Question

given: ∠1 and ∠2 are supplements, ∠3 and ∠4 are supplements, and ∠1 ≅ ∠4. prove: ∠2 ≅ ∠3. assemble the proof by dragging tiles to the statements and reasons columns. statements reasons m∠1 + m∠2 = 180 m∠3 + m∠4 = 180 ∠1 and ∠2 are supp. ∠3 and ∠4 are supp. ∠1 ≅ ∠4 m∠1 + m∠2 = m∠3 + m∠4

Explanation:

Step1: Recall the definition of supplementary angles

If two angles are supplementary, the sum of their measures is \(180^\circ\). So, since \(\angle1\) and \(\angle2\) are supplements, \(m\angle1 + m\angle2=180\) (Reason: Definition of supplementary angles). And since \(\angle3\) and \(\angle4\) are supplements, \(m\angle3 + m\angle4 = 180\) (Reason: Definition of supplementary angles).

Step2: Set the two equations equal

From Step 1, we have \(m\angle1 + m\angle2=180\) and \(m\angle3 + m\angle4 = 180\). By the Transitive Property of Equality (if \(a = c\) and \(b = c\), then \(a = b\)), we can set \(m\angle1 + m\angle2=m\angle3 + m\angle4\) (Reason: Transitive Property of Equality).

Step3: Use the congruent angles

We know that \(\angle1\cong\angle4\), which means \(m\angle1=m\angle4\) (Reason: Definition of congruent angles).

Step4: Substitute \(m\angle1\) with \(m\angle4\)

Substitute \(m\angle1\) in \(m\angle1 + m\angle2=m\angle3 + m\angle4\) with \(m\angle4\) (from Step 3). We get \(m\angle4 + m\angle2=m\angle3 + m\angle4\).

Step5: Subtract \(m\angle4\) from both sides

Subtract \(m\angle4\) from both sides of the equation \(m\angle4 + m\angle2=m\angle3 + m\angle4\). By the Subtraction Property of Equality (if \(a + b=a + c\), then \(b = c\)), we have \(m\angle2=m\angle3\) (Reason: Subtraction Property of Equality).

Step6: Conclude congruence

If the measures of two angles are equal, then the angles are congruent. So, since \(m\angle2=m\angle3\), we have \(\angle2\cong\angle3\) (Reason: Definition of congruent angles).

Answer:

To prove \(\angle2\cong\angle3\), we follow these steps:

Statements	Reasons
2. \(m\angle3 + m\angle4 = 180\)	2. \(\angle3\) and \(\angle4\) are supp. (Definition of supplementary angles)
3. \(m\angle1 + m\angle2 = m\angle3 + m\angle4\)	3. Transitive Property of Equality (from 1 and 2)
4. \(\angle1\cong\angle4\)	4. Given
5. \(m\angle1 = m\angle4\)	5. Definition of congruent angles (from 4)
6. \(m\angle4 + m\angle2 = m\angle3 + m\angle4\)	6. Substitution (replace \(m\angle1\) with \(m\angle4\) in 3)
7. \(m\angle2 = m\angle3\)	7. Subtraction Property of Equality (subtract \(m\angle4\) from both sides of 6)
8. \(\angle2\cong\angle3\)	8. Definition of congruent angles (from 7)