# 9. AD = 7; By the Parallelogram Opposite Sides Theorem (Thm. 7.3), AD = BC. 10. AE = 7; By the Parallelogram Diagonals Theorem (Thm. 7.6), AE = EC.

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1 3. Sample answer: Solve 5x = 3x + 1; opposite sides of a parallelogram are congruent; es; You could start b setting the two parts of either diagonal equal to each other b the Parallelogram Diagonals Theorem (Thm. 7.) Quiz (p. 3) x = 3 + x = 3 x =. (5 ) 1 = 3 1 = x = x = 5 x = x = 3. Interior angle = (1 ) 1 1 = 1 1 = 1 x + 3 = 3 x = 97 = 1 1 xterior angle = 3 1 = 3 In a regular decagon, the measure of each interior angle is 1 and the measure of each exterior angle is Interior angle = (15 ) 1 15 = 3 15 = 15 = xterior angle = 3 15 = In a regular 15-gon, the measure of each interior angle is 15 and the measure of each exterior angle is.. Interior angle = ( ) 1 = 39 = 15 = 1 xterior angle = 3 = 15 In a regular -gon, the measure of each interior angle is 15 and the measure of each exterior angle is Interior angle = ( ) 1 = 5 1 = 1, = 17 xterior angle = 3 = In a regular -gon, the measure of each interior angle is 17 and the measure of each exterior angle is.. CD = 1; B the Parallelogram Opposite Sides Theorem (Thm. 7.3), AB = CD. 9. AD = 7; B the Parallelogram Opposite Sides Theorem (Thm. 7.3), AD = BC. 1. A = 7; B the Parallelogram Diagonals Theorem (Thm. 7.), A = C. 11. BD = 1. =.; B the Parallelogram Diagonals Theorem (Thm. 7.), B = D. 1. m BCD = 1 ; B the Parallelogram Opposite Angles Theorem (Thm. 7.), m DAB = m BCD. 13. B the Parallelogram Consecutive Angles Theorem (Thm. 7.5), DAB and ABC are supplementar. So, m ABC = 1 1 =. 1. B the Parallelogram Consecutive Angles Theorem (Thm. 7.5), DAB and ADC are supplementar. So, m ADC = 1 1 =. 15. m ADB = 3 ; B the Alternate Interior Angles Theorem (Thm. 3.), m DBC = m ADB. 1. The quadrilateral is a parallelogram b the Opposite Sides Parallel and Congruent Theorem (Thm. 7.9). 17. The quadrilateral is a parallelogram b the Parallelogram Diagonals Converse (Thm. 7.1). 1. The quadrilateral is a parallelogram b the Parallelogram Opposite Angles Converse (Thm. 7.) Q T x R S Slope of QR ( ) = = + 3 ( 5) = = Slope of ST ( ) = = + = 7 1 = The slope of QR equals slope of ST, so QR ST. QR = (3 ( 5)) + ( ( )) = (3 + 5) + ( + ) = = = Because QR = ST =, QR ST. QR and ST are opposite sides that are both congruent and parallel. So, QRST is a parallelogram b the Opposite Sides Parallel and Congruent Theorem (Thm. 7.9). 5 Geometr Copright Big Ideas earning, C

2 . Z 5 W X 3 x Y Slope of WX 3 7 = 3 ( 3) = = = 3 Slope of XY = = = 3 Slope of YZ = 1 ( 3) 5 1 = = = 3 Slope of WZ 1 7 = 5 ( 3) = = = 3 Because the slope of WX equals the slope of YZ, WX YZ and because the slope of XY equals the slope of WZ, XY WZ. Because both pairs of opposite sides are parallel, WXYZ is a parallelogram b definition. 1. a. The stop sign is a regular octagon. b. ( ) 1 = 1 = 1 = 135 The measure of each interior angle is = 5 The measure of each exterior angle is 5.. a. b the Parallelogram Opposite Sides Theorem (Thm. 7.3). b the Parallelogram Opposite Sides Theorem (Thm. 7.3). b the Parallelogram Opposite Angles Theorem (Thm. 7.). b the Parallelogram Opposite Angles Theorem (Thm. 7.). b. Because QT RS and QT = RS, QRST is a parallelogram b the Opposite Sides Parallel and Congruent Theorem (Thm. 7.9). c. ST = 3 feet, because ST = QR b the Parallelogram Opposite Sides Theorem (Thm. 7.3). m QTS = 13, because m QTS = m QRS b the Parallelogram Opposite Angles Theorem (Thm. 7.). m TQR = 57, because TQR and QTS are consecutive interior angles and the are supplementar. So, m TQR = 1 13 = 57. Because TSR and TQR are opposite angles b the Parallelogram Opposite Angles Theorem (Thm. 7.), m TSR = xplorations (p. 37) 1. a. Check students work. b. Check students work. c. B D A C d. es; es; no; no; Because all points on a circle are the same distance from the center, AB A AC AD. So, the diagonals of quadrilateral BDC bisect each other, which means it is a parallelogram b the Parallelogram Diagonals Converse (Thm. 7.1). Because all angles of BDC are right angles, it is a rectangle. BDC is neither a rhombus nor a square because BD and C are not necessaril the same length as B and DC. e. Check students work. The quadrilateral formed b the endpoints of two diameters is a rectangle (and a parallelogram). In other words, a quadrilateral is a rectangle if and onl if its diagonals are congruent and bisect each other.. a. Check students work. b. A C D B c. es; no; es; no; Because the diagonals bisect each other, ABD is a parallelogram b the Parallelogram Diagonals Converse (Thm. 7.1). Because B = BD = AD = A, ABD is a rhombus. ABD is neither a rectangle nor a square because its angles are not necessaril right angles. d. Check students work. A quadrilateral is a rhombus if and onl if the diagonals are perpendicular bisectors of each other. 3. Because rectangles, rhombuses, and squares are all parallelograms, their diagonals bisect each other b the Parallelogram Diagonals Theorem (Thm. 7.). The diagonals of a rectangle are congruent. The diagonals of a rhombus are perpendicular. The diagonals of a square are congruent and perpendicular.. es; no; es; no; RSTU is a parallelogram because the diagonals bisect each other. RSTU is not a rectangle because the diagonals are not congruent. RSTU is a rhombus because the diagonals are perpendicular. RSTU is not a square because the diagonals are not congruent. 5. A rectangle has congruent diagonals that bisect each other. Copright Big Ideas earning, C Geometr 51

3 51. a. Given is an isosceles trapezoid., Prove STATNTS 1. is an isosceles trapezoid,, RASONS 1. Given.. Isosceles Trapezoid Base Angles Theorem (Thm. 7.1) Reflexive Propert of Congruence (Thm..1).. SAS Congruence Theorem (Thm. 5.5) Corresponding parts of congruent triangles are congruent. b. If the diagonals of a trapezoid are congruent, then the trapezoid is isosceles. et be a trapezoid, and. Construct line segments through and perpendicular to as shown below. A B Because, A and B are right angles, so BA is a rectangle and A B. Then B A b the H Congruence Theorem (Thm. 5.9). So, B A. b the Reflexive Propert of Congruence (Thm..1). So, b the SAS Congruence Theorem (Thm. 5.5). Then, and the trapezoid is isosceles b the Isosceles Trapezoid Base Angles Converse (Thm. 7.15). 5. You are given, is the midpoint of, F is the midpoint of, G is the midpoint of, and H is the midpoint of. B the definition of midsegment, F is a midsegment of, FG is a midsegment of, GH is a midsegment of, and H is a midsegment of. B the Triangle idsegment Theorem (Thm..), F, FG, GH, and H. You know that F GH and FG H b the Transitive Propert of Parallel ines (Thm. 3.9). FGH is a parallelogram b the definition of a parallelogram. B the Triangle idsegment Theorem (Thm..), F = 1, FG = 1, GH = 1, and H = 1. You can conclude = b the definition of congruent segments. Then b the Substitution Propert of qualit, FG = 1 and H = 1. It follows from the Transitive Propert of qualit that F = FG = GH = H. Then b the definition of congruent segments, F FG GH H.Therefore, b the definition of a rhombus, FGH is a rhombus. aintaining athematical Proficienc 53. Sample answer: A similarit transformation that maps the blue preimage to the green image is a translation 1 unit right followed b a dilation with a scale factor of. 5. Sample answer: A similarit transformation that maps the blue preimage to the green image is a reflection in the -axis followed b a dilation with a scale factor of What Did You earn? (p. 7) 1. The are congruent base angles of congruent isosceles triangles, DF and DGF.. If one tpe of quadrilateral is under another in the diagram, then a quadrilateral from the lower categor will alwas fit into the categor above it. 3. Find the difference between the length of the midsegment and the length of the given base, and then either add or subtract that amount to the midsegment to find the other base. Chapter 7 Review (pp. 1) 1. The sum of the measures of the interior angles of a regular 3-gon is (3 ) 1 = 1 = 5. The measure of each interior angle is (3 ) 1 = 1 = = 1. The measure of each exterior angle is 3 3 = 1.. The sum of the measures of the interior angles is ( ) 1 = 1 = x = x = 7 x = The sum of the measures of the interior angles is (7 ) 1 = 5 1 = 9. x x = 9 3x + 5 = 9 3x = x = x x = 3 13x + 15 = 3 13x = 195 x = a + 11 = 1 a = 79 b = 11. a 1 = 1 a = (b + 1) = 13 b = 7 Copright Big Ideas earning, C Geometr 5

4 7. c + 5 = 11 c = d + = 1 d = 1. idpoint of TR : ( , ) ( =, ) = (, 1) idpoint of QS : ( ( 3), ) = (, ) = (, 1) The coordinates of the intersection of the diagonals are (, 1). 9. x Slope of = 3 ( 3) = = Starting at, go down units and right 1 unit. So, the coordinates of are (, ). 1. Parallelogram Opposite Sides Converse (Thm. 7.7) 11. Parallelogram Diagonals Converse (Thm. 7.1) 1. Parallelogram Opposite Angles Converse (Thm. 7.) 13. B the Parallelogram Opposite Sides Converse (Thm. 7.7): x + 7 = 1x 1 x + 7 = 1 x = x = = = 11 = 1 = The quadrilateral is a parallelogram when x = 1 and =. 15. Z W Y X x Slope of WX = ( 1) = 3 Slope of XY = 1 = 1 = Slope of YZ = 1 = 3 = 3 Slope of WZ = ( 1) = + 1 = 1 = The slope of WX equals the slope of YZ, so WX YZ. The slope of XY equals the slope of WZ, so XY WZ. Because both pairs of opposite sides are parallel, WXYZ is a parallelogram b definition. 1. The special quadrilateral is a rhombus because it has four congruent sides. 17. The special quadrilateral is a parallelogram because it has two pairs of parallel sides. 1. The special quadrilateral is a square because it has four congruent sides and four right angles. 19. WY = XZ x + 3 = 3x 5x + 3 = 5x = x = 1 WY = = + 3 = 1 XZ = 3 1 = 3 = 1 1. B the Parallelogram Diagonals Converse (Thm. 7.1): x = x = x = x The quadrilateral is a parallelogram when x =. Geometr Copright Big Ideas earning, C

5 . x Slope of = 5 9 = = 1 Slope of = 9 7 = = Slope of = 7 3 = = 1 Slope of = 3 5 = = 1 Because the product of the two slopes is ( ( ) = 1, ) there are four right angles (,,, and ). So, quadrilateral is a rectangle. = (5 9 ) + ( ) = () + = 1 + = = 5 = 5 = (9 7 ) + ( ) = + = + 1 = = 5 = 5 Because =, and is a rectangle, which is also a parallelogram, opposite sides are equal. So, = = =. B the definitions of a rhombus and a square, quadrilateral is also a rhombus and a square. 1. m Z = m Y = 5 m X = 1 5 = 1 m W = m X = 1. The length of the midsegment is 1 ( ) = 1 5 = x = 3 7x + 55 = 3 7x = 15 x = 15 The two congruent angles are es; B the Isosceles Trapezoid Base Angles Converse (Thm. 7.15), if a trapezoid has a pair of congruent base angles, then the trapezoid is isosceles.. The quadrilateral is a trapezoid because it has exactl one pair of parallel sides. 7. The quadrilateral is a rhombus because it has four congruent sides.. The quadrilateral is a rectangle because it has four right angles. Chapter 7 Test (p. 11) 1. The diagonals bisect each other, so r = and s = In a parallelogram, opposite angles are equal. So, b = 11. Consecutive interior angles are supplementar, so a = 1 11 = In a parallelogram, opposite sides are congruent. So, p = 5 and q = + 3 = 9.. If consecutive interior angles are supplementar, then the lines that form those angles are parallel. So, the quadrilateral is a trapezoid. 5. The quadrilateral is a kite because it has two pairs of consecutive congruent sides, but opposite sides are not congruent.. The quadrilateral is an isosceles trapezoid because it has congruent base angles. 7. 3x + 5(x + 7) = 3 3x + 1x + 35 = 3 13x + 35 = 3 13x = 35 x = 5 x + 7 = = = 57 The measures of the exterior angles of the octagon are 5, 5, 5, 57, 57, 57, 57, and x = (1 ) + ( ) = + = + 1 = = 5 = 5 = ( ) + (1 ) = + = 1 + = = 1 5 = 5 idsegment = 1 ( + ) = 1 ( ) = 1 ( 5 ) = 3 5 Copright Big Ideas earning, C Geometr 7

6 . 1 S R Q 17. Design AB = DC and AD = BC, then ABCD is a parallelogram. 1. Sample answer: 1 1 P Slope of RP = = 1 = undefined Slope of SQ = 9 1 = = So, RP SQ, and quadrilateral PQRS is a rhombus. 9. es; the diagonals bisect each other. 1. no; and might not be parallel. x 11. es; m Y = 3 ( ) = 3 3 =. Because opposite angles are congruent, the quadrilateral is a parallelogram. 1. If one angle in a parallelogram is a right angle, then consecutive angles are supplementar. So, the parallelogram is a rectangle. 13. Show that a quadrilateral is a parallelogram with four congruent sides and four right angles, or show that a quadrilateral is both a rectangle and a rhombus. 1. x a. Slope of = 3 = 3 Starting at, go down 3 units and left units. So, the coordinates of are (, ). b. idpoint of = ( + ( ) 3 + ( 1), ) = (, = (1, 1) ) idpoint of = ( +, + ) = (, = (1, 1) ) The coordinates of the intersection of the diagonals are (1, 1). 15. idsegment = 1 ( + 15) = 1 (1) = 1.5 The middle shelf will have a diameter of 1.5 inches. 1. (n ) 1 = (5 ) 1 = 3 1 = 5 n = 1 The measure of each interior angle of a regular pentagon is 1. 3 Chapter 7 Standards Assessment (p. 1) 1. Definition of parallelogram; Alternate Interior Angles Theorem (Thm. 3.); Reflexive Propert of Congruence (Thm..1); Definition of congruent angles; Angle Addition Postulate (Post. 1.); Transitive Propert of qualit; Definition of congruent angles; ASA Congruence Theorem (Thm. 5.1); Corresponding parts of congruent triangles are congruent.. In steps 1 and, an angle bisector is drawn for A and C. The point of intersection D is the incenter of ABC and the center of the inscribed circle. B constructing a perpendicular segment to AB from D, the radius of the circle is D. An inscribed circle touches each side of the triangle. 3. no; No theorem can be used to prove itself.. UV = ( 1) + ( 1) = ( 3) + 1 = = 1 VQ = ( ( 5)) + (1 ) = ( + 5) + ( 1) = = 1 QR = ( ( 5)) + (5 ) = ( + 5) + 3 = = 1 RS = ( 1 ()) + ( 5) = ( 1 + ) + 1 = = 1 ST = ( ( 1)) + (5 ) = (3) + ( 1) = = 1 TU = ( 1) + (5 ) = (1) + (3) = = 1 Perimeter: UV + VQ + QR + RS + ST + TU = = 1 The perimeter is 1. The polgon QRSTUV is equilateral. For the hexagon to be equiangular each angle must be ( ) 1 = 1 = 7 = 1. Because m Q = 9, the hexagon is not equiangular. So, it is not a regular polgon. Geometr Copright Big Ideas earning, C

7 5. Given AB D, BC F, AC > DF Prove m B > m B 7. Given QRST is a parallelogram, QS RT Prove QRST is a rectangle. Q R A C D Indirect Proof Step 1: Assume temporaril that m B m. Then it follows that either m B = m or m B < m. Step : If m B < m, then AC < DF b the Hinge Theorem (Theorem.1). If m B = m, then B. So, ABC DF b the SAS Congruence Theorem (Theorem 5.5) and AC = DF. Step 3: Both conclusions contradict the given statement that AC > DF. So, the temporar assumption that m B m cannot be true. This proves that m B > m.. Given BC AD, BC CB, AB DC Prove ABCD is an isosceles trapezoid. A B C STATNTS 1. BC AD, BC BC, AB DC D F RASONS 1. Given. ABCD is a trapezoid.. Definition of trapezoid 3. m BC = m CB, m AB = m DC. m AB + m BC = m ABC, m DC + m CB = m DCB 5. m AB + m BC = m AB + m BC. m AB + m BC = m DC + m CB 3. Definition of congruent angles. Angle Addition Postulate (Post. 1.) 5. Reflexive Propert of qualit. Substitution Propert of qualit 7. m ABC = m DCB 7. Transititve Propert of qualit. ABC DCB. Definition of congruent angles 9. ABCD is an isosceles trapezoid. 9. Isosceles Trapezoid Base Angles Converse (Thm. 7.15) T Sample answer: STATNTS 1. QS RT. QT RS, QR TS S RASONS 1. Given. Parallelogram Opposite Sides Theorem (Thm. 7.3) 3. TQR ΔSRQ 3. SSS Congruence Theorem (Thm. 5.). TQR SRQ. Corresponding parts of congruent triangles are congruent. 5. m TQR + m SRQ = 1. m TQR = m SRQ = m QTS = m RST = 1. m QTS = 9 ; m RST = 9 9. TQR, SRQ, QTS, and RST are right angles. 5. Parallelogram Consecutive Angles Theorem (Thm. 7.5). Congruent supplementar angles have the same measure. 7. Parallelogram Consecutive Angles Theorem (Thm. 7.5). Subtraction Propert of qualit 9. Definition of a right angle 1. QRST is a rectangle. 1. Definition of a rectangle Copright Big Ideas earning, C Geometr 9

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