Question

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

Explanation:

Step1: Identify linear - pair angles

$\angle QUR$ and $\angle RUS$ form a linear pair, so $m\angle QUR + m\angle RUS=180^{\circ}$ (Angles forming a linear pair sum to $180^{\circ}$). Also, $\angle RUS$ and $\angle SUT$ form a linear pair, so $m\angle RUS + m\angle SUT = 180^{\circ}$ (Angles forming a linear pair sum to $180^{\circ}$).

Step2: Apply transitive property

Since $m\angle QUR + m\angle RUS=180^{\circ}$ and $m\angle RUS + m\angle SUT = 180^{\circ}$, by the transitive property of equality, $m\angle QUR + m\angle RUS=m\angle RUS + m\angle SUT$.

Step3: Use subtraction property

Subtract $m\angle RUS$ from both sides of the equation $m\angle QUR + m\angle RUS=m\angle RUS + m\angle SUT$. We get $m\angle QUR=m\angle SUT$ (Properties of addition, subtraction, multiplication, and division).

Step4: Convert to congruence

If two angles have equal measures, they are congruent. So, $\angle SUT\cong\angle QUR$.

Answer:

The proof is completed as shown above, demonstrating that $\angle SUT\cong\angle QUR$.