Question

prove: wxyz is a rhombus.
statements\treasons

1. wxyz is a parallelogram.\t1. given
2. xz ⊥ wy\t2. given
3. ∠wpx and ∠xpy are 90° ∠s.\t3. def. of ⊥ lines
4. ∠wpx and ∠xpy are rt. ∠s.\t4.
5. δwpx and δypx are rt. δs.\t5. def. of rt. δs
6. xp ≅ xp\t6.
7. xz bisects wy\t7. diags. of a ▱ bisect each other.
8. wp ≅ py\t8. def. of bisector
9. \t9. ll
10. wx ≅ xy\t10. cpctc
11. \t11. ▱ with one pair of consecutive sides ≅ -- rhombus

a. δwpx ≅ δypx
b. δwpx ≅ δypz
c. reflexive property of ≅
d. transitive property of ≅
e. def. right angle
f. wxyz is a rhombus
g. def. parallelogram

Explanation:

Step-by-Step Explanation for Each Blank:

Blank 4:

• Step 1: Analyze the statement and reason

Statement 3 says \( \angle WPX \) and \( \angle XPY \) are \( 90^\circ \) angles. Statement 4 rephrases them as right angles. The reason should define a right angle (since a \( 90^\circ \) angle is a right angle).

• Match with option: e. Def. right angle

Blank 6:

• Step 1: Analyze the statement and reason

Statement 6 is \( \overline{XP} \cong \overline{XP} \), which is a side congruent to itself. This is the Reflexive Property of Congruence.

• Match with option: c. Reflexive Property of \( \cong \)

Blank 9:

• Step 1: Analyze the statement and reason

Reason 9 is "LL" (Leg-Leg Congruence for right triangles). We have two right triangles \( \triangle WPX \) and \( \triangle YPX \):

• \( \overline{WP} \cong \overline{PY} \) (from step 8),
• \( \overline{XP} \cong \overline{XP} \) (from step 6),
• Both are right triangles (from step 5).

By LL, \( \triangle WPX \cong \triangle YPX \).

• Match with option: a. \( \triangle WPX \cong \triangle YPX \)

Blank 11:

• Step 1: Analyze the statement and reason

Reason 11 is " \( \parallel\text{ogram} \) with one pair of consecutive sides \( \cong \) → rhombus". We already proved \( \overline{WX} \cong \overline{XY} \) (step 10), and \( WXYZ \) is a parallelogram (step 1). So the conclusion is that \( WXYZ \) is a rhombus.

• Match with option: f. \( WXYZ \) is a rhombus

Final Answers:

• Blank 4: e. Def. right angle
• Blank 6: c. Reflexive Property of \( \cong \)
• Blank 9: a. \( \triangle WPX \cong \triangle YPX \)
• Blank 11: f. \( WXYZ \) is a rhombus

(If the question asks for each blank separately, list them as above. For example, if blank 4: e; blank 6: c; blank 9: a; blank 11: f.)

Answer:

Step-by-Step Explanation for Each Blank:

Blank 4:

• Step 1: Analyze the statement and reason

Statement 3 says \( \angle WPX \) and \( \angle XPY \) are \( 90^\circ \) angles. Statement 4 rephrases them as right angles. The reason should define a right angle (since a \( 90^\circ \) angle is a right angle).

• Match with option: e. Def. right angle

Blank 6:

• Step 1: Analyze the statement and reason

Statement 6 is \( \overline{XP} \cong \overline{XP} \), which is a side congruent to itself. This is the Reflexive Property of Congruence.

• Match with option: c. Reflexive Property of \( \cong \)

Blank 9:

• Step 1: Analyze the statement and reason

Reason 9 is "LL" (Leg-Leg Congruence for right triangles). We have two right triangles \( \triangle WPX \) and \( \triangle YPX \):

• \( \overline{WP} \cong \overline{PY} \) (from step 8),
• \( \overline{XP} \cong \overline{XP} \) (from step 6),
• Both are right triangles (from step 5).

By LL, \( \triangle WPX \cong \triangle YPX \).

• Match with option: a. \( \triangle WPX \cong \triangle YPX \)

Blank 11:

• Step 1: Analyze the statement and reason

Reason 11 is " \( \parallel\text{ogram} \) with one pair of consecutive sides \( \cong \) → rhombus". We already proved \( \overline{WX} \cong \overline{XY} \) (step 10), and \( WXYZ \) is a parallelogram (step 1). So the conclusion is that \( WXYZ \) is a rhombus.

• Match with option: f. \( WXYZ \) is a rhombus

Final Answers:

• Blank 4: e. Def. right angle
• Blank 6: c. Reflexive Property of \( \cong \)
• Blank 9: a. \( \triangle WPX \cong \triangle YPX \)
• Blank 11: f. \( WXYZ \) is a rhombus

(If the question asks for each blank separately, list them as above. For example, if blank 4: e; blank 6: c; blank 9: a; blank 11: f.)