Question

a partial proof is given below. use the information given to complete questions 45-47.
given: $overline{wx} \cong \overline{yz}$, $overline{xy} \cong \overline{wz}$
prove: $\delta wxz \cong \delta yzx$

statements	reasons
2. $\angle wxz \cong \angle yzx$; $\angle wzx \cong \angle xyz$	2.
3. $overline{xz} \cong \overline{zx}$	3.
4. $\delta wxz \cong \delta yzx$	4.

1. which reason justifies statement 2?

a. vertical angles
b. consecutive interior angles
c. corresponding angles
d. alternate interior angles

1. which reason justifies statement 3?

a. symmetric property
b. reflexive property
c. transitive property
d. definition of congruence

1. which reason justifies statement 4?

a. side-side-side
b. side-angle-side
c. angle-side-angle
d. angle-angle-side

Explanation:

Question 45

Step1: Recall Angle Theorems

We have parallel lines \( \overline{WX} \parallel \overline{YZ} \) and \( \overline{XY} \parallel \overline{WZ} \), so the figure is a parallelogram. For \( \angle WXZ \cong \angle YZX \) and \( \angle WZX \cong \angle XYZ \), we look at alternate interior angles. When a transversal (here \( XZ \)) crosses parallel lines (\( WX \parallel YZ \) and \( XY \parallel WZ \)), alternate interior angles are congruent. But wait, for \( \angle WXZ \) and \( \angle YZX \), the transversal is \( XZ \) and lines \( WX \parallel YZ \), so alternate interior angles? Wait, no, wait the angles in statement 2: \( \angle WXZ \cong \angle YZX \) and \( \angle WZX \cong \angle XYZ \). Wait, actually, when \( \overline{WX} \parallel \overline{YZ} \) and transversal \( XZ \), then \( \angle WXZ \) and \( \angle YZX \) are alternate interior angles. Similarly, for \( \overline{XY} \parallel \overline{WZ} \) and transversal \( XZ \), \( \angle WZX \) and \( \angle XYZ \) are alternate interior angles? Wait, no, maybe I misread. Wait the options: A is Vertical Angles, B is Consecutive Interior Angles, C is Corresponding Angles, D is Alternate Interior Angles. Alternate Interior Angles are congruent when lines are parallel. So since \( WX \parallel YZ \), transversal \( XZ \), so \( \angle WXZ \) (interior, alternate to \( \angle YZX \)) are congruent. So the reason is Alternate Interior Angles? Wait no, wait the options: D is Alternate Interior Angles. Wait let's check the options again. Option D: Alternate Interior Angles. So the correct reason for statement 2 (the angles being congruent) is Alternate Interior Angles because the lines are parallel, so alternate interior angles are congruent.

Step2: Match with Options

So the reason for \( \angle WXZ \cong \angle YZX \) and \( \angle WZX \cong \angle XYZ \) is that they are alternate interior angles formed by parallel lines cut by a transversal. So the correct option is D.

Step1: Recall Congruence Properties

Statement 3 is \( \overline{XZ} \cong \overline{ZX} \). This is the reflexive property, which states that any segment is congruent to itself. The reflexive property of congruence (for segments, \( \overline{AB} \cong \overline{AB} \)). So \( \overline{XZ} \cong \overline{ZX} \) is by reflexive property.

Step2: Match with Options

Option B is Reflexive Property, which matches because \( \overline{XZ} \) is congruent to itself, so reflexive property.

Step1: Recall Triangle Congruence Theorems

We have two angles and a side? Wait, no: from statement 1: \( \overline{WX} \cong \overline{YZ} \), \( \overline{XY} \cong \overline{WZ} \) (given, but wait no, statement 1 is \( \overline{WX} \cong \overline{YZ} \), \( \overline{XY} \cong \overline{WZ} \) (given). Statement 2: angles congruent (alternate interior). Statement 3: \( \overline{XZ} \cong \overline{ZX} \) (reflexive). So we have two angles and the included side? Wait, no: \( \angle WXZ \cong \angle YZX \), \( \overline{XZ} \cong \overline{ZX} \), \( \angle WZX \cong \angle XYZ \). Wait, that's Angle-Side-Angle (ASA)? Wait no, wait the triangles are \( \triangle WXZ \) and \( \triangle YZX \). Let's list the congruences: \( \angle WXZ \cong \angle YZX \) (from statement 2), \( \overline{XZ} \cong \overline{ZX} \) (statement 3), \( \angle WZX \cong \angle XYZ \) (statement 2). Wait, that's two angles and the included side (ASA). But wait the options: A is SSS, B is SAS, C is ASA, D is AAS. Wait, no, wait the order: in \( \triangle WXZ \) and \( \triangle YZX \), we have \( \angle WXZ \cong \angle YZX \), \( \overline{XZ} \cong \overline{ZX} \), \( \angle WZX \cong \angle XYZ \). So that's ASA (Angle-Side-Angle), but wait the options: C is Angle-Side-Angle? Wait the options are A. Side-Side-Side, B. Side-Angle-Side, C. Angle-Side-Angle, D. Angle-Angle-Side. Wait, but in our case, we have two angles and the included side (the side between the two angles is \( XZ \)), so that's ASA. But wait, let's check the options again. Wait the problem says "Which reason justifies statement 4: \( \triangle WXZ \cong \triangle YZX \)". So with \( \angle WXZ \cong \angle YZX \), \( \overline{XZ} \cong \overline{ZX} \), \( \angle WZX \cong \angle XYZ \), that's ASA. But wait the options: C is Angle-Side-Angle. Wait, but maybe I made a mistake. Wait, no, wait the sides: \( \overline{WX} \cong \overline{YZ} \) (given), \( \overline{XY} \cong \overline{WZ} \) (given), and \( \overline{XZ} \cong \overline{ZX} \). Wait, that's SSS? But no, statement 2 is angles. Wait, no, the given is \( \overline{WX} \cong \overline{YZ} \), \( \overline{XY} \cong \overline{WZ} \) (given), then angles congruent (alternate interior), then \( \overline{XZ} \cong \overline{ZX} \). Wait, maybe it's SSS? No, because we have angles. Wait, no, the correct congruence criterion here is ASA? Wait, no, let's re-express:

In \( \triangle WXZ \) and \( \triangle YZX \):

• \( \angle WXZ \cong \angle YZX \) (statement 2)
• \( \overline{XZ} \cong \overline{ZX} \) (statement 3)
• \( \angle WZX \cong \angle XYZ \) (statement 2)

So that's two angles and the included side (ASA), which is option C: Angle-Side-Angle. Wait, but the options: C is Angle-Side-Angle. So the reason for \( \triangle WXZ \cong \triangle YZX \) is ASA.

Step2: Match with Options

So the congruence criterion is Angle-Side-Angle (ASA), which is option C.

Answer:

D. Alternate Interior Angles

Question 46