complete the paragraph proof given: <1 and <2 are supplementary and <2 and <3 are supplementary Prove: <1 is congruent to <3
Given: ∠1 and ∠2 are supplementary (meaning ∠1 + ∠2 = 180 degrees) and ∠2 and ∠3 are supplementary (meaning ∠2 + ∠3 = 180 degrees).
To Prove: ∠1 is congruent to ∠3.
Proof:
Since ∠1 and ∠2 are supplementary, we can write the equation ∠1 + ∠2 = 180 degrees. (1)
Similarly, since ∠2 and ∠3 are supplementary, we can write the equation ∠2 + ∠3 = 180 degrees. (2)
We want to prove ∠1 is congruent to ∠3, which means that ∠1 and ∠3 have the same measure.
We can rewrite equation (1) as ∠2 = 180 - ∠1. (3)
Plugging equation (3) into equation (2), we have (180 - ∠1) + ∠3 = 180.
Simplifying, we get 180 - ∠1 + ∠3 = 180.
Rearranging, we have ∠3 - ∠1 = 0.
Adding ∠1 to both sides, we have ∠3 = ∠1.
Therefore, we have proven that ∠1 is congruent to ∠3.
To prove that <1 is congruent to <3 given that <1 and <2 are supplementary, and <2 and <3 are supplementary, we can use the transitive property of congruence.
Given:
<1 and <2 are supplementary (1 + 2 = 180 degrees)
<2 and <3 are supplementary (2 + 3 = 180 degrees)
To prove:
<1 is congruent to <3 (1 ≅ 3)
Proof:
1. <1 and <2 are supplementary (Given)
2. <2 and <3 are supplementary (Given)
3. 1 + 2 = 180 degrees (Definition of supplementary angles)
4. 2 + 3 = 180 degrees (Definition of supplementary angles)
5. 1 + 2 = 2 + 3 (Transitive property of equality)
6. 1 = 3 (Subtracting 2 from both sides)
7. <1 is congruent to <3 (Definition of congruent angles)
To prove that <1 is congruent to <3, given that <1 and <2 are supplementary and <2 and <3 are supplementary, we can use the transitive property of congruence.
Here is a complete paragraph proof:
Statement 1: <1 and <2 are supplementary. (Given)
Statement 2: <2 and <3 are supplementary. (Given)
Statement 3: m<1 + m<2 = 180°. (Definition of supplementary angles)
Statement 4: m<2 + m<3 = 180°. (Definition of supplementary angles)
Statement 5: m<1 + m<2 = m<2 + m<3. (Transitive property of equality, combining statements 3 and 4)
Statement 6: m<1 = m<3. (Subtracting m<2 from both sides of statement 5)
Statement 7: <1 is congruent to <3. (Definition of congruent angles)
Therefore, we have proven that <1 is congruent to <3, using the given information that <1 and <2 are supplementary and <2 and <3 are supplementary.