Question

which statement proves that quadrilateral hijk is a kite?
○ $overline{hi}perpoverline{ij}$, and $mangle h = mangle j$.
○ $ih = ij = 3$ and $jk = hk=sqrt{29}$, and $ih
eq jk$ and $ij
eq hk$.
○ $overline{ik}$ intersects $overline{hj}$ at the mid - point of $overline{hj}$ at $(-1.5,2.5)$.
○ the slope of $overline{hk}=-\frac{2}{5}$ and the slope of $overline{jk}=-\frac{5}{2}$.

Explanation:

Step1: Recall kite - definition

A kite is a quadrilateral with two pairs of adjacent sides equal and the unequal sides not equal.

Step2: Analyze each option

• Option 1: $\overline{HI}\perp\overline{IJ}$ and $m\angle H = m\angle J$ does not prove it's a kite.
• Option 2: $IH = IJ=3$ and $JK = HK=\sqrt{29}$, and $IH

eq JK$ and $IJ
eq HK$ satisfies the property of a kite (two pairs of adjacent - sides are equal and the non - adjacent equal sides are not equal).

• Option 3: $\overline{IK}$ intersecting $\overline{HJ}$ at the mid - point of $\overline{HJ}$ does not prove it's a kite.
• Option 4: Just knowing the slopes of $\overline{HK}$ and $\overline{JK}$ does not prove it's a kite.

Answer:

IH = IJ = 3 and JK = HK = $\sqrt{29}$, and IH ≠ JK and IJ ≠ HK.