Question

geometry midterm study guide
given: ∠1 and ∠2 are supplementary
∠1 ≅ ∠4
prove: ∠2 and ∠4 are supplementary
statements reasons

1. ∠1 and ∠2 are supplementary 1. ?
2. ∠1 ≅ ∠4 2. given
3. m∠1 = m∠4 3. ?
4. m∠1 + m∠2 = 180° 4. ?
5. ? 5. substitution property of equality
6. ∠2 and ∠4 are supplementary 6. ?

Explanation:

Step1: Reason for Statement 1

The first statement is given in the problem, so the reason is "Given".

Step2: Reason for Statement 3

If two angles are congruent ($\angle1\cong\angle4$), their measures are equal. This is the definition of congruent angles, so the reason is "Definition of Congruent Angles".

Step3: Reason for Statement 4

Supplementary angles are defined as two angles whose measures add up to $180^\circ$. Since $\angle1$ and $\angle2$ are supplementary (from Statement 1), this means $m\angle1 + m\angle2 = 180^\circ$. So the reason is "Definition of Supplementary Angles".

Step4: Statement for Step 5

Using the Substitution Property of Equality, we substitute $m\angle1$ with $m\angle4$ (from Statement 3) in the equation $m\angle1 + m\angle2 = 180^\circ$ (from Statement 4). So the statement is $m\angle4 + m\angle2 = 180^\circ$.

Step5: Reason for Statement 6

If the sum of the measures of two angles ($\angle2$ and $\angle4$) is $180^\circ$ (from Statement 5), then by the definition of supplementary angles, the two angles are supplementary. So the reason is "Definition of Supplementary Angles".

Answer:

1. Reason: Given
2. (Already given)
3. Reason: Definition of Congruent Angles
4. Reason: Definition of Supplementary Angles
5. Statement: $m\angle4 + m\angle2 = 180^\circ$
6. Reason: Definition of Supplementary Angles