overline(W Y) perp overline(V X) and overline(W X) cong overline(V W) . Complete the proof that overline(X Y) cong overline(V Y) . & Statement & Reason 1 & overline(W Y) perp overline(V X) & Given 2 & overline(W X) simeq overline(V W) & Given 3 & angle W V X cong angle V X W & 4 & angle V Z W cong angle W Z X & All right angles are congruent 5 & Delta V W Z cong triangle X W Z & AAS 6 & angle V W Y cong angle X W Y & CPCTC 7 & overline(W Y) cong overline(W Y) & 8 & Delta V W Y cong triangle X W Y & SAS 9 & overline(X Y) cong overline(V Y) & CPCTC

Question

overline(W Y) perp overline(V X) and overline(W X) cong overline(V W) . Complete the proof that overline(X Y) cong overline(V Y) . & Statement & Reason 1 & overline(W Y) perp overline(V X) & Given 2 & overline(W X) simeq overline(V W) & Given 3 & angle W V X cong angle V X W & 4 & angle V Z W cong angle W Z X & All right angles are congruent 5 & Delta V W Z cong triangle X W Z & AAS 6 & angle V W Y cong angle X W Y & CPCTC 7 & overline(W Y) cong overline(W Y) & 8 & Delta V W Y cong triangle X W Y & SAS 9 & overline(X Y) cong overline(V Y) & CPCTC

Answer 1

#ExplanationThe proof provided is a geometric proof that uses a variety of postulates, theorems, and definitions to prove that the segments

and

are congruent. The proof begins with two given statements:

and

. In step 3, the proof needs to show that

. This can be done using the definition of perpendicular lines, which states that if two lines are perpendicular, then they form congruent adjacent angles. In step 4, the proof states that

because all right angles are congruent. This is a postulate that states that all right angles (angles measuring 90 degrees) are congruent to each other.In step 5, the proof shows that

using the Angle-Angle-Side (AAS) Congruence Postulate. This postulate states that if two angles and the non-included side of one triangle are congruent to two angles and the non-included side of another triangle, then the triangles are congruent.In step 6, the proof shows that

using the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) theorem. This theorem states that if two triangles are congruent, then all corresponding parts (sides and angles) are congruent.In step 7, the proof states that

. This is a reflexive property of congruence, which states that a quantity is congruent to itself.In step 8, the proof shows that

using the Side-Angle-Side (SAS) Congruence Postulate. This postulate states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.Finally, in step 9, the proof shows that

using the CPCTC theorem again.#AnswerThe missing reasons in the proof are as follows:For step 3, the reason is "Definition of Perpendicular lines".For step 7, the reason is "Reflexive Property of Congruence".

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