Question

complete the proof that (angle sutcongangle qur).

1 (mangle qur + mangle rus = 180^{circ})
2 (mangle rus + mangle sut = 180^{circ}) angles forming a linear pair sum to (180^{circ})
3 (mangle qur + mangle rus = mangle rus + mangle sut)
4 (mangle qur = mangle sut) properties of addition, subtraction, multiplication, and division
5 (angle sutcongangle qur)

Explanation:

Step1: Reason for statement 1

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

Step2: Reason for statement 3

Substitution property of equality. Because \(m\angle QUR + m\angle RUS = 180^{\circ}\) (from step 1) and \(m\angle RUS + m\angle SUT=180^{\circ}\) (from step 2), we can substitute \(180^{\circ}\) in the first equation with \(m\angle RUS + m\angle SUT\) from the second equation. So \(m\angle QUR + m\angle RUS=m\angle RUS + m\angle SUT\).

Step3: Reason for statement 5

Definition of congruent angles. If \(m\angle QUR = m\angle SUT\) (from step 4), then by the definition that two angles are congruent if their measures are equal, \(\angle SUT\cong\angle QUR\).

Answer:

1. Angles forming a linear pair sum to \(180^{\circ}\)
2. Substitution property of equality
3. Definition of congruent angles