6-6 TRAPEZOIDS and KITES

A kite is a quadrilateral with two pairs of consecutive sides and no opposite sides .

A trapezoid is a quadrilateral with exactly one pair of parallel sides.

An isosceles trapezoid is a trapezoid with legs that are

A midsegment of a trapezoid is the segment that joins the midpoints of its legs.

The two s that share a base of a trapezoid are called base angles. A trapezoid has two pairs of base s.

The nonparallel sides are called legs.

The parallel sides of a trapezoid are called bases.

VOCABULARY

6-6 TRAPEZOIDS and KITESWord or Word Phrase

Defintion Picture or Example

trapezoidA trapezoid is a quadrilateral with onepair of parallel sides.

legs of a trapezoid

bases of a trapezoid

isosceles trapezoidAn isosceles trapezoid is a trapezoidwith legs that are congruent.

base angles A and B or C and D

kiteA kite is a quadrilateral with 2 pairs of consecutive, sides. In a kite, no opposite sides are .

midsegment of a trapezoid

The legs of a trapezoid are the non-parallel sides.The bases of a trapezoid are the parallel sides.

The base angles are the s that share the base of a trapezoid.

The midsegment of a trapezoid is the segment that joins the midpoints of the legs.

6-6 TRAPEZOIDS and KITESOBJECTIVES: To verify and use the properties of trapezoids and kites.

Two isosceles triangles form the figure below. Each white segment is a midsegment of a triangle. What can you determine about the angles in region 2? In region 3? Explain.

In each region, the s are either supplementary or

The midsegment of each isosceles is ‖ to its base,so same-side interior s are supplementary.

Since base s in an isosceles are so the s sharing the midsegment of each are .

6-6 TRAPEZOIDS and KITESOBJECTIVES: To verify and use the properties of trapezoids and kites.

Theorem 6-20If a quadrilateral is an isosceles trapezoid, then each pair of base angles is congruent.

Theorem 6-21If a quadrilateral is an isosceles trapezoid, then its diagonals are congruent.

6-6 TRAPEZOIDS and KITESOBJECTIVES: To verify and use the properties of trapezoids and kites.

Theorem 6-22If a quadrilateral is a trapezoid, then (1) the midsegment is parallel to the bases, and (2) the length of the midsegment is half the sum of the lengths of the bases.

6-6 TRAPEZOIDS and KITES

Theorem 6-23If a quadrilateral is a kite, thenits diagonals are perpendicular.

OBJECTIVES: To verify and use the properties of trapezoids and kites.

Concept Summary - Relationships Among Quadrilaterals

6-6 TRAPEZOIDS and KITES

Finding Angle Measures in Trapezoids

a. In the diagram, PQRS is an isosceles trapezoid and mR = 106. What are mP, mQ, and mS?

b. In Problem 1, if CDEF were not an isosceles trapezoid, would C and D still be supplementary? Explain.

OBJECTIVE: To verify and use the properties of trapezoids and kites.

𝒎 𝑷=𝒎𝑸=𝟕𝟒 𝒎𝑺=𝟏𝟎𝟔

𝐘𝐞𝐬 ; so same-side interior s are supplementary.

6-6 TRAPEZOIDS and KITESOBJECTIVE: To verify and use the properties of trapezoids and kites.

Finding Angle Measures in Isosceles TrapezoidsA fan like the one in Problem 2 has 15 congruent angles meeting at the

center.What are the measures of the base angles of the trapezoids in its second ring?

Q: What is the measure of each one of the 15 s meeting at the center?𝟑𝟔𝟎°

𝟏𝟓 =𝟐𝟒°

obtuse angles measure

acute angles measure

24

78 78

102 102

6-6 TRAPEZOIDS and KITESOBJECTIVE: To verify and use the properties of trapezoids and kites.

Investigating the Diagonals of Isosceles TrapezoidsChoose from a variety of tools (such as a protractor, a ruler, or a compass) to

investigate patterns in the diagonals of isosceles trapezoid PQRS. Explain your choice. Do your observations support your conjecture in Problem 3? Explain your reasoning.

In Problem 3 (HH): Use a protractor to measure the s formed by the diagonals and a compass to check if the diagonals are

Answer: Use a ruler to measure the segments. This supports the conjecture that if a quadrilateral is an isosceles trapezoid, then the diagonals are congruent.

6-6 TRAPEZOIDS and KITESOBJECTIVE: To verify and use the properties of trapezoids and kites.

Using the Midsegment of a Trapezoid

a. is the midsegment of trapezoid PQRS. What is x? What is MN?

b. How many midsegments can a triangle have? How many midsegments can a trapezoid have? Explain.

𝒙=𝟔23

𝟑𝟏

A has 3 midsegments joining any pair of the side midpoints.

A trapezoid has 1 midsegment joining the midpoints of the two legs.

6-6 TRAPEZOIDS and KITESOBJECTIVE: To verify and use the properties of trapezoids and kites.

Finding Angle Measures in Kites

Quadrilateral KLMN is a kite. What are m1, m2, and m3?

𝒎𝟏=𝟗𝟎°𝒎𝟑=𝟑𝟔°𝒎𝟐=𝟓𝟒°

6-6 TRAPEZOIDS and KITESOBJECTIVE: To verify and use the properties of trapezoids and kites.

1. What are the measures of the numbered angles?

2. Quadrilateral WXYZ is an isosceles trapezoid. Are the two trapezoids formed by drawing midsegment QR isosceles trapezoids? Explain.

𝒎𝟏=𝟕𝟖°𝒎𝟐=𝟗𝟎°

𝒎𝟏=𝟗𝟒°𝒎𝟐=𝟏𝟑𝟐°

Yes, the midsegment is ‖ to both bases and bisects each of the two congruent legs.

𝒎𝟑=𝟏𝟐°

6-6 TRAPEZOIDS and KITESOBJECTIVE: To verify and use the properties of trapezoids and kites.

3. Find the length of the perimeter of trapezoid LMNP with midsegment .

Perimeter of

𝑸𝑹=𝟏𝟐 ( 𝑳𝑴+𝑷𝑵 )

78

Solve for PN :

𝟐𝟓=𝟏𝟐 (𝟏𝟔+𝑷𝑵 )

16 + PN PN

Perimeter of

6-6 TRAPEZOIDS and KITESOBJECTIVE: To verify and use the properties of trapezoids and kites.

4. Vocabulary Is a kite a parallelogram? Explain.

5. Analyze Mathematical Relationships (1)(F) How is a kite similar to a rhombus? How is it different? Explain.

6. Evaluate Reasonableness (1)(B) Since a parallelogram has two pairs of parallel sides, it certainly has one pair of parallel sides. Therefore, a parallelogram must also be a trapezoid. Is this reasoning correct? Explain.

No, a kite’s opposite sides are not ‖ or .

Similar: Their diagonals are .⏊

Different: Only one diagonal of a kite bisects opposite s;

a rhombus has all sides .

No. A trapezoid is defined as a quad. with exactly 1 pair of ‖ sides and a parallelogram has exactly 2 pairs of ‖ sides, so a parallelogram is not a trapezoid.