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CHAPTER 5 Parallelograms, Trapezoids, Medians, and Midpoints
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5.13 Applying principles 5 and 6 to the medians of a triangle Find x and y in each part of Fig. 5-27.
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(a) Since AM x
2 63
MB, CM is the median to hypotenuse AB. Hence by Principle 5, 3x 60. 8 and y
20 and 3y
20. Thus,
and y
1 (b) BD and AF are medians of nABC. Hence by Principle 6, x = 2 (16)
3(7)
(c) CD is the median to hypotenuse AB; hence by Principle 5, CD CD and AF are medians of nABC; hence by Principle 6, x
15. 5 and y
2 3 (15)
1 3 (15)
5.14 Proving a midpoint problem
Given: Quadrilateral ABCD E, F, G, and H are midpoints of AD, AB, BC, and CD, respectively. To Prove: EFGH is a ~. Plan: Prove EF and GH are congruent and parallel.
PROOF:
1. 2. 3. 4. 5. 6.
Statements Draw BD. E, F, G, and H are midpoints. EF i BD and GH i BD 1 1 EF 2 BD and GH 2 BD EF i GH EF > GH EFGH is a ~.
1. 2. 3. 4. 5. 6.
Reasons A segment may be drawn between any two points. Given A line segment joining the midpoints of two sides of a n is parallel to the third side and equal in length to half the third side. Two lines parallel to a third line are parallel to each other. Segments of the same length are congruent. If two sides of a quadrilateral are > and i, the quadrilateral is a ~.
SUPPLEMENTARY PROBLEMS
5.1. Find x and y in each part of Fig. 5-28. (5.1)
Fig. 5-28
CHAPTER 5 Parallelograms, Trapezoids, Medians, and Midpoints
5.2. 5.3. Prove that if the base angles of a trapezoid are congruent, the trapezoid is isosceles.
(5.2)
Prove that (a) the diagonals of an isosceles trapezoid are congruent; (b) if the nonparallel sides AB and CD of an isosceles trapezoid are extended until they meet at E, triangle ADE thus formed is isosceles. (5.2) Name the parallelograms in each part of Fig. 5-29. (5.4)
Fig. 5-29
State why ABCD in each part of Fig. 5-30 is a parallelogram.
(5.5)
Fig. 5-30
Assuming ABCD in Fig. 5-31 is a parallelogram, find x and y if (a) AD (b) AB (c) m/A (d) m/A 5x, AB 2x, BC 4y 2x, CD 3y y, perimeter 7x 84 5y 10
(5.3)
8, CD
25, AD x y
60, m/C 10x
2y, m/D 15, m/C
3x, m/B
Fig. 5-31
Fig. 5-32
Assuming ABCD in Fig. 5-32 is a parallelogram, find x and y if (a) AE (b) AE (c) AE (d) AE x y, EC 20, BE x x y, ED 9 7, ED 24 x y 8
(5.3)
x, EC 3x 2x
4y, BE 4, EC y, AC x
2y, ED 2y
12, BE x
30, BE
y, BD
CHAPTER 5 Parallelograms, Trapezoids, Medians, and Midpoints
Provide the proofs requested in Fig. 5-33. (5.6)
Fig. 5-33
Prove each of the following: (a) The opposite sides of a parallelogram are congruent (Principle 3). (b) If the opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram (Principle 8). (c) If two sides of a quadrilateral are congruent and parallel, the quadrilateral is a parallelogram (Principle 9). (d) The diagonals of a parallelogram bisect each other (Principle 6). (e) If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram (Principle 11).
5.10. Assuming ABCD in Fig. 5-34 is a rhombus, find x and y if
(5.7)
Fig. 5-34
(a) BC (b) AB (c) AB (d) AB (e) m/B ( f ) m/1
35, CD 43, AD 7x, AD x
8x 4x 3x
5, BD 3, BD 10, BC 2x 3x 5x y, BC
5y, m/C y y 12 2y
60 120
8, m/B
y, AD 130 , m/1 8x
10, m/A 4, m/D
29, m/2
y (5.8)
5.11. Provide the proofs requested in Fig. 5-35.
Fig. 5-35
CHAPTER 5 Parallelograms, Trapezoids, Medians, and Midpoints
5.12. Prove each of the following: (a) If the diagonals of a parallelogram are congruent, the parallelogram is a rectangle.
(5.9)
(b) If the diagonals of a parallelogram are perpendicular to each other, the parallelogram is a rhombus. (c) If a diagonal of a parallelogram bisects a vertex angle, then the parallelogram is a rhombus. (d) The diagonals of a rhombus divide it into four congruent triangles. (e) The diagonals of a rectangle are congruent. 5.13. Find x and y in each part of Fig. 5-36. (5.10)
Fig. 5-36
5.14. Find x and y in each part of Fig. 5-37.
(5.11)
Fig. 5-37
5.15. If MP is the median of trapezoid ABCD in Fig. 5-38 (a) Find m if b (b) Find b if b (c) Find b if b 23 and b 46 and m 51 and m 15. 41. 62.
(5.12)
Fig. 5-38
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