8.6 warm up warm up lesson quiz lesson quiz lesson presentation lesson presentation identify special...

8.6

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Identify Special Quadrilaterals

8.6 Warm-Up

ANSWER 65

Find the value of x.

1.

ANSWER 120

2.

8.6 Warm-Up

Find the value of x.

3.

ANSWER 70

8.6 Example 1

Quadrilateral ABCD has at least one pair of opposite angles congruent. What types of quadrilaterals meet this condition?

SOLUTION

There are many possibilities.

8.6 Example 2

SOLUTION

The diagram shows AE CE and BE DE . So, the diagonals bisect each other. By Theorem 8.10, ABCD is a parallelogram.

8.6 Example 2

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

8.6 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.

Is enough information given in the diagram to show that quadrilateral PQRS is an isosceles trapezoid? Explain.

8.6 Example 3

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

8.6 Guided Practice

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.

8.6 Guided Practice

Give the most specific name for the quadrilateral.Explain your reasoning.

ANSWER

Kite: there are two pairs of consecutive congruent sides.

8.6 Guided Practice

Give the most specific name for the quadrilateral.Explain your reasoning.

ANSWER

Trapezoid: there is one pair of parallel sides.

8.6 Guided Practice

Give the most specific name for the quadrilateral.Explain your reasoning.

ANSWER

Quadrilateral; there is not enough information to be more specific.

8.6 Guided Practice

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.

8.6 Lesson Quiz

1. The diagonals of a rectangle are always perpendicular.

Write true or false.

ANSWER False

2. The diagonals of a rhombus are always congruent.

ANSWER False

3. One pair of opposite angles of a kite are congruent.

ANSWER True

8.6 Lesson 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 . = =

8.6 Lesson 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.

=