Question

use the drop - down menus to complete the proof.
step 1: set the compass width between x and j. draw arc xy through ∠jkl.
step 2: keeping the compass width, draw arcs from x and y inside the angle. label the intersection z.
step 3: use the straightedge to connect k and z.
click the arrows to choose an answer from each menu.
if $overline{xz}$ and $overline{yz}$ are drawn on the construction from step 3, lengths $kx = ky$ and $xz=$ choose... since they are choose... in addition, $overline{kz}cong$ choose... by the reflexive property of congruence. so, $\triangle kxzcong$ choose... by the side - side - side (sss) postulate and $angle jkzcongangle lkz$ since they are corresponding parts of congruent triangles.

Explanation:

Step1: Analyze compass - set lengths

In the construction, when we set the compass width between \(X\) and \(J\) and draw arc \(XY\), we get \(KX = KY\) because of the way the compass is used (equal - radius construction). When we draw arcs from \(X\) and \(Y\) with the same compass width to get point \(Z\), we have \(XZ=YZ\) since the compass width is the same for both arcs, and they are radii of congruent circles (formed by the same - width compass).

Step2: Identify congruent sides

We know that \(\overline{KZ}\cong\overline{KZ}\) by the Reflexive Property of Congruence.

Step3: Apply SSS postulate

In \(\triangle KXZ\) and \(\triangle KYZ\), we have \(KX = KY\), \(XZ = YZ\), and \(KZ=KZ\). By the Side - Side - Side (SSS) postulate, \(\triangle KXZ\cong\triangle KYZ\).

Answer:

If \(\overline{XZ}\) and \(\overline{YZ}\) are drawn on the construction from step 3, lengths \(KX = KY\) and \(XZ = YZ\) since they are radii of congruent circles formed by the same - width compass. In addition, \(\overline{KZ}\cong\overline{KZ}\) by the Reflexive Property of Congruence. So, \(\triangle KXZ\cong\triangle KYZ\) by the Side - Side - Side (SSS) postulate and \(\angle JKZ\cong\angle LKZ\) since they are corresponding parts of congruent triangles.