1. 1.
Find the value of x.
ANSWER 65 2.
ANSWER
120

2. Find the value of x. 3.
ANSWER 70

3. 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.

4. 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.

5. 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

6. EXAMPLE 3
Identify a quadrilateral
Is enough information given in the diagram to show that quadrilateral PQRS is an isosceles trapezoid? Explain.
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.

7. 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

8. 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.

9. GUIDED PRACTICE
for Examples 1, 2, and 3
Give the most specific name for the quadrilateral. Explain your reasoning.
ANSWER
Kite: there are two pairs of consecutive congruent sides.

10. GUIDED PRACTICE
for Examples 1, 2, and 3
Give the most specific name for the quadrilateral. Explain your reasoning.
ANSWER
Trapezoid: there is one pair of parallel sides, and because the diagonals do not bisect each other, it is not a parallelogram.

11. 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.

12. 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
It’s possible that MNPQ could be a rectangle or a square since you don’t know the relationship between MQ and NP.

13. Daily Homework Quiz
Write true or false.
1. The diagonals of a rectangle are perpendicular.
ANSWER
False
2. The diagonals of a rhombus are congruent.
ANSWER
False
3. One pair of opposite angles of a kite are congruent.
ANSWER
True

14. 4. Give the most specific name for the quadrilateral. Explain.
Daily Homework Quiz
4. Give the most specific name for the quadrilateral. Explain.
ANSWER
Rhombus ;
It is a since two pairs of opp are . s =
Since two consec. sides are , all sides are .

15. Daily Homework Quiz
5. Points A(1, 4), B(6, –1), C(1, –6), D(–4, –1) are the vertices of a quadrilateral. Give the most specific name for ABCD. Explain.
ANSWER
Square; slope of AB = slope of CD = – 1, slope of BC = slope of AD = 1. Opp. sides are ||, so ABCD is a Two consec. sides are and AB = BC = 5 2, so ABCD is a square.