The parallel sides of a trapezoid are called bases. VOCABULARY 6-6 TRAPEZOIDS and KITES A trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides of a trapezoid are called bases. The nonparallel sides are called legs. The two s that share a base of a trapezoid are called base angles. A trapezoid has two pairs of base s. An isosceles trapezoid is a trapezoid with legs that are ≅. A kite is a quadrilateral with two pairs of consecutive sides ≅ and no opposite sides ≅. A midsegment of a trapezoid is the segment that joins the midpoints of its legs.
The legs of a trapezoid are the non-parallel sides. 6-6 TRAPEZOIDS and KITES Word or Word Phrase Defintion Picture or Example trapezoid A trapezoid is a quadrilateral with one pair of parallel sides. legs of a trapezoid 𝑻𝑷 𝒐𝒓 𝑹𝑨 bases of a trapezoid 𝑻𝑹 𝒐𝒓 𝑷𝑨 isosceles trapezoid An isosceles trapezoid is a trapezoid with legs that are congruent. base angles A and B or C and D kite A 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.
∴ In each region, the s are either supplementary or ≅. 6-6 TRAPEZOIDS and KITES OBJECTIVES: 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. 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 ≅. ∴ In each region, the s are either supplementary or ≅.
If a quadrilateral is an isosceles trapezoid, 6-6 TRAPEZOIDS and KITES OBJECTIVES: To verify and use the properties of trapezoids and kites. Theorem 6-20 If a quadrilateral is an isosceles trapezoid, then each pair of base angles is congruent. Theorem 6-21 If a quadrilateral is an isosceles trapezoid, then its diagonals are congruent.
If a quadrilateral is a trapezoid, then 6-6 TRAPEZOIDS and KITES OBJECTIVES: To verify and use the properties of trapezoids and kites. Theorem 6-22 If 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.
Concept Summary - Relationships Among Quadrilaterals 6-6 TRAPEZOIDS and KITES OBJECTIVES: To verify and use the properties of trapezoids and kites. Theorem 6-23 If a quadrilateral is a kite, then its diagonals are perpendicular. Concept Summary - Relationships Among Quadrilaterals
𝑫𝑬 ‖ 𝑪𝑭 , so same-side interior s are supplementary. 6-6 TRAPEZOIDS and KITES OBJECTIVE: To verify and use the properties of 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? 𝒎𝑷=𝒎𝑸=𝟕𝟒 𝒎𝑺=𝟏𝟎𝟔 b. In Problem 1, if CDEF were not an isosceles trapezoid, would C and D still be supplementary? Explain. 𝐘𝐞𝐬; 𝑫𝑬 ‖ 𝑪𝑭 , so same-side interior s are supplementary.
Finding Angle Measures in Isosceles Trapezoids 6-6 TRAPEZOIDS and KITES OBJECTIVE: To verify and use the properties of trapezoids and kites. Finding Angle Measures in Isosceles Trapezoids A 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? 24 acute angles measure 𝟕𝟖 obtuse angles measure 𝟏𝟎𝟐 102 102 78 78 Q: What is the measure of each one of the 15 s meeting at the center? 𝟑𝟔𝟎° 𝟏𝟓 =𝟐𝟒°
Investigating the Diagonals of Isosceles Trapezoids 6-6 TRAPEZOIDS and KITES OBJECTIVE: To verify and use the properties of trapezoids and kites. Investigating the Diagonals of Isosceles Trapezoids Choose 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. 𝑷𝑹=𝑸𝑺, 𝐭𝐡𝐮𝐬 𝑷𝑹 ≅ 𝑸𝑺. This supports the conjecture that if a quadrilateral is an isosceles trapezoid, then the diagonals are congruent.
b. How many midsegments can a triangle have? 6-6 TRAPEZOIDS and KITES OBJECTIVE: To verify and use the properties of trapezoids and kites. Using the Midsegment of a Trapezoid a. 𝑴𝑵 is the midsegment of trapezoid PQRS. What is x? What is MN? 𝒙=𝟔 𝑴𝑵=23 b. How many midsegments can a triangle have? How many midsegments can a trapezoid have? Explain. 𝟑 𝟏 A has 3 midsegments joining any pair of the side midpoints. A trapezoid has 1 midsegment joining the midpoints of the two legs.
Finding Angle Measures in Kites 6-6 TRAPEZOIDS and KITES OBJECTIVE: 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°
1. What are the measures of the numbered angles? 6-6 TRAPEZOIDS and KITES OBJECTIVE: To verify and use the properties of trapezoids and kites. 1. What are the measures of the numbered angles? 𝒎𝟏=𝟕𝟖° 𝒎𝟏=𝟗𝟒° 𝒎𝟐=𝟗𝟎° 𝒎𝟐=𝟏𝟑𝟐° 𝒎𝟑=𝟏𝟐° 2. Quadrilateral WXYZ is an isosceles trapezoid. Are the two trapezoids formed by drawing midsegment QR isosceles trapezoids? Explain. Yes, the midsegment is ‖ to both bases and bisects each of the two congruent legs.
6-6 TRAPEZOIDS and KITES OBJECTIVE: To verify and use the properties of trapezoids and kites. 3. Find the length of the perimeter of trapezoid LMNP with midsegment 𝑄𝑅 . Solve for PN : 7 𝑸𝑹= 𝟏 𝟐 𝑳𝑴+𝑷𝑵 𝟐𝟓= 𝟏 𝟐 𝟏𝟔+𝑷𝑵 8 𝟓𝟎= 16 + PN 𝟑𝟒= PN Perimeter of 𝑳𝑴𝑵𝑷=𝟐 𝟖 +𝟏𝟔+𝟐 𝟕 +𝟑𝟒 Perimeter of 𝑳𝑴𝑵𝑷=𝟖𝟎
No. No, a kite’s opposite sides are not ‖ or ≅ . 6-6 TRAPEZOIDS and KITES OBJECTIVE: To verify and use the properties of trapezoids and kites. 4. Vocabulary Is a kite a parallelogram? Explain. No, a kite’s opposite sides are not ‖ or ≅ . 5. Analyze Mathematical Relationships (1)(F) How is a kite similar to a rhombus? How is it different? Explain. Similar: Their diagonals are ⏊. Different: Only one diagonal of a kite bisects opposite s; a rhombus has all sides ≅. 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 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.