Question

Using Common Parts Write a plan for a proof that does not use overlapping triangles. Given: angle ZXW congruent to angle YWX, angle ZWX congruent to angle YXW Prove: ZW congruent to YX Plan: Label point M: where ZX intersects WY. Observation: MW is congruent to MX due to the Converse of the Isosceles Triangle Theorem. Note: angle ZMW is congruent to angle YMX because vertical angles are congruent. Apply the Angle Addition Postulate: angle ZWM is congruent to angle YXM. Triangle Congruence: Conclude that triangle ZWM is congruent to triangle YXM by Angle-Angle-Side (AAS). Final Conclusion: State that ZW is congruent to YX by the Corresponding Parts of Congruent Triangles are Congruent (CPCTC). Choose the correct answers for the blanks: Blank A: angles W and X are congruent, as vertical angles are always congruent. Blank B: vertical angles are congruent. Blank C: AAS (Angle-Angle-Side) is the congruence criterion. Blank D: CPCTC (Corresponding Parts of Congruent Triangles are Congruent) is the justification for ZW congruent to YX.

Answer 1

ZW is congruent to YX, justified by CPCTC.

Step-by-step explanation:

In the given proof, we start by establishing that angle ZWX is congruent to angle YXW and angle ZXW is congruent to angle YWX. We introduce point M, the intersection of ZX and WY. By applying the Converse of the Isosceles Triangle Theorem, we deduce that MW is congruent to MX.

Additionally, vertical angles ZMW and YMX are congruent. Using the Angle Addition Postulate, we then conclude that angle ZWM is congruent to angle YXM. Applying the Angle-Angle-Side (AAS) criterion, we establish the congruence of triangles ZWM and YXM.

Now, the Corresponding Parts of Congruent Triangles are Congruent (CPCTC) comes into play. CPCTC states that if two triangles are congruent, then their corresponding parts (sides and angles) are also congruent.

Therefore, we can conclude that ZW is congruent to YX. This is our final answer, supported by the logical sequence of statements and the application of geometric principles. The proof avoids overlapping triangles and relies on well-established theorems, ensuring a sound and valid argument.

In summary, the proof navigates through the given angles, introduces the point of intersection, establishes congruence, and ultimately employs CPCTC to assert the congruence of ZW and YX. The logical flow and adherence to geometric principles reinforce the validity of the conclusion.