Question

use the drop - down menus to complete the proof.

step 1: set the compass width between k 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 $xz \cong$ choose... since they are choose.... in addition, $overline{kz} \cong$ choose... by the reflexive property of congruence. so, $\triangle kxz \cong$ choose... by the side - side - side (sss) postulate and $\angle jkz \cong \angle lkz$ since they are corresponding parts of congruent triangles.

Explanation:

Step 1: Analyze \( XZ \cong \) ?

In the construction, the compass width is kept the same when drawing arcs from \( X \) and \( Y \), so \( XZ \cong YZ \) because they are radii of arcs drawn with the same compass width (i.e., they are congruent as arcs drawn from equal radii).

Step 2: Analyze \( \overline{KZ} \cong \) ?

By the Reflexive Property of Congruence, a segment is congruent to itself. So \( \overline{KZ} \cong \overline{KZ} \).

Step 3: Analyze \( \triangle KXZ \cong \) ?

We know \( KX = KY \), \( XZ = YZ \), and \( KZ = KZ \). By SSS (Side - Side - Side) postulate, \( \triangle KXZ \cong \triangle KYZ \).

Answer:

• For \( XZ \cong \): \( YZ \) (since they are radii of arcs drawn with the same compass width, or congruent arcs' radii)
• For \( \overline{KZ} \cong \): \( \overline{KZ} \) (by Reflexive Property of Congruence)
• For \( \triangle KXZ \cong \): \( \triangle KYZ \) (by SSS postulate)

So filling in the blanks:

• First "Choose...": \( YZ \)
• Second "Choose...": radii of arcs drawn with the same compass width (or congruent arcs' radii, depending on the options, but the key congruence is \( XZ\cong YZ\))
• Third "Choose...": \( \overline{KZ} \)
• Fourth "Choose...": \( \triangle KYZ \)