22. (l2) identify the converse of the following statement, and determine whether it is true or false
if a triangle is equiangular, then it has three angles with equal measures.
a. if a triangle has less than three angles with equal measures, then it is equiangular; false
b. if a triangle has three angles with equal measures, then it is equiangular; true
c. if a triangle has three angles with equal measures, then it is equiangular; false
d. if a triangle has less than three angles with equal measures, then it is equiangular; true
e. if three angles of a triangle are measured, then it is an equiangular triangle; false
23. select the correct proof from the options listed.
given: $ \triangle j l m $ is equilateral; $ z $ is the midpoint of $ \overline{j m} $
prove: $ \triangle j z l \cong \triangle m z l $
a.
\begin{tabular}{l|l}
statements & reasons \\
\hline
1. $ \triangle j l m $ is equilateral. & 1. given \\
2. $ \overline{j l} \cong \overline{m l} $ & 2. definition of an equilateral triangle \\
3. $ z $ is the midpoint of $ \overline{j m} $. & 3. given \\
4. $ \overline{j z} \cong \overline{m z} $ & 4. definition of a midpoint \\
5. $ \overline{l z} \cong \overline{l z} $ & 5. reflexive property \\
6. $ \triangle j z l \cong \triangle m z l $ & 6. side-side-side postulate \\
\end{tabular}
b.
\begin{tabular}{l|l}
statements & reasons \\
\hline
1. $ \triangle j l m $ is equilateral. & 1. given \\
2. $ \overline{j l} \cong \overline{l z} $ & 2. definition of an equilateral triangle \\
3. $ z $ is the midpoint of $ \overline{j m} $. & 3. given \\
4. $ \overline{j z} \cong \overline{m z} $ & 4. definition of parallel lines \\
5. $ \overline{l z} \cong \overline{l z} $ & 5. substitution property \\
6. $ \triangle j z l \cong \triangle m z l $ & 6. angle-side-angle postulate \\
\end{tabular}
c.
\begin{tabular}{l|l}
statements & reasons \\
\hline
1. $ \triangle j l m $ is equilateral. & 1. given \\
2. $ z $ is the midpoint of $ \overline{j m} $. & 2. given \\
3. $ \overline{l z} \perp \overline{j m} $ & 3. definition of an equilateral triangle \\
4. $ \overline{j z} \cong \overline{m z} $ & 4. definition of a midpoint \\
5. $ \overline{l z} \cong \overline{l z} $ & 5. reflexive property \\
6. $ \triangle j z l \cong \triangle m z l $ & 6. side-angle-side postulate \\
\end{tabular}
d.
\begin{tabular}{l|l}
statements & reasons \\
\hline
1. $ \triangle j l m $ is equilateral. & 1. given \\
2. $ \angle j l z \cong \angle m l z $ & 2. congruent angles are created by two lines cut by \\
& a transversal. \\
3. $ \overline{l z} \perp \overline{j m} $ & 3. definition of perpendicular lines \\
4. $ \overline{l z} \cong \overline{l z} $ & 4. same line \\
5. $ \triangle j z l \cong \triangle m z l $ & 5. angle-side-angle postulate \\
\end{tabular}

Question

22. (l2) identify the converse of the following statement, and determine whether it is true or false
if a triangle is equiangular, then it has three angles with equal measures.
a. if a triangle has less than three angles with equal measures, then it is equiangular; false
b. if a triangle has three angles with equal measures, then it is equiangular; true
c. if a triangle has three angles with equal measures, then it is equiangular; false
d. if a triangle has less than three angles with equal measures, then it is equiangular; true
e. if three angles of a triangle are measured, then it is an equiangular triangle; false
23. select the correct proof from the options listed.
given: $ \triangle j l m $ is equilateral; $ z $ is the midpoint of $ \overline{j m} $
prove: $ \triangle j z l \cong \triangle m z l $
a.
\begin{tabular}{l|l}
statements & reasons \\
\hline
1. $ \triangle j l m $ is equilateral. & 1. given \\
2. $ \overline{j l} \cong \overline{m l} $ & 2. definition of an equilateral triangle \\
3. $ z $ is the midpoint of $ \overline{j m} $. & 3. given \\
4. $ \overline{j z} \cong \overline{m z} $ & 4. definition of a midpoint \\
5. $ \overline{l z} \cong \overline{l z} $ & 5. reflexive property \\
6. $ \triangle j z l \cong \triangle m z l $ & 6. side-side-side postulate \\
\end{tabular}
b.
\begin{tabular}{l|l}
statements & reasons \\
\hline
1. $ \triangle j l m $ is equilateral. & 1. given \\
2. $ \overline{j l} \cong \overline{l z} $ & 2. definition of an equilateral triangle \\
3. $ z $ is the midpoint of $ \overline{j m} $. & 3. given \\
4. $ \overline{j z} \cong \overline{m z} $ & 4. definition of parallel lines \\
5. $ \overline{l z} \cong \overline{l z} $ & 5. substitution property \\
6. $ \triangle j z l \cong \triangle m z l $ & 6. angle-side-angle postulate \\
\end{tabular}
c.
\begin{tabular}{l|l}
statements & reasons \\
\hline
1. $ \triangle j l m $ is equilateral. & 1. given \\
2. $ z $ is the midpoint of $ \overline{j m} $. & 2. given \\
3. $ \overline{l z} \perp \overline{j m} $ & 3. definition of an equilateral triangle \\
4. $ \overline{j z} \cong \overline{m z} $ & 4. definition of a midpoint \\
5. $ \overline{l z} \cong \overline{l z} $ & 5. reflexive property \\
6. $ \triangle j z l \cong \triangle m z l $ & 6. side-angle-side postulate \\
\end{tabular}
d.
\begin{tabular}{l|l}
statements & reasons \\
\hline
1. $ \triangle j l m $ is equilateral. & 1. given \\
2. $ \angle j l z \cong \angle m l z $ & 2. congruent angles are created by two lines cut by \\
 & a transversal. \\
3. $ \overline{l z} \perp \overline{j m} $ & 3. definition of perpendicular lines \\
4. $ \overline{l z} \cong \overline{l z} $ & 4. same line \\
5. $ \triangle j z l \cong \triangle m z l $ & 5. angle-side-angle postulate \\
\end{tabular}

Accepted Answer

### 22.
# Brief Explanations:
For a conditional statement “If $p$, then $q$” ($p
ightarrow q$), the converse is “If $q$, then $p$” ($q
ightarrow p$).
In the statement “If a triangle is equiangular ($p$), then it has three angles with equal measures ($q$)”, the converse is “If a triangle has three angles with equal measures ($q$), then it is equiangular ($p$)”.
By the definition of an equiangular triangle (a triangle with all angles equal),  the converse is a true statement.

# Answer:
B. If a triangle has three angles with equal measures, then it is equiangular; true

### 23.
# Brief Explanations:
- **Option A**:
    - Since $\triangle JLM$ is equilateral, by the definition of an equilateral triangle, $JL = ML$ (so $\overline{JL}\cong\overline{ML}$).
    - $Z$ is the mid - point of $\overline{JM}$, so by the definition of a mid - point, $JZ = MZ$ (so $\overline{JZ}\cong\overline{MZ}$).
    - $\overline{LZ}\cong\overline{LZ}$ by the reflexive property ($a = a$ for any geometric segment $a$).
    - Then, by the Side - Side - Side (SSS) postulate ($\triangle ABC\cong\triangle DEF$ if $AB = DE$, $BC=EF$, and $AC = DF$), $\triangle JZL\cong\triangle MZL$ since $JL = ML$, $JZ = MZ$, and $LZ = LZ$.
 - **Option B**:
    - In an equilateral triangle $\triangle JLM$, $JL
eq LZ$ (the side of an equilateral triangle is not equal to  a segment from a vertex to a mid - point of a non - adjacent side). Also, the reason for $\overline{JZ}\cong\overline{MZ}$ should be the definition of a mid - point, not parallel lines. And the postulate used is incorrect as we have three pairs of  sides, not angles - sides - angles.
 - **Option C**:
    - Just because $Z$ is the mid - point of $\overline{JM}$ in an equilateral triangle $\triangle JLM$, we cannot directly say $\overline{LZ}\perp\overline{JM}$ (this is a property of an isosceles triangle's median, but we need to prove it first. Also, the postulate used is incorrect as we have three pairs of sides for SSS, not side - angle - side).
 - **Option D**:
    - $\angle JL Z\cong\angle MLZ$ is not a given property (we need to prove it using the SSS congruence of triangles, not assume it from some transversal property). Also, $\overline{LZ}\perp\overline{JM}$ is not  established, and the steps and reasons are jumbled.

# Answer:
A.
| Statements | Reasons |
|--|--|
| 1. $\triangle JLM$ is equilateral. | 1. Given |
| 2. $\overline{JL}\cong\overline{ML}$ | 2. Definition of an equilateral triangle |
| 3. $Z$ is the midpoint of $\overline{JM}$. | 3. Given |
| 4. $\overline{JZ}\cong\overline{MZ}$ | 4. Definition of a midpoint |
| 5. $\overline{LZ}\cong\overline{LZ}$ | 5. Reflexive Property |
| 6. $\triangle JZL\cong\triangle MZL$ | 6. Side - Side - Side Postulate |

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QUESTION IMAGE

Question

Explanation:

22.

Answer:

23.