Question

∠t and ∠r are supplementary and ∠r and ∠s are supplementary. complete the proof that ∠s ≅ ∠t.

	statement	reason
1	∠t and ∠r are supplementary	given
2	∠r and ∠s are supplementary	given
3	( mangle t + mangle r = 180^circ )	definition of supplementary angles
4	( mangle r + mangle s = 180^circ )
5	( mangle t + mangle r = mangle r + mangle s )
6	( mangle s = mangle t )
7	( angle s cong angle t )	definition of congruence

Explanation:

Step1: Reason for Statement 4

By the definition of supplementary angles (if two angles are supplementary, the sum of their measures is \(180^\circ\)), since \(\angle R\) and \(\angle S\) are supplementary (given in Statement 2), we have \(m\angle R + m\angle S = 180^\circ\). So the reason is "Definition of supplementary angles".

Step2: Reason for Statement 5

From Statement 3, we know \(m\angle T + m\angle R = 180^\circ\), and from Statement 4, \(m\angle R + m\angle S = 180^\circ\). By the Transitive Property of Equality (if \(a = c\) and \(b = c\), then \(a = b\)), we can set \(m\angle T + m\angle R = m\angle R + m\angle S\). So the reason is "Transitive Property of Equality" (or "Substitution Property" since both equal \(180^\circ\)).

Step3: Reason for Statement 6

Starting with \(m\angle T + m\angle R = m\angle R + m\angle S\) (Statement 5), we can subtract \(m\angle R\) from both sides of the equation. By the Subtraction Property of Equality (if \(a + c = b + c\), then \(a = b\)), we get \(m\angle S = m\angle T\). So the reason is "Subtraction Property of Equality".

Filling the Table:

• Statement 4 Reason: Definition of supplementary angles
• Statement 5 Reason: Transitive Property of Equality (or Substitution Property)
• Statement 6 Reason: Subtraction Property of Equality

(If we were to present just the answers for each blank in order as per the table's empty reason cells):

For row 4 (Statement 4) reason: \(\boldsymbol{\text{Definition of supplementary angles}}\)
For row 5 (Statement 5) reason: \(\boldsymbol{\text{Transitive Property of Equality (or Substitution Property)}}\)
For row 6 (Statement 6) reason: \(\boldsymbol{\text{Subtraction Property of Equality}}\)

Answer:

Step1: Reason for Statement 4

By the definition of supplementary angles (if two angles are supplementary, the sum of their measures is \(180^\circ\)), since \(\angle R\) and \(\angle S\) are supplementary (given in Statement 2), we have \(m\angle R + m\angle S = 180^\circ\). So the reason is "Definition of supplementary angles".

Step2: Reason for Statement 5

From Statement 3, we know \(m\angle T + m\angle R = 180^\circ\), and from Statement 4, \(m\angle R + m\angle S = 180^\circ\). By the Transitive Property of Equality (if \(a = c\) and \(b = c\), then \(a = b\)), we can set \(m\angle T + m\angle R = m\angle R + m\angle S\). So the reason is "Transitive Property of Equality" (or "Substitution Property" since both equal \(180^\circ\)).

Step3: Reason for Statement 6

Starting with \(m\angle T + m\angle R = m\angle R + m\angle S\) (Statement 5), we can subtract \(m\angle R\) from both sides of the equation. By the Subtraction Property of Equality (if \(a + c = b + c\), then \(a = b\)), we get \(m\angle S = m\angle T\). So the reason is "Subtraction Property of Equality".

Filling the Table:

• Statement 4 Reason: Definition of supplementary angles
• Statement 5 Reason: Transitive Property of Equality (or Substitution Property)
• Statement 6 Reason: Subtraction Property of Equality

(If we were to present just the answers for each blank in order as per the table's empty reason cells):

For row 4 (Statement 4) reason: \(\boldsymbol{\text{Definition of supplementary angles}}\)
For row 5 (Statement 5) reason: \(\boldsymbol{\text{Transitive Property of Equality (or Substitution Property)}}\)
For row 6 (Statement 6) reason: \(\boldsymbol{\text{Subtraction Property of Equality}}\)