Question

∠u and ∠s are complementary and ∠s and ∠t are complementary. complete the proof that ∠u ≅ ∠t.

statement	reason
2 ∠s and ∠t are complementary
3 ( mangle u + mangle s = 90^circ )
4 ( mangle s + mangle t = 90^circ )
5 ( mangle u + mangle s = mangle s + mangle t )
6 ( mangle u = mangle t )
7 ( angle u cong angle t )

Explanation:

Step1: Identify Given Information

The problem states that \(\angle U\) and \(\angle S\) are complementary, and \(\angle S\) and \(\angle T\) are complementary. This is the given information for the proof.

Step2: Define Complementary Angles

By the definition of complementary angles, if two angles are complementary, the sum of their measures is \(90^\circ\). So for \(\angle U\) and \(\angle S\) being complementary, we have \(m\angle U + m\angle S = 90^\circ\) (Reason: Definition of complementary angles). Similarly, for \(\angle S\) and \(\angle T\) being complementary, \(m\angle S + m\angle T = 90^\circ\) (Reason: Definition of complementary angles).

Step3: Transitive Property of Equality

Since \(m\angle U + m\angle S = 90^\circ\) and \(m\angle S + m\angle T = 90^\circ\), we can set them equal to each other: \(m\angle U + m\angle S = m\angle S + m\angle T\) (Reason: Transitive property of equality, as both equal \(90^\circ\)).

Step4: Subtraction Property of Equality

Subtract \(m\angle S\) from both sides of the equation \(m\angle U + m\angle S = m\angle S + m\angle T\). This gives \(m\angle U = m\angle T\) (Reason: Subtraction property of equality, subtracting the same quantity from both sides preserves equality).

Step5: Definition of Congruent Angles

If the measures of two angles are equal, then the angles are congruent. So since \(m\angle U = m\angle T\), we have \(\angle U \cong \angle T\) (Reason: Definition of congruent angles, angles with equal measures are congruent).

Now, filling in the reasons for each step:

1. \(\angle U\) and \(\angle S\) are complementary: Given
2. \(\angle S\) and \(\angle T\) are complementary: Given
3. \(m\angle U + m\angle S = 90^\circ\): Definition of complementary angles
4. \(m\angle S + m\angle T = 90^\circ\): Definition of complementary angles
5. \(m\angle U + m\angle S = m\angle S + m\angle T\): Transitive property of equality (or Substitution, since both equal \(90^\circ\))
6. \(m\angle U = m\angle T\): Subtraction property of equality (subtract \(m\angle S\) from both sides)
7. \(\angle U \cong \angle T\): Definition of congruent angles (angles with equal measures are congruent)

Answer:

1. Given
2. Given
3. Definition of complementary angles
4. Definition of complementary angles
5. Transitive property of equality (or Substitution)
6. Subtraction property of equality
7. Definition of congruent angles