Exercise 8 Page 587

Again we need to consider a diagram that describes pentagon $ABCDE,$ which is inscribed in a circle.

This time we want to find a relationship between $\angle CBE$ and $\angle CAE.$ When we create $\angle CBE$ we draw the segment $BE.$ This segment divides the pentagon into a quadrilateral and an isosceles triangle. By the Base Angles Theorem the triangle's base angles are congruent. Knowing this and by using the Triangle Sum Theorem, we can find their measure. Let's now use that $m\angle EBA$ and $m\angle CBE$ add to $108.$ Let's now study $\angle CAE.$ When we create this angle we need to draw a segment which divides the pentagon into a quadrilateral and an isosceles triangle. The triangle, $△CBA,$ has two sides and the included angle congruent with $△BAE.$ By the Side-Angle-Side Congruence Theorem the triangles are congruent. By the definition of congruence we know that both triangles' base angles have the same measure. Let's use that $m\angle BAC$ and $m\angle CAE$ add to $108.$ We have found that $\angle CBE$ and $\angle CAE$ have the same measure.