Question

the diagram shows two triangles. in \\( \triangle jkl, jk = f, kl = g \\), and \\( \angle j \\) is a right angle. in \\( \triangle wxy, wx = f, xy = g, wy = h \\), and \\( h ^ { 2 } = f ^ { 2 } + g ^ { 2 } \\).

select all of the statements that are true.

\\( \angle x \\) is a right angle.

\\( \triangle wxy \\) is an equilateral triangle.

\\( m \angle w + m \angle y = 90 ^ { \circ } \\)

the length of \\( \overline { x y } \\) is greater than the length of \\( \overline { j l } \\).

Explanation:

Step1: Check if $\angle X$ is a right angle

In $\triangle WXY$, since $h^{2}=f^{2}+g^{2}$ (by the Pythagorean theorem converse), $\angle X$ is a right angle.

Step2: Check if $\triangle WXY$ is an equilateral triangle

An equilateral triangle has all sides equal. Here, $WX = f$, $XY = g$, $WY = h$ and $h^{2}=f^{2}+g^{2}$ (unless $f = g = 0$, which is not possible for triangle side - lengths), so $\triangle WXY$ is not equilateral.

Step3: Check $m\angle W+m\angle Y$

In a right - angled triangle $\triangle WXY$ (right - angled at $X$), by the angle - sum property of a triangle ($m\angle W + m\angle X+m\angle Y=180^{\circ}$) and $m\angle X = 90^{\circ}$, we get $m\angle W + m\angle Y=90^{\circ}$.

Step4: Check the length of $\overline{XY}$ and $\overline{JL}$

In $\triangle JKL$, using the Pythagorean theorem $JL=\sqrt{g^{2}-f^{2}}$ (since $\angle J = 90^{\circ}$, $KL^{2}=JK^{2}+JL^{2}$, so $JL=\sqrt{KL^{2}-JK^{2}}=\sqrt{g^{2}-f^{2}}$) and $XY = g$. Since $\sqrt{g^{2}-f^{2}}0$.

Answer:

• $\angle X$ is a right angle.
• $m\angle W + m\angle Y = 90^{\circ}$
• The length of $\overline{XY}$ is greater than the length of $\overline{JL}$.