1. An isosceles trapezoid does not have perpendicular diagonals, and a rectangle and a rhombus are both parallelograms.


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1 Quadrilaterals  Answers 1. A 2. C 3. A 4. C 5. C 6. B 7. B 8. B 9. B 10. C 11. D 12. B 13. A 14. C 15. D Quadrilaterals  Explanations 1. An isosceles trapezoid does not have perpendicular diagonals, and a rectangle and a rhombus are both parallelograms. So, a quadrilateral with perpendicular diagonals that is not a parallelogram must be a kite. 2. Since the diagonals of a parallelogram bisect one another, set the lengths of segments BE and ED equal, and solve for x. BE = ED 3x inches = 84 inches x = 28 Now, use the value of x to find the length of segment EC, which is equal to AE. EC = AE 3.5x inches = AE 3.5(28) inches = AE 98 inches = AE 3. Since PQRS is a rectangle, the diagonals are congruent. Since a rectangle is a parallelogram, the diagonals bisect each other.
2 Therefore, SF RF and SFR is an isosceles triangle with congruent base angles E and G. So, m E + m G + m F = m F = m F = 180 m F = m F = A square and a rhombus are both equilateral, and an isosceles trapezoid cannot have four right angles. So, a parallelogram that is not equilateral but has congruent diagonals and four right angles must be a rectangle. 5. A rectangle is a parallelogram, and neither an isosceles trapezoid nor a kite could have exactly two adjacent right angles. So, a quadrilateral that is not a parallelogram and could have exactly two adjacent right angles must be a trapezoid. 6. The Alternate Interior Angle Theorem states that if two parallel lines are cut by a transversal, then the alternate interior angles will be congruent. Since m ABC is 90 and m BCD is 90, it can be assumed that AB is parallel to DC. It is seen that the two parallel segments are cut by the transversal BD; therefore, m ABE = m CDE. Since m ABE = m CDE and m ABE = 38, m CDE = 38. Since AEB and DEC are vertical angles, their measures will be congruent. Since m AEB = 102, m DEC = 102. To find the measure of ECD, use triangle DEC. m CDE + m DEC + m ECD = m ECD = m ECD = 180 m ECD = 40 The measure of ECD equals A trapezoid is a polygon that consists of only one pair of parallel sides. A rectangle is a polygon that consists of exactly two pairs of parallel sides, but this describes squares and rhombuses, as well. A quadrilateral is a polygon with four sides that does not necessarily have two pairs of parallel sides. So, if a polygon consists of exactly two pairs of parallel sides, the polygon with the strongest condition is a parallelogram.
3 8. For ABCD to be a parallelogram, opposite sides need to be congruent. Set up and solve the following system of equations. Solve the system of equations using the elimination method. Multiply the second equation by 8. This will allow the y term to have the same coefficient in each equation, differing only by sign. Add the first equation to the newly multiplied second equation, and solve for x. Substitute x into the first equation, and solve for y. 9. Rhombus', rectangles, and parallelograms will always have diagonals that bisect each other. 10. An isosceles trapezoid will have diagonals of equal length but will never contain right angles by definition. A square and rectangle will have diagonals of equal length but will contain 4 right angles. A rhombus and any other parallelogram that does not contain right angles will not have diagonals of equal length. 11. Segments AB and DC are parallel lines and segments AD and BC are parallel lines. Since line segments AD and BC are parallel, they will form the same angles with line segments AB and DC. So, when discussing angles, segments AD and BC can be viewed as the same line cutting across parallel lines. When viewed in this way, B and D can be viewed as alternate interior angles. Therefore, B D. 12. The sum of the angles in a quadrilateral is 360. First, find the sum of the angles given.
4 = 221 Then, subtract the sum from = 139 Therefore, the value of x is For RSTU to be a parallelogram, opposite sides need to be congruent. So, set up and solve the following system of equations. Solve the second equation for x. Substitute x into the first equation, and solve for y. Substitute y into the first equation, and solve for x. 14. The sum of the angles in a quadrilateral is 360. First, find the sum of the angles given = 226 Then, subtract the sum from = 134
5 Therefore, the value of x is Rectangles and isosceles trapezoids will always have diagonals of equal length, but rhombuses and parallelograms will not always have diagonals of equal length.
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