Question

complete the proof that ∠uyv ≅ ∠wyx. diagram with points w, v, y, x, u
statement | reason
--- | ---

1. ( mangle uyv + mangle vyw = 180^circ ) | angles forming a linear pair sum to ( 180^circ )
2. ( mangle vyw + mangle wyx = 180^circ ) | angles forming a linear pair sum to ( 180^circ )
3. ( mangle uyv + mangle vyw = mangle vyw + mangle wyx ) | transitive property of equality
4. ( mangle uyv = mangle wyx ) | blank
5. ( angle uyv cong angle wyx ) | dropdown options: properties of addition, subtraction, multiplication, and division; reflexive property of congruence; reflexive property of equality; substitution; transitive property of congruence; ...

Explanation:

Step1: Recall Angle Congruence Definition

If two angles have equal measures, then they are congruent. So if \( m\angle UYW = m\angle WYX \), then \( \angle UYW \cong \angle WYX \).

Step2: Identify the Reason

The step before (step 4) shows \( m\angle UYW = m\angle WYX \), and step 5 is the congruence. The reason that if two angles have equal measures, they are congruent is related to the definition of congruent angles, but from the options, the "Definition of Congruent Angles" (or in the given options, the property that equal measures imply congruence, which is part of the properties leading to congruence. Wait, looking at the options, the correct reason here is that if two angles have equal measures, they are congruent. The option that fits is based on the fact that \( m\angle A = m\angle B \implies \angle A \cong \angle B \), which is the definition (or the property) that equal measures imply congruence. Among the given options, the relevant one is the idea that if two angles have the same measure, they are congruent. So the reason for step 5 is that "If two angles have equal measures, then they are congruent" (which is the definition of congruent angles). From the dropdown, the correct choice is related to the property that equal measures lead to congruence. So the reason is that angles with equal measures are congruent, which is the definition of congruent angles. So the reason for step 5 is "Definition of Congruent Angles" (or in the given options, the appropriate property. Looking at the options, the correct one is that if \( m\angle UYW = m\angle WYX \), then \( \angle UYW \cong \angle WYX \) by the definition of congruent angles (angles with equal measures are congruent). So the reason is the property that equal angle measures imply congruence.

Answer:

The reason for step 5 is that if two angles have equal measures, they are congruent (i.e., the Definition of Congruent Angles). From the given options, the correct choice is the one that states angles with equal measures are congruent (usually phrased as "If \( m\angle A = m\angle B \), then \( \angle A \cong \angle B \)"). So the reason is the definition of congruent angles, which would correspond to the option that uses the fact that equal measures imply congruence.