Given: ∠1 and ∠2 are supplementary, and ∠2 and ∠3 are supplementary. Prove: ∠1 ≅ ∠3 By definition of supplementary angles, m∠1 + m∠2 = _____ (a) and m∠2 + m∠3 = _____ (b). Then, m∠1 + m∠2 = m∠2 + m∠3 by the __________ (c). Subtract m∠2 from each side. You get m∠1 = ______ (d), or ∠1 ≅ _____ (e).
By definition of supplementary angles, m∠1 + m∠2 = 180° (a) and m∠2 + m∠3 = 180° (b). Then, m∠1 + m∠2 = m∠2 + m∠3 by the Transitive Property of Equality (c). Subtract m∠2 from each side. You get m∠1 = m∠3 (d), or ∠1 ≅ ∠3 (e).
By definition of supplementary angles, m∠1 + m∠2 = 180° (a) and m∠2 + m∠3 = 180° (b).
Then, since ∠2 is common to both equations, we can equate the two equations:
m∠1 + m∠2 = m∠2 + m∠3 by the Transitive Property of Equality (c).
Subtracting m∠2 from each side, we get:
m∠1 = m∠3 (d)
From (d), we can conclude that ∠1 and ∠3 have equal measures.
Therefore, ∠1 ≅ ∠3 (e) by the definition of congruent angles.
To prove that ∠1 ≅ ∠3, we need to show that the measures of ∠1 and ∠3 are equal.
First, let's use the given information to set up equations for the measures of the angles. We are told that ∠1 and ∠2 are supplementary, which means that the sum of their measures is 180 degrees. So, we can write this as:
m∠1 + m∠2 = 180 (a)
Similarly, we are also given that ∠2 and ∠3 are supplementary, so their measures also add up to 180 degrees:
m∠2 + m∠3 = 180 (b)
Now, we can see that both equations have a common term, m∠2. To prove that ∠1 ≅ ∠3, we need to show that their measures are equal. So, let's rearrange equation (b) by subtracting m∠2 from each side:
m∠2 + m∠3 - m∠2 = 180 - m∠2
This simplifies to:
m∠3 = 180 - m∠2
Now, let's substitute this value for m∠3 in equation (a):
m∠1 + m∠2 = m∠3 + m∠2
Since we know that m∠3 = 180 - m∠2, we can substitute this in the equation:
m∠1 + m∠2 = (180 - m∠2) + m∠2
By grouping like terms, we can simplify this to:
m∠1 = 180
From this equation, we can conclude that m∠1 is equal to 180 degrees. Now, recall that we are trying to prove that ∠1 ≅ ∠3, which means that the measures of both angles should be equal. Since m∠1 = 180 degrees, we can conclude that ∠1 is a straight angle, which is equal to 180 degrees.
So, we have shown that ∠1 ≅ ∠3, as both angles have a measure of 180 degrees.
By definition of supplementary angles, m∠1 + m∠2 = 180° (a) and m∠2 + m∠3 = 180° (b). Then, m∠1 + m∠2 = m∠2 + m∠3 by the Transitive Property of Equality (c). Subtract m∠2 from each side. You get m∠1 = m∠3 (d), or ∠1 ≅ ∠3 (e).
Your original answer is correct. Here's the proof written out more formally:
Given: ∠1 and ∠2 are supplementary, and ∠2 and ∠3 are supplementary.
We are trying to prove: ∠1 ≅ ∠3
Proof:
By the definition of supplementary angles, we have:
m∠1 + m∠2 = 180° (a)
m∠2 + m∠3 = 180° (b)
Since ∠2 is common to both equations, we can equate them:
m∠1 + m∠2 = m∠2 + m∠3
Using the Transitive Property of Equality, we can simplify the equation to:
m∠1 = m∠3 (c)
This means that the measures of ∠1 and ∠3 are equal. By the definition of congruent angles, we can conclude that ∠1 ≅ ∠3.
What is the value of x? Justify each step. AC = 32
Without further information or context, it is not possible to determine the value of x or justify any steps. The given information of AC = 32 does not provide any information about x or the shape or diagram in which it is used. Please provide additional relevant information or context to solve for x.
Write a two-column proof. Given: 7y = 8x – 14; y = 6 Prove: x = 7
Statements Reasons
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1. 7y = 8x - 14 Given
2. y = 6 Given
3. 7(6) = 8x - 14 Substitution (Substitute y with 6 in statement 1)
4. 42 = 8x - 14 Simplification
5. 42 + 14 = 8x Addition (Add 14 to both sides of statement 4)
6. 56 = 8x Simplification
7. 56/8 = x Division (Divide both sides of statement 6 by 8)
8. 7 = x Simplification