Chapter 9.5, Problem 10WE

Solution

To prove: Arcs between parallel cords are equal.

Given:

The figure:

  $\overline{AB}\parallel \overline{CD}$

Concept Used:

Theorem 3-2: Alternate interior angles between parallel lines are equal.

Explanation:

In the given figure $\overline{AD}$ and $\overline{BC}$ .

It is given that $\overline{AB}\parallel \overline{CD}$ so, by theorem 3-2,

  $m\angle ABC=m\angle BCD$ [Alternate interior angles] ….. (1)

Now by theorem 9-7, the measure of an inscribed angle is equal to half the measure of its intercepted, so

  $m\angle ABC=\frac{1}{2}m\stackrel{⌢}{A}C$ ….. (2)

And

  $m\angle BCD=\frac{1}{2}m\stackrel{⌢}{B}D$ ….. (3)

From (1), (2) and (3),

  $\begin{array}{c}m\angle ABC=m\angle BCD\\ \frac{1}{2}m\stackrel{⌢}{A}C=\frac{1}{2}m\stackrel{⌢}{B}D\\ m\stackrel{⌢}{A}C=m\stackrel{⌢}{B}D\end{array}$

This gives that,

  $\stackrel{⌢}{A}C\cong \stackrel{⌢}{B}D$

Hence proved that, arcs between parallel cords are equal.