Question

1. given: $overline{zw} cong overline{wx}$

$overline{wy} perp overline{zx}$
prove: $angle z cong angle x$

possible statements:
(a) segment $zw \cong$ segment $wx$, segment $wy \perp$ segment $zx$,
(b) segment $wy \perp$ segment $zx$, (c) segment $wy \cong$ segment $wy$,
(d) segment $zw \cong$ segment $wx$

possible reasons:
(e) given, (f) sss, (g) sas, (h) asa, (i) hl, (j) cpctc, (k) prove

statement #1: ____

statement #2: ____

reason #3: ____

reason #4: ____

Explanation:

Step1: Determine Statement #1

The given information is $\overline{ZW} \cong \overline{WX}$ and $\overline{WY} \perp \overline{ZX}$. So Statement #1 should be the given statements, which is option (A): segment $ZW \cong$ segment $WX$, segment $WY \perp$ segment $ZX$.

Step2: Determine Statement #2

The reflexive property states that a segment is congruent to itself, so segment $WY \cong$ segment $WY$, which is option (C).

Step3: Determine Reason #3

We have $\overline{ZW} \cong \overline{WX}$ (given), $\overline{WY} \cong \overline{WY}$ (reflexive), and since $\overline{WY} \perp \overline{ZX}$, $\angle ZYW$ and $\angle XYW$ are right angles and thus congruent. So we can use the Hypotenuse - Leg (HL) theorem for right triangles. So Reason #3 is (I) HL.

Step4: Determine Reason #4

Once we have $\triangle ZWY \cong \triangle XWY$, the corresponding parts of congruent triangles are congruent (CPCTC), so $\angle Z \cong \angle X$. So Reason #4 is (J) CPCTC.

Answer:

Statement #1: A. segment \( ZW \cong \) segment \( WX \), segment \( WY \perp \) segment \( ZX \)
Statement #2: C. segment \( WY \cong \) segment \( WY \)
Reason #3: I. HL
Reason #4: J. CPCTC