practice & problem solving
understand
13. communicate and justify fill in the missing reasons in the proof of theorem 5-9. (hint: the comparison property of inequality states that if a = b + c and c 0, then a b.)
given: ab ac, $\overline{ac} \cong \overline{am}$
prove: $m\angle acb m\angle b$
triangle diagram with points a, b, c, m and angles 1, 2, 3
statements | reasons
--- | ---
1) $\overline{ac} \cong \overline{am}$ | 1) given
2) $m\angle 1 = m\angle 2$ | 2) isosc. triangle thm.
3) $m\angle acb = m\angle 2 + m\angle 3$ | 3)
4) $m\angle acb m\angle 2$ | 4)
5) $m\angle acb m\angle 1$ | 5)
6) $m\angle 1 = m\angle b + m\angle 3$ | 6) ext. angles thm.
7) $m\angle 1 m\angle b$ | 7)
8) $m\angle acb m\angle b$ | 8)
14. cor pro anc
15.

Question

practice & problem solving
understand
13. communicate and justify fill in the missing reasons in the proof of theorem 5-9. (hint: the comparison property of inequality states that if a = b + c and c > 0, then a > b.)
given: ab > ac, $\overline{ac} \cong \overline{am}$
prove: $m\angle acb > m\angle b$
triangle diagram with points a, b, c, m and angles 1, 2, 3
statements | reasons
--- | ---
1) $\overline{ac} \cong \overline{am}$ | 1) given
2) $m\angle 1 = m\angle 2$ | 2) isosc. triangle thm.
3) $m\angle acb = m\angle 2 + m\angle 3$ | 3)
4) $m\angle acb > m\angle 2$ | 4)
5) $m\angle acb > m\angle 1$ | 5)
6) $m\angle 1 = m\angle b + m\angle 3$ | 6) ext. angles thm.
7) $m\angle 1 > m\angle b$ | 7)
8) $m\angle acb > m\angle b$ | 8)
14. cor pro anc
15.

Accepted Answer

# Explanation:
## Step1: Angle Addition Postulate
The Angle Addition Postulate states that if a point lies in the interior of an angle, the measure of the whole angle is equal to the sum of the measures of the two smaller angles. So, for $\angle ACB$, since $m\angle ACB=m\angle2 + m\angle3$, the reason for statement 3 is the Angle Addition Postulate.
## Step2: Comparison Property of Inequality
Given $m\angle ACB=m\angle2 + m\angle3$ and $m\angle3>0$ (because it is an angle measure), by the Comparison Property of Inequality (if $a = b + c$ and $c>0$, then $a>b$), we have $m\angle ACB>m\angle2$. So the reason for statement 4 is the Comparison Property of Inequality.
## Step3: Substitution
Since $m\angle1 = m\angle2$ (from statement 2: Isosceles Triangle Theorem, which states that in an isosceles triangle, the base angles are equal. Here, $\triangle ACM$ is isosceles with $AC\cong AM$), we can substitute $m\angle2$ with $m\angle1$ in the inequality $m\angle ACB>m\angle2$. So the reason for statement 5 is Substitution.
## Step4: Comparison Property of Inequality
Given $m\angle1=m\angle B + m\angle3$ and $m\angle3>0$ (because it is an angle measure), by the Comparison Property of Inequality (if $a = b + c$ and $c>0$, then $a>b$), we have $m\angle1>m\angle B$. So the reason for statement 7 is the Comparison Property of Inequality.
## Step5: Transitive Property of Inequality
We know that $m\angle ACB>m\angle1$ (statement 5) and $m\angle1>m\angle B$ (statement 7). By the Transitive Property of Inequality (if $a > b$ and $b>c$, then $a>c$), we have $m\angle ACB>m\angle B$. So the reason for statement 8 is the Transitive Property of Inequality.

# Answer:
3) Angle Addition Postulate; 4) Comparison Property of Inequality; 5) Substitution; 7) Comparison Property of Inequality; 8) Transitive Property of Inequality.

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QUESTION IMAGE

Question

Explanation:

Step1: Angle Addition Postulate

Step2: Comparison Property of Inequality

Step3: Substitution

Step4: Comparison Property of Inequality

Step5: Transitive Property of Inequality

Answer: