Question

which statement proves that quadrilateral hijk is a kite?
hi ⊥ ij, and m∠h = m∠j.
ih = ij = 3 and jk = hk = √29, and ih ≠ jk and ij ≠ hk.
ik intersects hj at the midpoint of hj at (-1.5, 2.5).
the slope of hk = -2/5 and the slope of jk = -5/2.

Explanation:

Step1: Recall the definition of a kite

A kite is a quadrilateral with two pairs of adjacent sides that are equal in length.

Step2: Analyze each option

• Option 1: \(\overline{HI}\perp\overline{IJ}\), and \(m\angle H = m\angle J\). This does not satisfy the definition of a kite.
• Option 2: \(IH = IJ = 3\) and \(JK = HK=\sqrt{29}\), and \(IH

eq JK\) and \(IJ
eq HK\). This satisfies the definition of a kite (two pairs of adjacent - sides equal).

• Option 3: \(\overline{IK}\) intersects \(\overline{HJ}\) at the mid - point of \(\overline{HJ}\) at \((-1.5,2.5)\). This property is not the defining property of a kite.
• Option 4: The slope of \(\overline{HK}=-\frac{2}{5}\) and the slope of \(\overline{JK}=-\frac{5}{2}\). This only shows that \(HK\) and \(JK\) are not perpendicular (since \((-\frac{2}{5})\times(-\frac{5}{2}) = 1

eq - 1\)), and it does not prove the kite property.

Answer:

\(IH = IJ = 3\) and \(JK = HK=\sqrt{29}\), and \(IH
eq JK\) and \(IJ
eq HK\)