Answer:
By definition of supplementary angles, m∠1 + m∠2 = 180° and m∠2 + m∠3 = 180°. Then, m∠1 + m∠2 = m∠2 + m∠3 by the addition property of equality. Subtract m∠2 from each side. You get m∠1 = m∠3 or ∠1 ≅ ∠3.
Step-by-step explanation:
Let's first understand what supplementary angles mean.
Supplementary angles are an angle pair that add up to 180 degrees.
So, if we know that the measure of angle one and the measure of angle 2 are supplementary, we know that their angles sum up to 180 degrees. Same thing with measures of angles 2 and 3 which are supplementary.
This falls into the reason of "Given" which supports the statement of m∠1 and m∠2 are supplementary and m∠2 and m∠3 are also supplementary.
So if we are given those angle pairs are supplementary, their angle measures add up to 180 degrees.
Now, when they say:
"Then, m∠1 + m∠2 = m∠2 + m∠3 by the ______"
They are asking for which definition, property, postulate, or theorem this is proven by (the reason).
In geometry, there are so many proof reasons that fall into each of those categories.
When we focus on the additional property of equality, they mean:
If a = b, then a + c = b + c
In this case, angle measures 1 and 3 are congruent, or equal to each other, which is also what we're trying to prove in this proof. So, we can use m∠1 and m∠3 as "a" and "b".
So if we use m∠1 and m∠3 for a + c = b + c, we see that the "c" value is added to both angle measures (which have the same value, previously said). In this case, the "c" value represents m∠2. So, when we add measure angle 2 to both those angles, we know that the value on both sides are the same since angle measures 1 and 3 have the same value and anything added to both sides(which have to be the same) of the equation will remain the same value on both sides.
Therefore, we can say that :
Then, m∠1 + m∠2 = m∠2 + m∠3 by the addition property of equality.