Chapter 1.3, Problem 53PPS

(a)

To determine

To sketch: the figure and label $\overline{AB}\cong \overline{CD,}\overline{AB}\cong \overline{BD,}$ and $\overline{AB}\cong \overline{CA}$ .

Explanation of Solution

Given information:

A proof is an argument in which given statement make is supported by a true statement. It is given that $\overline{AB}\cong \overline{CD,}\overline{AB}\cong \overline{BD,}$ , and $\overline{AB}\cong \overline{CA,}$ P wants to prove $\Delta ADB\cong \Delta ADC$ , to do this, he must show that $\overline{AD}\cong \overline{AD},\overline{AB}\cong \overline{DC}$ and $\overline{BD}\cong \overline{AC}$ .

  

Calculation:

Drawn figure as per the given problem,

  

(b)

To determine

To prove: $\Delta ADB\cong \Delta ADC$ .

Explanation of Solution

Given information:

A proof is an argument in which given statement make is supported by a true statement. It is given that $\overline{AB}\cong \overline{CD,}\overline{AB}\cong \overline{BD,}$ , and $\overline{AB}\cong \overline{CA,}$ P wants to prove $\Delta ADB\cong \Delta ADC$ , to do this, he must show that $\overline{AD}\cong \overline{AD},\overline{AB}\cong \overline{DC}$ and $\overline{BD}\cong \overline{AC}$ .

  

Calculation:

  $\overline{AD}\cong \overline{AD}$ by the reflexive property. The transitive property show that if $\overline{AB}\cong \overline{AC}$ and $\overline{AC}\cong \overline{DC}$ , then $\overline{AB}\cong \overline{DC}$ , and if $\overline{AB}\cong \overline{AC}$ , then $\overline{BD}\cong \overline{AC}$ .

So, prove that $\Delta ADB\cong \Delta ADC$ .

(c)

To determine

To write: an equation for the perimeter of $ACDB$ .

Answer to Problem 53PPS

The equation is $P=x+x+x+x$ .

Explanation of Solution

Given information:

A proof is an argument in which given statement make is supported by a true statement. It is given that $\overline{AB}\cong \overline{CD,}\overline{AB}\cong \overline{BD,}$ , and $\overline{AB}\cong \overline{CA,}$ P wants to prove $\Delta ADB\cong \Delta ADC$ , to do this, he must show that $\overline{AD}\cong \overline{AD},\overline{AB}\cong \overline{DC}$ and $\overline{BD}\cong \overline{AC}$ .

  

Calculation:

Let us consider,

  $AC$ is $x$ centimeter.

To find the perimeter of $ABCD$ .

So,

  $\begin{array}{c}\text{Perimeter}=AB+AC+BD+DC\left(\text{all}\text{sides}\text{are}\text{equal}\right)\\ P=x+x+x+x\end{array}$

So, the equation is $P=x+x+x+x$ .