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Geometry 300 6.6-Trapezoids Name_________________________ Chapter 6 Day 7 Objectives: By the end of this lesson, you will be able to… Recognize the properties of trapezoids. Use the properties of trapezoids in proofs and other problems. Key Definitions: base Trapezoid: A quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases. leg leg The non parallel sides are called legs. Base angles are the two consecutive angles whose common side is a base. Name the pairs of base angles and the common base they share. base A trapezoid is an Isosceles Trapezoid if its legs are ________________________ Theorem #1: Both pairs of base angles of an isosceles trapezoid are _________________________ What about ∠𝐴 𝑎𝑛𝑑 ∠𝐵? They are _____________________, as are ∠𝐶 𝑎𝑛𝑑 ∠𝐷. Example 1: An isosceles trapezoid LAKE has A = 76. Find the measures of the other 3 angles. Example 2: Find the perimeter of isosceles trapezoid ABCD if the bases AB and CD are 12 and 22 respectively and the height is 12. A D B C Geometry 300 6.6-Trapezoids Name_________________________ Chapter 6 Day 7 Given: Trapezoid NYPD with vertices N(2, -9), Y(-1, 1), P(3, 8), D(10, 5). Step 1: Graph the trapezoid and find the slopes of all four sides. mNY mYP mPD mND Which pair of sides are the bases? Why? Step 2: Determine the lengths of bases. Are the bases congruent? Step 3: Determine the coordinates for the midpoints of both legs, plot them, and then connect them with a segment. This is called the median. Label the endpoints of the median, A and B. Step 4: Find the slope of median AB Geometry 300 6.6-Trapezoids Name_________________________ Chapter 6 Day 7 What do you notice about the slope of AB and the other sides of the trapezoid? Step 5: Find the length of median AB Step 6: Calculate YP + ND. What conclusion can you make about sum of the measures of the bases (YP + ND) and the length of the median? This gives us our last theorem… Theorem 3: The median of a trapezoid is parallel to the bases and its measure is __________________ the sum of the measures of the bases. Example 3: Given trapezoid TRIP, with median EF, find the value of x. Geometry 300 6.6-Trapezoids Name_________________________ Chapter 6 Day 7 Example 4: ABCD is an isosceles trapezoid with bases AD and BC. Use the figure and the given information to solve each problem. a) If BA = 9, find CD b)If AC = 21, find BD b) If AC = 4y – 5, and BD = 2y + 3, find AC and BD. c) If mBAD 123, find mCBA d)If mADC 105, find mDAB Example 5: Find the length of the median of a trapezoid with vertices at (1, 0), (3, -1), (6, 2), and (7, 6). Practice 1) Given trapezoid TRAP with median EZ, a) Find EZ if TR = 42 and PA = 30 b) Find TR if EZ = 17 and PA = 14 Geometry 300 6.6-Trapezoids Name_________________________ Chapter 6 Day 7 2) In isosceles trapezoid SNAP with median OH, find all the missing angle measures: m∠1 = ______ m∠2 = ______ m∠3 = ______ m∠4 = ______ m∠5 = ______ m∠6 = ______ m∠7 = ______ 3) If possible, draw a trapezoid that has the following characteristics. If the trapezoid cannot be drawn, explain why. a) Three congruent sides b) Congruent bases c) a leg longer than both bases d) bisecting diagonals e) Two right angles f) four acute angles g) One pair of opposite angles congruent 4) Given trapezoid GHIJ with median KL, find the value of x Geometry 300 6.6-Trapezoids Name_________________________ Chapter 6 Day 7 5) Given 𝑚∠𝐸 = (2𝑦 2 − 25)° and 𝑚∠𝐻 = (𝑦 2 + 24)° find the value of y so that EFGH is isosceles. 6) Find the measure of each angle. 7) PQRS is an isosceles trapezoid with bases PS and QR. Use the figure and the given information to solve each problem. a. If PS = 20 and QR = 14, find TV b. If QR = 14.3, and TV = 23.2, find PS c. If TV = x + 7, and PS + QR = 5x + 2, find x. d. If mRVT 57 , find mQTV . Geometry 300 6.6-Trapezoids Name_________________________ Chapter 6 Day 7 8) In the accompanying figure, isosceles trapezoid ABCD has bases of length 5 and 11 and an altitude of length 4. Find AB. 5 B C 4 A 11 D 10) Show that QRST is a trapezoid if Q(-1,-2), R(6,-2), S (10,1), T (7,4). a. What are the coordinates of the median of QRST? b. Prove that the median of the trapezoid is parallel to the bases c. Prove that the median has a length equal to the average of the base lengths.