Question

complete the proof that ∠uyv≅∠wyx.
statement reason
1 m∠uyv + m∠vyw = 180° angles forming a linear pair sum to 180°
2 m∠vyw + m∠wyx = 180° angles forming a linear pair sum to 180°
3 m∠uyv + m∠vyw = m∠vyw + m∠wyx transitive property of equality
4 m∠uyv = m∠wyx properties of addition, subtraction, multiplication, and division
5 ∠uyv≅∠wyx

Explanation:

Step1: Recall linear - pair angle property

We know from statements 1 and 2 that $\angle UYV$ and $\angle VYW$ form a linear pair, and $\angle VYW$ and $\angle WYX$ form a linear pair, so $m\angle UYV + m\angle VYW=180^{\circ}$ and $m\angle VYW + m\angle WYX = 180^{\circ}$.

Step2: Apply transitive property

By the transitive property of equality, since $m\angle UYV + m\angle VYW=180^{\circ}$ and $m\angle VYW + m\angle WYX = 180^{\circ}$, we have $m\angle UYV + m\angle VYW=m\angle VYW + m\angle WYX$.

Step3: Use subtraction property of equality

Subtract $m\angle VYW$ from both sides of the equation $m\angle UYV + m\angle VYW=m\angle VYW + m\angle WYX$. We get $m\angle UYV=m\angle WYX$.

Step4: Apply congruence definition

Two angles are congruent if and only if they have the same measure. Since $m\angle UYV=m\angle WYX$, by the definition of congruence, $\angle UYV\cong\angle WYX$.

Answer:

Definition of congruence