1. EXAMPLE 1 Identify quadrilaterals Quadrilateral ABCD has at least one pair of opposite angles congruent. What types of quadrilaterals meet this condition? SOLUTION There are many possibilities.

2. EXAMPLE 2 Standardized Test Practice SOLUTION The diagram shows AE CE and BE DE. So, the diagonals bisect each other. By Theorem 8.10, ABCD is a parallelogram.

3. EXAMPLE 2 Standardized Test Practice Rectangles, rhombuses and squares are also parallelograms. However, there is no information given about the side lengths or angle measures of ABCD. So,you cannot determine whether it is a rectangle, a rhombus, or a square. The correct answer is A. ANSWER

4. EXAMPLE 3 SOLUTION STEP 1 Show that PQRS is a trapezoid. R and S are supplementary,but P and S are not. So, PS QR, but PQ is not parallel to SR. By definition, PQRS is a trapezoid. Identify a quadrilateral Is enough information given in the diagram to show that quadrilateral PQRS is an isosceles trapezoid? Explain.

5. EXAMPLE 3 Identify a quadrilateral STEP 2 Show that trapezoid PQRS is isosceles. P and S are a pair of congruent base angles. So, PQRS is an isosceles trapezoid by Theorem 8.15. Yes, the diagram is sufficient to show that PQRS is an isosceles trapezoid. ANSWER

6. GUIDED PRACTICE for Examples 1, 2 and 3 1. Quadrilateral DEFG has at least one pair of opposite sides congruent. What types of quadrilaterals meet this condition? ANSWER Parallelogram, Rectangle, Square, Rhombus, Trapezoid. In all these quadrilaterals at least one pair of opposite sides is congruent.

7. GUIDED PRACTICE for Examples 1, 2 and 3 Give the most specific name for the quadrilateral. Explain your reasoning. ANSWER It is a kite as kite is a quadrilateral that has two pair of consecutive congruent sides, but opposite sides are not congruent.

8. GUIDED PRACTICE for Examples 1, 2 and 3 Give the most specific name for the quadrilateral. Explain your reasoning. ANSWER VWXY is a trapezoid. one pair of opposite sides are parallel, and the diagonals do not bisect each other, therefore it is a trapezoid.

9. GUIDED PRACTICE for Examples 1, 2 and 3 Give the most specific name for the quadrilateral. Explain your reasoning. ANSWER Quadrilateral; there is not enough information to be more specific.

10. GUIDED PRACTICE for Examples 1, 2 and 3 5. Error Analysis A student knows the following information about quadrilateral MNPQ:MN PQ, MP NQ, and P Q. The student concludes that MNPQ is an isosceles trapezoid. Explain why the student cannot make this conclusion. ANSWER MNPQ could be a rectangle or a square since you do not know the relationship between MQ and NP. There is not enough information to conclude it is an isosceles trapezoid.