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Chapter 6 Section 4

Chapter 6 Section 4. Squares and Rhombi. Warm-Up. Quadrilateral SURE is a rectangle. Find the values of v, w, x, y, and z . 1) SR = 4v + 2, EU = 6V -8 2) EC = 2w + 3, CU = 3w – 1 3) m<1 = x, m<2 = 2x 4) UR = 6y -7, SE = 4y 5) m<1 = 2z, m<3 = 8z . S. U. 2. C. 3. 1. E. R.

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Chapter 6 Section 4

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  1. Chapter 6Section 4 Squares and Rhombi

  2. Warm-Up • Quadrilateral SURE is a rectangle. Find the values of v, w, x, y, and z. • 1) SR = 4v + 2, EU = 6V -8 • 2) EC = 2w + 3, CU = 3w – 1 • 3) m<1 = x, m<2 = 2x • 4) UR = 6y -7, SE = 4y • 5) m<1 = 2z, m<3 = 8z S U 2 C 3 1 E R

  3. Warm-Up • Quadrilateral SURE is a rectangle. Find the values of v, w, x, y, and z. • 1) SR = 4v + 2, EU = 6v -8 • Diagonals of a rectangle are congruent. • SR = EU • 4v + 2 = 6v – 8 • 2 = 2v – 8 • 10 = 2v • 5 = v • 2) EC = 2w + 3, CU = 3w – 1 • Diagonals of a rectangle bisect each other. • EC = CU • 2w + 3 = 3w – 1 • 3 = w – 1 • 4 = w S U 2 C 3 1 E R

  4. Warm-Up Quadrilateral SURE is a rectangle. Find the values of v, w, x, y, and z. • 3) m<1 = x, m<2 = 2x • Triangle SCE is an isosceles triangle. So <2 is congruent to <SEC. Also <SEC + <1 = 90. • 90 = m<SEC + m<1 • 90 = m<2 + m<1 • 90 = 2x + x • 90 = 3x • 30 = x • 4) UR = 6y -7, SE = 4y -1 • Opposite sides are congruent. • UR = SE • 6y – 7 = 4y - 1 • -7 = -2y – 1 • -6 = -2y • 3 = y S U 2 C 3 1 E R • 5) m<1 = 2z, m<3 = 8z • Triangle ECR is an isosceles triangle. So <1 is congruent to <CRE. • 180 = m<1 + m<3 + m<CRE • 180 = 2z + 8z + 2z • 180 = 12z • 15 = z

  5. Vocabulary Rhombus-A quadrilateral with four congruent sides. Theorem 6-11- The diagonals of a rhombus are perpendicular. Theorem 6-12- If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

  6. Vocabulary cont. Theorem 6-13- Each diagonal of a rhombus bisects a pair of opposite angles. Square-A quadrilateral with four right angles and four congruent sides.

  7. Example 1:Use rhombus DLMP with DM = 26 to determine whether each statement is true or false. Justify your answers. D L O • A) OM = 13 • True; Since a rhombus is a parallelogram, the diagonals bisect each other. Thus, if DM = 26, OM = ½(26) = 13. • B) MD is congruent to PL • False; The diagonals of a rhombus are not congruent unless the rhombus is a square. • C) m<DLO = m<LDO • False; These angles of the rhombus are complementary. P M

  8. Example 2: Use rhombus BCDE and the given information to find each missing value. B C 1 2 3 • A) If m<1 = 2x + 20 and m<2 = 5x – 4, find the value of x. • The diagonals of a rhombus bisect a pair of opposite angles. • m<1 = m<2 • 2x + 20 = 5x – 4 • 20 = 3x – 4 • 24 = 3x • 8 = x • B) If BD = 15, find BF • Since a rhombus is a parallelogram, the diagonals bisect each other. Thus, if BD = 15, BF = ½(BD) = 7.5 F E D C) If m<3 = y2 + 26, find y. The diagonals of a rhombus are perpendicular. 90 = m<3 90 = y2+ 26 64 = y2 8 = y

  9. Example 3: Use rhombus RSTV and the given information to find each missing value. S T • A) If m<RST = 67, find m<RSW. • The diagonals of a rhombus bisect a pair of opposite angles. • m<RST = m<RSW + m<WST • m<RST = 2(m<RSW) • 67 = 2(m<RSW) • 33.5 = m<RSW • B) Find m<SVT if m<STV = 135. • Consecutive angles are supplementary. • 180 = m<STV + m<TVR • 180 = m<STV + m<SVT + m<SVR • 180 = m<STV + 2(m<SVT) • 180 = 135 + 2(m<SVT) • 45 = 2(m<SVT) • 22.5 = m<SVT W R V C) If m<SWT = 2x + 8, find the value of x. The diagonals of a rhombus are perpendicular. 90 = m<SWT 90 = 2x + 8 82 = 2x 41 = x

  10. Example 4: Determine whether the quadrilateral PARK is a parallelogram, rhombus, a rectangle, or a square for P(-1, 0), A(1, -1), R(2, 1), and K(0, 2). First find if it is a parallelogram by finding the slope of each side. The slope formula is m = (y2 – y1)/(x2 – x1) PA m = (y2 – y1)/(x2 – x1) m = (0 – -1)/(-1 – 1) m = (0 + 1)/(-1 -1) m = (1)/(-2) m = -1/2 AR m = (y2 – y1)/(x2 – x1) m = (-1 – 1)/(1 – 2) m = -2/-1 m = 2 PK m = (y2 – y1)/(x2 – x1) m = (0– 2)/(-1– 0) m = -2/(-1) m = 2 RK m = (y2 – y1)/(x2 – x1) m = (1 – 2)/(2 – 0) m = -1/2 Since opposite sides are parallel, quad PARK is a parallelogram. Also since PA is perpendicular to AR, AR is perpendicular to RK, RK is perpendicular to PK, and PK is perpendicular to PA, quad PARK is a rectangle.

  11. Example 4 cont: Determine whether the quadrilateral PARK is a parallelogram, rhombus, a rectangle, or a square for P(-1, 0), A(1, -1), R(2, 1), and K(0, 2). Now check if all the sides are congruent. The distance formula is d=√((x2 – x1)2 + (y2 – y1)2) PA d=√((x2 – x1)2 + (y2 – y1)2) d=√((-1 – 1)2+ (0 – -1)2) d=√((-1 – 1)2+ (0 + 1)2) d=√((-2)2+ (1)2) d=√(4+ 1) d=√(5) AR d=√((x2 – x1)2 + (y2 – y1)2) d=√((1 – 2)2+ (-1 – 1)2) d=√((-1)2+ (-2)2) d=√(1 + 4) d=√(5) PK d=√((x2 – x1)2 + (y2 – y1)2) d=√((-1 – 0)2+ (0 – 2)2) d=√((-1)2+ (-2)2) d=√(1 + 4) d=√(5) RK d=√((x2 – x1)2 + (y2 – y1)2) d=√((2 – 0)2+ (1 – 2)2) d=√((2)2+ (-1)2) d=√(4+ 1) d=√(5) Since all the sides are congruent, quad PARK is a square. Since it is a square it is also a rhombus. Quadrilateral PARK is a parallelogram, a rectangle, a square, and a rhombus.

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