Chapter 7.5, Problem 1MRE

a.

To determine

To complete the given statement.

Answer to Problem 1MRE

  $GF\cong GC$ and $ED\cong EF$ .

Explanation of Solution

Given:

  $AF$ is parallel to is parallel to $CD$ .

Also, $AB\cong BC$.

  

Given statement: $GF\cong ?$ and $ED\cong ?$

Calculation:

In $\Delta ACF$ ,

  $AB\cong BC$ (given)

  $\Rightarrow B$ is the midpoint of $AC$ .

Now, $AF$ is parallel to $BE$ ,

  $\Rightarrow BG$ is parallel to $AF$ (parts of parallel lines are also parallel)

So, by mid − point theorem,

  $G$ is the mid − point of $FC$ .

  $\Rightarrow GF\cong GC$ .

In $\Delta DCF$ ,

  $GF\cong GC$ (proved above)

  $\Rightarrow G$ is the midpoint of $CF$ .

Now, $BE$ is parallel to $CD$ ,

  $\Rightarrow GE$ is parallel to $CD$ (parts of parallel lines are also parallel)

So, by Midpoint Theorem,

  $E$ is the midpoint of $DF$ .

  $\Rightarrow ED\cong EF$ .

Conclusion:

Therefore, the complete statement is: $GF\cong GC$ and $ED\cong EF$ .

b.

To determine

To complete the given statement.

Answer to Problem 1MRE

If $AB=9$ , then $AC=18$ .

Explanation of Solution

Given:

  $AF$ is parallel to $BE$ is parallel to $CD$ .

  $AB\cong BC$.

  

  $AB=9$.

Given statement: if $AB=9$ , then $AC=?$

Calculation:

Since, $AC=AB+BC$ , and

  $AB\cong BC$ (given),

  $\Rightarrow AC=AB+AB$ ,

  $\Rightarrow AC=9+9$ ( $\text{Since, }AB=9$ ),

  $\Rightarrow AC=18$ .

Conclusion:

Therefore, the complete statement is: if $AB=9$ , then $AC=18.$

c.

To determine

To complete the given statement.

Answer to Problem 1MRE

If $FG=3x+2$ and $GC=7x-10$ , then $x=1.2$ .

Explanation of Solution

Given:

  $AF$ is parallel to $BE$ is parallel to $CD$ .

  $AB\cong BC$.

  

  $FG=3x+2$ , and

  $GC=7x-10$.

Given statement: if $FG=3x+2$ and $GC=7x-10$ , then $x=?$

Calculation:

Since $FG\cong GC$ (proved above).

So,

  $3x+2=7x-10$ ,

  $\Rightarrow 7x+3x=10+2$,$\Rightarrow 10x=12$,$\Rightarrow x=\frac{12}{10}$,$\Rightarrow x=1.2$.

Conclusion:

Therefore, the complete statement is: if $FG=3x+2$ and $GC=7x-10$ , then $x=1.2$ .

d.

To determine

To find the measure of $m\angle FEG.$

Answer to Problem 1MRE

  $m\angle FEG=90°$ .

Explanation of Solution

Given:

  $AB$ is parallel to $BE$ is parallel to $CD$ .

  $AB\cong BC$.

  

  $m\angle EDC=90°$.

Given statement: $m\angle FEG=?$

Concept used:

Properties of parallel and transversal lines.

Calculation:

Since $BE$ is parallel to $DC$ (given), and $FD$ is transversal.

Then, $\angle FEG\cong \angle EDC$ (corresponding angles on the same side of the transversal are congruent).

  $\Rightarrow m\angle FEG=m\angle EDC$

  $\Rightarrow m\angle FEG=90°$. ( $\text{Since }m\angle EDC=90°$ )Conclusion:

Therefore, the complete statement is: $m\angle FEG=90°$ .