Question

statements

1. δabc with ac > ab; \\(\overline{bd}\\) is drawn so that \\(\overline{ad} \cong \overline{ab}\\)
2. \\(\angle 1 \cong \angle 3\\)
3. \\(m\angle 1 = m\angle 3\\)
4. \\(m\angle abc = m\angle 1 + m\angle 2\\)
5. \\(m\angle abc > m\angle 1\\)
6. \\(m\angle abc > m\angle 3\\)
7. \\(m\angle 3 = m\angle 2 + m\angle c\\)
8. \\(m\angle 3 > m\angle c\\)
9. \\(m\angle abc > m\angle c\\)

reasons

1. given
2. isosceles δ thrm.

3.

1. post.
2. comp. prop. of \\(\

eq\\)
6.

1. ext. \\(\angle\\) thrm.
2. prop. of \\(\

eq\\);

1. prop. of \\(\

eq\\);

a. int. \\(\angle\\) thrm.
b. def. \\(\cong \angle\\)s
c. substitution
d. comp. prop. of \\(\
eq\\)
e. scalene δ thrm.
f. trans. prop. of \\(\
eq\\)
g. \\(\angle\\) addition postulate

Explanation:

Step 1: Analyze Reason 3

We know that if two angles are congruent ($\angle 1 \cong \angle 3$), then their measures are equal. This is the definition of congruent angles. So the reason for $m\angle 1 = m\angle 3$ is "Def. $\cong \angle$s" (option b).

Step 2: Analyze Reason 4

The statement $m\angle ABC = m\angle 1 + m\angle 2$ is about the sum of two angles forming a larger angle, which is the Angle Addition Postulate. So the reason here is " $\angle$ addition Postulate" (option g).

Step 3: Analyze Reason 6

We have $m\angle ABC > m\angle 1$ (from step 5) and $m\angle 1 = m\angle 3$ (from step 3). So we substitute $m\angle 1$ with $m\angle 3$ in the inequality, which is the Substitution property. Wait, no, actually, since $m\angle 1 = m\angle 3$ and $m\angle ABC > m\angle 1$, by the Comparison Property of Inequality (or Substitution in inequality), we get $m\angle ABC > m\angle 3$. Wait, the options: the reason for $m\angle ABC > m\angle 3$ is because we know $m\angle ABC > m\angle 1$ and $m\angle 1 = m\angle 3$, so we can substitute $m\angle 1$ with $m\angle 3$ in the inequality. But looking at the options, "Substitution" (option c) or maybe "Trans. Prop. of $
eq$"? Wait, no, let's re - check. We have $m\angle ABC > m\angle 1$ and $m\angle 1 = m\angle 3$, so by the Substitution property (since $m\angle 1$ and $m\angle 3$ are equal, we can replace one with the other in the inequality), so reason 6 is "Substitution" (option c)? Wait, no, the options: the reason for step 6: we have $m\angle ABC > m\angle 1$ (step 5) and $m\angle 1 = m\angle 3$ (step 3), so using the Substitution property (substituting $m\angle 1$ with $m\angle 3$ in the inequality $m\angle ABC > m\angle 1$) gives $m\angle ABC > m\angle 3$. So reason 6 is option c (Substitution).

Step 4: Analyze Reason 8

We have $m\angle 3 = m\angle 2 + m\angle C$. Since $m\angle 2$ is a positive measure (angle measure is positive), then $m\angle 3$ must be greater than $m\angle C$. This is the Comparison Property of Inequality (if $a = b + c$ and $c>0$, then $a > b$). So the reason for $m\angle 3 > m\angle C$ is "Comp. Prop. of $
eq$" (option d)? Wait, no, the option for reason 8: we have $m\angle 3=m\angle 2 + m\angle C$, so $m\angle 3$ is equal to the sum of $m\angle 2$ and $m\angle C$. Since $m\angle 2>0$, then $m\angle 3 > m\angle C$ by the Comparison Property of Inequality. So reason 8 is option d (Comp. Prop. of $
eq$).

Step 5: Analyze Reason 9

We have $m\angle ABC > m\angle 3$ (step 6) and $m\angle 3 > m\angle C$ (step 8). By the Transitive Property of Inequality (if $a > b$ and $b > c$, then $a > c$), we get $m\angle ABC > m\angle C$. So the reason for step 9 is "Trans. Prop. of $
eq$" (option f).

Answer:

1. b. Def. $\cong \angle$s
2. g. $\angle$ addition Postulate
3. c. Substitution
4. d. Comp. Prop. of $

eq$

1. f. Trans. Prop. of $

eq$