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ParallelogramsParallelograms 5-1
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EXAMPLE 1 Use properties of parallelograms ALGEBRA Find the values of x and y. ABCD is a parallelogram by the definition of a parallelogram. Use Theorem 8.3 to find the value of x. Opposite sides of a are. AB = CD Substitute x + 4 for AB and 12 for CD. x + 4 = 12 Subtract 4 from each side. x =8 By Theorem 8.4, A C, or m A = m C. So, y ° = 65°. In ABCD, x = 8 and y = 65. ANSWER
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GUIDED PRACTICE for Example 1 Find FG and m G. 1. SOLUTION Opposite sides of a are. FG =HE x =8 By Theorem 8.4, E G, or m E = m G. So, G ° = 60°. In FEHG, FG = 8 and m G = 60°. ANSWER
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GUIDED PRACTICE for Example 1 Find the values of x and y. 2. SOLUTION Opposite sides of a are. JK =ML Substitute 18 for JK and y + 3 for ML. 18 =y + 3 Subtract 3 from each side. 15 =y By Theorem 8.4, J L, or m J = m L. Substitute 2x =50 Divide 2 from each side. x =25 In JKLM, x = 25 and y = 15. ANSWER
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EXAMPLE 2 Use properties of parallelograms So, m ADC + m BCD = 180°. Because m ADC = 110°, m BCD =180° –110° = 70°. SOLUTION By Theorem 8.5, the consecutive angle pairs in ABCD are supplementary. Desk Lamp As shown, part of the extending arm of a desk lamp is a parallelogram. The angles of the parallelogram change as the lamp is raised and lowered. Find m BCD when m ADC = 110°.
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EXAMPLE 3 Standardized Test Practice SOLUTION By Theorem 8.6, the diagonals of a parallelogram bisect each other. So, P is the midpoint of diagonals LN and OM. Use the Midpoint Formula. The correct answer is A. ANSWER = Coordinates of midpoint P of OM 4 + 0 2 7 + 0 2, ( ) = 7 2, 2 ( )
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GUIDED PRACTICE for Examples 2 and 3 NM 3. SOLUTION By Theorem 8.6, the diagonals of a parallelogram bisect each other. So, N is the midpoint of diagonals KM. KN =NM Substitute 2 =NM Find the indicated measure in JKLM.
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GUIDED PRACTICE for Examples 2 and 3 Find the indicated measure in JKLM. KM 4. SOLUTION By theorem 8.6 KM =KN + NM Substitute KM =2 + 2 Add KM =4
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GUIDED PRACTICE for Examples 2 and 3 m JML 5. SOLUTION By Theorem 8.5, the consecutive angle pairs in JKLM are supplementary. So, m KJM + m JML = 180°. Because m KJM = 110°, m JML =180° –110° = 70°. Find the indicated measure in JKLM.
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GUIDED PRACTICE for Examples 2 and 3 Find the indicated measure in JKLM. m KML 6. SOLUTION m JML =m KMJ + m KNL Substitute 70° =30° + m KML Subtract 40° =m KML
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EXAMPLE 2 Identify a parallelogram ARCHITECTURE The doorway shown is part of a building in England. Over time, the building has leaned sideways. Explain how you know that SV = TU. SOLUTION In the photograph, ST UV and ST UV. By Theorem 8.9, quadrilateral STUV is a parallelogram. By Theorem 8.3, you know that opposite sides of a parallelogram are congruent. So, SV = TU.
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EXAMPLE 3 Use algebra with parallelograms ALGEBRA For what value of x is quadrilateral CDEF a parallelogram? SOLUTION By Theorem 8.10, if the diagonals of CDEF bisect each other, then it is a parallelogram. You are given that CN EN. Find x so that FN DN.
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EXAMPLE 3 Use algebra with parallelograms Set the segment lengths equal. FN = DN Substitute 5x –8 for FN and 3x for DN. 5x – 8 = 3x3x Subtract 3x from each side. 2x – 8 = 0 Add 8 to each side. 2x = 8 Divide each side by 2. x = 4 When x = 4, FN = 5(4) –8 = 12 and DN = 3(4) =12. Quadrilateral CDEF is a parallelogram when x = 4. ANSWER
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GUIDED PRACTICE for Examples 2 and 3 What theorem can you use to show that the quadrilateral is a parallelogram? 2. In the graphic, two opposite sides are equal, i.e, 30 m each and parallel, Therefore, the quadrilateral is a parallelogram. By theorem 8.9. ANSWER
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GUIDED PRACTICE for Examples 2 and 3 What theorem can you use to show that the quadrilateral is a parallelogram? 3. Two pairs of opposite sides are equal. Therefore, the quadrilateral is a parallelogram. By theorem 8.7 ANSWER
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GUIDED PRACTICE for Examples 2 and 3 What theorem can you use to show that the quadrilateral is a parallelogram? 4. By theorem 8.8, if the opposite angles are Congruent, the quadrilateral is a parallelogram. ANSWER
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GUIDED PRACTICE for Examples 2 and 3 For what value of x is quadrilateral MNPQ a parallelogram? Explain your reasoning. 5. SOLUTION [ Diagonals in bisect each other ] By Theorem 8.6 2x =10 – 3x Add 3x to each side 5x =10 Divide each side by 5 x =2
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EXAMPLE 2 Identify a parallelogram ARCHITECTURE The doorway shown is part of a building in England. Over time, the building has leaned sideways. Explain how you know that SV = TU. SOLUTION In the photograph, ST UV and ST UV. By Theorem 8.9, quadrilateral STUV is a parallelogram. By Theorem 8.3, you know that opposite sides of a parallelogram are congruent. So, SV = TU.
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EXAMPLE 3 Use algebra with parallelograms ALGEBRA For what value of x is quadrilateral CDEF a parallelogram? SOLUTION By Theorem 8.10, if the diagonals of CDEF bisect each other, then it is a parallelogram. You are given that CN EN. Find x so that FN DN.
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EXAMPLE 3 Use algebra with parallelograms Set the segment lengths equal. FN = DN Substitute 5x –8 for FN and 3x for DN. 5x – 8 = 3x3x Subtract 3x from each side. 2x – 8 = 0 Add 8 to each side. 2x = 8 Divide each side by 2. x = 4 When x = 4, FN = 5(4) –8 = 12 and DN = 3(4) =12. Quadrilateral CDEF is a parallelogram when x = 4. ANSWER
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GUIDED PRACTICE for Examples 2 and 3 What theorem can you use to show that the quadrilateral is a parallelogram? 2. In the graphic, two opposite sides are equal, i.e, 30 m each and parallel, Therefore, the quadrilateral is a parallelogram. By theorem 8.9. ANSWER
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GUIDED PRACTICE for Examples 2 and 3 What theorem can you use to show that the quadrilateral is a parallelogram? 3. Two pairs of opposite sides are equal. Therefore, the quadrilateral is a parallelogram. By theorem 8.7 ANSWER
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GUIDED PRACTICE for Examples 2 and 3 What theorem can you use to show that the quadrilateral is a parallelogram? 4. By theorem 8.8, if the opposite angles are Congruent, the quadrilateral is a parallelogram. ANSWER
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GUIDED PRACTICE for Examples 2 and 3 For what value of x is quadrilateral MNPQ a parallelogram? Explain your reasoning. 5. SOLUTION [ Diagonals in bisect each other ] By Theorem 8.6 2x =10 – 3x Add 3x to each side 5x =10 Divide each side by 5 x =2
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