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State Whether Each Sentence Is True Or False. If False, Replace The Chapter 6 Study Guide and Review State whether each sentence is true or false. If 4. The base of a trapezoid is one of the parallel sides. false, replace the underlined word or phrase to SOLUTION: make a true sentence. 1. No angles in an isosceles trapezoid are congruent. One of the parallel sides of a trapezoid is its base. The statement is true. SOLUTION: By definition, an isosceles trapezoid is a trapezoid in ANSWER: which the legs are congruent, both pairs of base true angles are congruent, and the diagonals are 5. The diagonals of a rhombus are perpendicular. congruent. SOLUTION: Thus, the statement "No angles in an isosceles A rhombus has perpendicular diagonals. The trapezoid are congruent." is false. It should be "both statement is true. pairs of base angles". ANSWER: ANSWER: true false, both pairs of base angles 6. The diagonal of a trapezoid is the segment that connects the midpoints of the legs. 2. If a parallelogram is a rectangle, then the diagonals are congruent. SOLUTION: SOLUTION: A diagonal is a segment that connects any two nonconsecutive vertices. The midsegment of a A rectangle is a parallelogram with four right angles, trapezoid is the segment that connects the midpoint of opposite sides parallel, opposite sides congruent, the legs of the trapezoid. opposite angles congruent, consecutive angles are supplementary, and the diagonals bisect each other. Thus, the statement "The diagonal of a trapezoid is The statement is true. the segment that connects the midpoints of the legs." ANSWER: is false. It should be the "midsegment". true ANSWER: 3. A midsegment of a trapezoid is a segment that false, midsegment connects any two nonconsecutive vertices. 7. A rectangle is not always a parallelogram. SOLUTION: SOLUTION: The midsegment of a trapezoid is the segment that By definition, a rectangle is a parallelogram with four connects the midpoints of the legs of the trapezoid. A right angles. diagonal is a segment that connects any two nonconsecutive vertices. Thus, the statement "A rectangle is not always a parallelogram." is false. It should be "always". Thus, the statement "A midsegment of a trapezoid is a segment that connects any two nonconsecutive ANSWER: vertices." is false. It should be "diagonal". false, is always ANSWER: false, diagonal eSolutions Manual - Powered by Cognero Page 1 Chapter 6 Study Guide and Review 8. A quadrilateral with only one set of parallel sides is a Find the sum of the measures of the interior parallelogram. angles of each convex polygon. 11. decagon SOLUTION: A parallelogram is a quadrilateral with both pairs of SOLUTION: opposite sides parallel. A trapezoid is a quadrilateral A decagon has ten sides. Use the Polygon Interior with exactly one pair of parallel sides. Angles Sum Theorem to find the sum of its interior angle measures. Thus, the statement "A quadrilateral with only one set of parallel sides is a parallelogram." is false. It should Substitute n = 10 in . be "trapezoid". ANSWER: false, trapezoid 9. Explain how to prove that a given parallelogram is a ANSWER: square. 1440 SOLUTION: Show that the parallelogram is both a rectangle and 12. 15-gon a rhombus. SOLUTION: ANSWER: A 15-gon has fifteen sides. Use the Polygon Interior Show that the parallelogram is both a rectangle and Angles Sum Theorem to find the sum of its interior a rhombus. angle measures. 10. Explain how to determine the number of sides in a Substitute n = 15 in . regular polygon given the measure of one of its interior angles. SOLUTION: Multiply the given angle measure by n. Set this expression equal to (n – 2) · 180 and solve for n. ANSWER: ANSWER: 2340 Multiply the given angle measure by n. Set this expression equal to (n – 2) · 180 and solve for n. eSolutions Manual - Powered by Cognero Page 2 Chapter 6 Study Guide and Review 13. SNOWFLAKES The snowflake decoration shown 15. ≈ 166.15 is a regular hexagon. Find the sum of the measures of SOLUTION: the interior angles of the hexagon. Let n = the number of sides in the polygon. Since all angles of a regular polygon are congruent, the sum of the interior angle measures is about 166.15n. By the Polygon Interior Angles Sum Theorem, the sum of the interior angle measures can also be expressed as . SOLUTION: A hexagon has six sides. Use the Polygon Interior ANSWER: Angles Sum Theorem to find the sum of its interior angle measures. 26 Use ABCD to find each measure. Substitute n = 6 in . ANSWER: 16. 720 SOLUTION: We know that consecutive angles in a parallelogram The measure of an interior angle of a regular are supplementary. polygon is given. Find the number of sides in So, the polygon. Substitute. 14. 135 SOLUTION: Let n = the number of sides in the polygon. Since all ANSWER: angles of a regular polygon are congruent, the sum of 65° the interior angle measures is 135n. By the Polygon Interior Angles Sum Theorem, the sum of the interior angle measures can also be expressed as . 17. AD SOLUTION: We know that opposite sides of a parallelogram are congruent. So, ANSWER: ANSWER: 8 18 eSolutions Manual - Powered by Cognero Page 3 Chapter 6 Study Guide and Review 18. AB ALGEBRA Find the value of each variable in each parallelogram. SOLUTION: We know that opposite sides of a parallelogram are congruent. So, 20. ANSWER: SOLUTION: 12 Since the opposite sides of a parallelogram are congruent, 2x + 9 = 4x – 5. 19. SOLUTION: Solve for x. We know that opposite angles of a parallelogram are congruent. So, ANSWER: 115° Since the sum of the measures of the angles in a triangle is 180, write and solve an equation. So, x = 7 and y = 8. ANSWER: x = 7, y = 8 eSolutions Manual - Powered by Cognero Page 4 Chapter 6 Study Guide and Review 22. DESIGN What type of information is needed to determine whether the shapes that make up the stained glass window below are parallelograms? 21. SOLUTION: Since the opposite sides are congruent, 3y + 13 = 2y + 19. SOLUTION: Solve for y. If both pairs of opposite sides are the same length or 3y + 13 = 2y + 19 if one pair of opposite sides is congruent and parallel, y = 6 then the shapes are parallelograms. The shapes can also be parallelograms if both pairs of opposite angles Since the opposite angles are congruent, 2x + 41 = are congruent or if the diagonals bisect each other. 115. ANSWER: Solve for x. Sample answer: If both pairs of opposite sides are the 2x + 41 = 115 same length or if one pair of opposite sides is 2x = 74 congruent and parallel, then the shapes are x = 37 parallelograms. The shapes can also be ANSWER: parallelograms if both pairs of opposite angles are x = 37, y = 6 congruent or if the diagonals bisect each other. Determine whether each quadrilateral is a parallelogram. Justify your answer. 23. SOLUTION: One pair of opposite sides are parallel and congruent. By Theorem 6.12, if one pair of opposite sides of a quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. No other information is needed to determine that the figure is a parallelogram. ANSWER: yes, Theorem 6.12 eSolutions Manual - Powered by Cognero Page 5 Chapter 6 Study Guide and Review 2. AE = CF (Def. of segs) 3. (Opp. sides of a ) 4. BC = AD (Def. of segs) 24. 5. BC = BF + CF, AD = AE +ED (Seg. Add. Post.) SOLUTION: 6. BF + CF = AE + ED (Subst.) The diagonals of the figure bisect each other. By 7. BF + AE = AE + ED (Subst.) Theorem 6.11, if the diagonals of a quadrilateral 8. BF = ED (Subt. Prop.) bisect each other, then the quadrilateral is a 9. (Def. of segs) parallelogram. No other information is needed to 10. (Def. of ) determine that the figure is a parallelogram. 11. Quadrilateral EBFD is a parallelogram. (If one pair of opposite sides is parallel and congruent then it ANSWER: is a parallelogram.) yes, Theorem 6.11 ANSWER: 25. PROOF Write a two-column proof. Given: Prove: Quadrilateral EBFD is a parallelogram. Given: Prove: Quadrilateral EBFD is a parallelogram. Proof: SOLUTION: 1. ABCD is a parallelogram, (Given) You need to walk through the proof step by step. 2. AE = CF (Def. of segs) Look over what you are given and what you need to 3. (Opp. sides of a ) prove. Here, you are given . You 4. BC = AD (Def. of segs) need to prove that EBFD is a parallelogram. Use the 5. BC = BF + CF, AD = AE +ED (Seg. Add. Post.) properties that you have learned about parallelograms 6. BF + CF = AE + ED (Subst.) to walk through the proof. 7. BF + AE = AE + ED (Subst.) 8. BF = ED (Subt. Prop.) Given: 9. (Def. of segs) Prove: Quadrilateral EBFD is a parallelogram. 10. (Def. of ) 11. Quadrilateral EBFD is a parallelogram. (If one pair of opposite sides is parallel and congruent then it is a parallelogram.) Proof: 1. ABCD is a parallelogram, (Given) eSolutions Manual - Powered by Cognero Page 6 Chapter 6 Study Guide and Review ALGEBRA Find x and y so that the quadrilateral is a parallelogram. 27. SOLUTION: 26. We know that diagonals of a parallelogram bisect each other. So, . SOLUTION: We know that opposite angles of a parallelogram are Solve for x. congruent. So, 12x + 72 = 25x + 20 and 3y + 36 = 9y – 12. Solve for x. 12x + 72 = 25x + 20 Alternate interior angles in a parallelogram are 72 = 13x + 20 congruent. 52 = 13x 4 = x Solve for y. Solve for y. 5y = 60 y = 12 3y + 36 = 9y – 12 36 = 6y – 12 When x = 5 and y = 12, the quadrilateral is a 48 = 6y parallelogram.
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