Chapter 6.1, Problem 18CE

Solution

(a)

To Complete: the statement $m\angle 1\underset{\_}{?}m\angle 3$ by $<,=,\text{ or >}$ for the figure shown.

  $=$

Given:

Figure is shown below.

  

Calculation:

Since, vertically opposite angles are equal therefore.

  $m\angle 1=m\angle 3$

Therefore, the answer is $=$ .

(b)

To Complete: the statement $m\angle 2\underset{\_}{?}m\angle 3$ by $<,=,\text{ or >}$ for the figure shown.

  $<$

Given:

Figure is shown below.

  

Calculation:

Mark angle 4 as shown below.

  

Since, vertically opposite angles are equal therefore.

  $m\angle 1=m\angle 3$

  $m\angle 3=m\angle 2+\angle 4$

If $a=b+c\text{and }c>0,\text{then}a>b$

  $m\angle 3>\angle 2$

It can be written as.

  $m\angle 2<m\angle 3$

Therefore, the answer is $<$ .

(c)

To Complete: the statement $m\angle 1\underset{\_}{?}m\angle 2$ by $<,=,\text{ or >}$ for the figure shown.

  $>$

Given:

Figure is shown below.

  

Calculation:

Mark angle 4 as shown below.

  

Since, vertically opposite angles are equal therefore.

  $m\angle 1=m\angle 3$

  $\begin{array}{l}m\angle 3=m\angle 2+\angle 4\\ \Rightarrow m\angle 1=m\angle 2+\angle 4\end{array}$

If $a=b+c\text{and }c>0,\text{then}a>b$

  $m\angle 1>\angle 2$

Therefore, the answer is $>$ .