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Name _______________________________________ Date ___________________ Class __________________ LESSON 6-6 Reteach Properties of Kites and Trapezoids A kite is a quadrilateral with exactly two pairs of congruent consecutive sides. If a quadrilateral is a kite, such as FGHJ, then it has the following properties. Properties of Kites FH GJ G J Exactly one pair of opposite angles is congruent. The diagonals are perpendicular. A trapezoid is a quadrilateral with exactly one pair of parallel sides. If the legs of a trapezoid are congruent, the trapezoid is an isosceles trapezoid. Each nonparallel side is called a leg. Each || side is called a base. Base angles are two consecutive angles whose common side is a base. Isosceles Trapezoid Theorems • In an isosceles trapezoid, each pair of base angles is congruent. • If a trapezoid has one pair of congruent base angles, then it is isosceles. • A trapezoid is isosceles if and only if its diagonals are congruent. In kite ABCD, mBCD 98, and mADE 47. Find each measure. 1. mDAE ________________________________________ 2. mBCE ________________________________________ 3. mABC ________________________________________ 5. In trapezoid EFGH, FH 9. Find AG. 4. Find mJ in trapezoid JKLM. ________________________________________ ________________________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 6-46 Holt Geometry Name _______________________________________ Date ___________________ Class __________________ LESSON 6-6 Reteach Properties of Kites and Trapezoids continued Trapezoid Midsegment Theorem The midsegment of a trapezoid is the segment whose endpoints are the midpoints of the legs. • The midsegment of a trapezoid is parallel to each base. AB MN and AB LP • The length of the midsegment is one-half the sum of the length of the bases. 1 AB (MN LP) 2 AB is the midsegment of LMNP. Find each value so that the trapezoid is isosceles. 7. AC 2z 9, BD 4z 3. Find the value of z. 6. Find the value of x. ________________________________________ ________________________________________ Find each length. 8. KL 9. PQ ________________________________________ 10. EF ________________________________________ 11. WX ________________________________________ ________________________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 6-47 Holt Geometry Name _______________________________________ Date ___________________ Class __________________ LESSON 6-6 Practice A Properties of Kites and Trapezoids Fill in the blanks to complete each theorem or definition. 1. If a quadrilateral is a kite, then its ________________________ are perpendicular. 2. If a quadrilateral is a kite, then exactly one pair of opposite ________________________ are congruent. 3. A kite is a quadrilateral with exactly two pairs of congruent consecutive ________________________. ABCD is a kite. Use the figure to find each measure in Exercises 4–6. 4. mD 5. AB ________________________ 6. CD ________________________ ________________________ A trapezoid is a quadrilateral with exactly one pair of parallel sides. Name the parts of trapezoid PQRS asked for in Exercises 7–9. 7. both bases ____________________ 8. both legs ____________________ 9. one pair of base angles ____________________ Fill in the blanks to complete each theorem or definition. 10. A trapezoid is isosceles if and only if its ____________________ are congruent. 11. If a trapezoid has one pair of congruent base angles, then the trapezoid is ____________________. 12. If the legs of a trapezoid are ____________________, then the trapezoid is an isosceles trapezoid. 13. If a quadrilateral is an isosceles trapezoid, then each pair of ____________________ is congruent. In an art museum, a statue sits on a pedestal with sides that are isosceles trapezoids. Name the parts of isosceles trapezoid EFGH asked for in Exercises 14 and 15. 14. both pairs of congruent angles _____________________________________ 15. both pairs of congruent segments _____________________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 6-44 Holt Geometry Name _______________________________________ Date ___________________ Class __________________ LESSON 6-6 Practice B Properties of Kites and Trapezoids In kite ABCD, mBAC 35 and mBCD 44. For Exercises 1–3, find each measure. 1. mABD ________________________ 2. mDCA 3. mABC ________________________ ________________________ 4. Find the area of EFG. _____________________ 6. KM 7.5, and NM 2.6. Find LN. 5. Find mZ. ________________________________________ ________________________________________ 7. Find the value of n so that PQRS is isosceles. ________________________________________ 8. Find the value of x so that EFGH is isosceles. ________________________________________ 9. BD 7a 0.5, and AC 5a 2.3. Find the value of a so that ABCD is isosceles. ________________________________________ 10. QS 8z2, and RT 6z2 38. Find the value of z so that QRST is isosceles. ________________________________________ Use the figure for Exercises 11 and 12. The figure shows a ziggurat. A ziggurat is a stepped, flat-topped pyramid that was used as a temple by ancient peoples of Mesopotamia. The dashed lines show that a ziggurat has sides roughly in the shape of a trapezoid. 11. Each “step” in the ziggurat has equal height. Give the vocabulary term for MN. ________________________________________ 12. The bottom of the ziggurat is 27.3 meters long, and the top of the ziggurat is 11.6 meters long. Find MN. ________________________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. 6-45 Holt Geometry LESSON 6-6 Challenge Investigating Darts A dart is another type of geometric figure. Examples of darts are shown in the table. Darts Not Darts 1. Based on your observations of the figures in the table, write a definition of the term dart. ________________________________________________________________________________________ ________________________________________________________________________________________ 2. Describe the similarities and differences between the properties of darts and kites. ________________________________________________________________________________________ ________________________________________________________________________________________ 3. Find the values of x and y so that GHJK is a dart. ________________________________________ 4. Make a conjecture about the line that contains the diagonal from the tip to the vertex of the tail of a dart and the diagonal that joins the fin angles. ________________________________________ ________________________________________ ________________________________________ 5. Make a conjecture about the two triangles formed by the tip-to-tail diagonal. ________________________________________________________________________________________ Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor. A61 Holt Geometry