Question

What is the missing reason in the proof? (Reason 9)

Answer 1

transitive property

Explanation:

Generally proofs flow naturally from one step to the next. Consequently, each step usually follows from the previous step, and the Reason given justifies why that change is valid.

From statement 8, note that
`90^(o)=m\angle{ABD}`

For statement 9, the proof asserts that
`m\angle{ABC}=m\angle{ABD}`

What changed between line 8 and 9? The left side of the equation lost it's 90 degrees, and it became the measure of angle ABC.

So, what justifies changing the 90degrees into a measure of an angle (specifically, the measure of angle ABC)?

Recall from statement 3, that
`m\angle{ABC}=90^(o)`.

The "substitution property" would be a valid reason for step 9, (but of the given options, that isn't one, so we must look for another valid reason).

Recall that the transitive property states:

`\text{If }a=b\text{ and }b=c, \text{then }a=c`

There are three parts,

1. the first part needs a=b

2. the second part needs b=c

3. the third part produces a=c

Notice that in the second part, b=c, and then for the third part, on the left side, the "b" disappears, and the "a" sort of appears out of nowhere.

Statement 8 is like the second part, and statement 9 is like the third part.

Replacing "a" with
`m\angle{ABC}`, replacing "b" with
`90^(o)`, and replacing "c" with
`m\angle{ABD}`, we can update the transitive property and see how it applies to our situation:

Original transitive property:
`\text{If }a=b\text{ and }b=c, \text{then }a=c`

Updated transitive property:
`\text{If }m\angle{ABC}=90^(o)\text{ and }90^(o)=m\angle{ABD}, \text{then }m\angle{ABC}=m\angle{ABD}`

In order to use the transitive property, we need the first part and the second part to be true, and then it will be a valid reason for the last part to be true. The first part was already proven back in statement 3, the second part was just proven in statement 8, so the conclusion (the third part) is valid and can be statement 9, because of the transitive property.