Chapter 10.3, Problem 16PSB

To determine

To Prove : ABCD is a parallelogram.

Answer to Problem 16PSB

1

Explanation of Solution

Given information : $\text{Two equal circles with centres B and D and }\stackrel{⌢}{\text{AE}}\cong \stackrel{⌢}{CE}.$

  

  $\begin{array}{l}\stackrel{⌢}{AE}\cong \stackrel{⌢}{CE}\Rightarrow \overline{AE}=\overline{CE}\\ \text{because equal arcs intercepts equal chords}\text{.}\\ \text{Also, }\overline{\text{BE}}\cong \overline{DE}\text{ [Radii of equal circles]}\\ \text{Now, }\overline{\text{BD}}and\overline{AC}\text{ are two diagonal of parallelogram ABCD}\\ \text{and}\overline{\text{BE}}\cong \overline{DE},\overline{AE}\cong \widetilde{CE}\\ \Rightarrow \overline{BD}and\overline{AC}\text{bisector each other}\text{.}\\ \text{Using the property}\text{. If diagonals of a quadrilateral}\\ \text{bisect each other then it is a parallelogram}\text{.}\\ \Rightarrow \text{ABCD}\text{is a}\text{parallelogram}\text{.}\end{array}$