \(TU=c\), so \(TU\) is the hypotenuse. Wait, no, wait the user's second statement says \(\overline{TV}\) is the hypotenuse. Wait, no, there is a mistake. Wait, in \(\triangle WXY\), by Pythagorean theorem \(WY^{2}=a^{2}+b^{2}\). Since \(a^{2}+b^{2}=c^{2}\) in \(\triangle TUV\), then \(WY = c\). And in \(\triangle TUV\), since \(a^{2}+b^{2}=c^{2}\), \(\angle V=90^{\circ}\).

Step3: Analyze the third statement

Since \(a^{2}+b^{2}=c^{2}\) in \(\triangle TUV\), by the converse of the Pythagorean theorem, \(\angle V = 90^{\circ}\)

Step4: Analyze the fourth statement

In \(\triangle WXY\), by the Pythagorean theorem \(WY^{2}=XY^{2}+WX^{2}\). Given \(XY = a\), \(WX = b\), and \(a^{2}+b^{2}=c^{2}\) (from \(\triangle TUV\)), so \(WY^{2}=c^{2}\), then \(WY = c\)

Answer:

\(m\angle TUV + m\angle WXY = 180^{\circ}\), \(\angle V\) is a right angle, \(WY = c\)