Question

a kite is a quadrilateral which has 2 sides next to each other that are congruent and where the other 2 sides are also congruent. given kite wxyz, which of the following shows that at least one of the diagonals of a kite decomposes the kite into 2 congruent triangles? in kite wxyz, wx is congruent to wz and yx is congruent to yz. wy is congruent to wy since segments are congruent to themselves. triangle wyz is congruent to triangle wyx by the side - side - side triangle congruence theorem; thus, at least one diagonal of the kite decomposes the kite into two congruent triangles. in kite wxyz, wx is congruent to wz and yx is congruent to yz. triangle wyz is congruent to triangle wyx by the side - side triangle congruence theorem; thus, at least one diagonal of the kite decomposes the kite into two congruent triangles. in kite wxyz, wx is congruent to wz and yx is congruent to yz. angle wxy is congruent to angle wzy. triangle wyz is congruent to triangle wyx by the side - angle - side triangle congruence theorem; thus, at least one diagonal of the kite decomposes the kite into two congruent triangles. in kite wxyz, angle wxy is congruent to angle wzy and angle xwz is congruent to angle xyz. wy is congruent to wy since segments are congruent to themselves. triangle wyz is congruent to triangle wyx by the angle - side - angle triangle congruence theorem; thus, at least one diagonal of the kite decomposes the kite into two congruent triangles.

Explanation:

Step1: Recall kite properties

A kite has two - pairs of adjacent congruent sides. In kite \(WXYZ\), \(WX\cong WZ\) and \(YX\cong YZ\), and the diagonal \(WY\) is common to \(\triangle WYZ\) and \(\triangle WYX\).

Step2: Apply SSS congruence theorem

The Side - Side - Side (SSS) Triangle Congruence Theorem states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. Here, in \(\triangle WYZ\) and \(\triangle WYX\), \(WX = WZ\), \(YX=YZ\), and \(WY = WY\). So, \(\triangle WYZ\cong\triangle WYX\) by SSS.

Answer:

The first option: In kite \(WXYZ\), \(WX\) is congruent to \(WZ\) and \(YX\) is congruent to \(YZ\). \(WY\) is congruent to \(WY\) since segments are congruent to themselves. Triangle \(WYZ\) is congruent to triangle \(WYX\) by the Side - Side - Side Triangle Congruence Theorem, thus, at least one diagonal of the kite decomposes the kite into two congruent triangles.