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Quadrilaterals

Quadrilaterals. 5-2. Ride.

Gabriel
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Quadrilaterals

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  1. Quadrilaterals 5-2

  2. Ride An amusement park ride has a moving platform attached to four swinging arms. The platform swings back and forth, higher and higher, until it goes over the top and around in a circular motion. In the diagram below, ADand BCrepresent two of the swinging arms, and DCis parallel to the ground (line l). Explain why the moving platform ABis always parallel to the ground. EXAMPLE 1 Solve a real-world problem

  3. By the definition of a parallelogram, AB DC. Because DCis parallel to line l, ABis also parallel to line l by the Transitive Property of Parallel Lines. So, the moving platform is parallel to the ground. EXAMPLE 1 Solve a real-world problem SOLUTION The shape of quadrilateral ABCDchanges as the moving platform swings around, but its side lengths do not change. Both pairs of opposite sides are congruent, so ABCDis a parallelogram by Theorem 8.7.

  4. 1. In quadrilateral WXYZ, m W = 42°,m X =138°, m Y = 42°. Find m Z. Is WXYZa parallelogram? Explain your reasoning. 360° m W + m K + m Y + m Z = 42° + 138° + 42° + m Z = 360° 360° m Z + 222° = m Z = 138° for Example 1 GUIDED PRACTICE SOLUTION Corollary to Theorem 8.1 Substitute Combine like terms. Subtract. Yes, since the opposite angles of the quadrilateral are congruent, WXYZ is a parallelogram.

  5. Show that quadrilateral ABCDis a parallelogram. First use the Distance Formula to show that ABand CDare congruent. 29 [2 – (–3)]2 + (5 – 3)2 = 29 (5 – 0)2 + (2 – 0)2 = EXAMPLE 4 Use coordinate geometry SOLUTION One way is to show that a pair of sides are congruent and parallel. Then apply Theorem 8.9. AB = CD =

  6. 29 , AB BecauseAB = CD = CD. AB CD. Then use the slope formula to show that 5 – (3) 2 – 0 Slope of CD = = Slope of AB = = 5 – 0 2 – (–3) Because ABand CDhave the same slope, they are parallel. ANSWER ABand CDare congruent and parallel. So, ABCDis a parallelogram by Theorem 8.9. 2 2 5 5 EXAMPLE 4 Use coordinate geometry

  7. EXAMPLE 4 for Example 4 GUIDED PRACTICE 6.Refer to the Concept Summary. Explain how other methods can be used to show that quadrilateral ABCDin Example 4 is a parallelogram. SOLUTION Find the Slopes of all 4 sides and show that each opposite sides always have the same slope and, therefore, are parallel. Find the lengths of all 4 sides and show that the opposite sides are always the same length and, therefore, are congruent. Find the point of intersection of the diagonals and show the diagonals bisect each other.

  8. Q Q a. S S a. By definition, a rhombus is a parallelogram with four congruent sides.By Theorem 8.4, opposite angles of a parallelogram are congruent. So, .The statement is always true. EXAMPLE 1 Use properties of special quadrilaterals For any rhombus QRST, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning. SOLUTION

  9. Q Q b. R R EXAMPLE 1 Use properties of special quadrilaterals For any rhombus QRST, decide whether the statement is always or sometimes true. Draw a sketch and explain your reasoning. SOLUTION b. If rhombus QRSTis a square, then all four angles are congruent right angles. So, If QRSTis a square. Because not all rhombuses are also squares, the statement is sometimes true.

  10. Classify the special quadrilateral. Explain your reasoning. EXAMPLE 2 Classify special quadrilaterals SOLUTION The quadrilateral has four congruent sides. One of the angles is not a right angle, so the rhombus is not also a square. By the Rhombus Corollary, the quadrilateral is a rhombus.

  11. 1. For any rectangle EFGH, is it always or sometimes true that Explain your reasoning. FG FG GH ? GH ANSWER Adjacent sides of a rectangle can be congruent . If, it is a square. A square is also a rectangle with four right angles but rectangle is not always a square. Therefore , in EFGH , only if EFGH is a square. for Examples 1 and 2 GUIDED PRACTICE

  12. ANSWER D C Square A B for Examples 1 and 2 GUIDED PRACTICE 2. A quadrilateral has four congruent sides and four congruent angles. Sketch the quadrilateral and classify it.

  13. EXAMPLE 1 Identify quadrilaterals Quadrilateral ABCDhas at least one pair of opposite angles congruent. What types of quadrilaterals meet this condition? SOLUTION There are many possibilities.

  14. The diagram shows AE CEand BE DE. So, the diagonals bisect each other. By Theorem 8.10, ABCD is a parallelogram. EXAMPLE 2 Standardized Test Practice SOLUTION

  15. The correct answer is A. ANSWER EXAMPLE 2 Standardized Test Practice Rectangles, rhombuses and squares are also parallelograms. However, there is no information given about the side lengths or angle measures of ABCD. So,you cannot determine whether it is a rectangle, a rhombus, or a square.

  16. Is enough information given in the diagram to show that quadrilateral PQRSis an isosceles trapezoid? Explain. STEP 1 Show that PQRSis a trapezoid. Rand Sare supplementary,but P and S are not. So, PS QR , but PQis not parallel to SR. By definition, PQRSis a trapezoid. EXAMPLE 3 Identify a quadrilateral SOLUTION

  17. STEP 2 Show that trapezoid PQRS is isosceles. P and Sare a pair of congruent base angles. So, PQRSis an isosceles trapezoid by Theorem 8.15. ANSWER Yes, the diagram is sufficient to show that PQRS is an isosceles trapezoid. EXAMPLE 3 Identify a quadrilateral

  18. ANSWER Parallelogram, Rectangle, Square, Rhombus, Trapezoid. In all these quadrilaterals at least one pair of opposite sides is congruent. for Examples 1, 2 and 3 GUIDED PRACTICE 1. Quadrilateral DEFGhas at least one pair of opposite sides congruent. What types of quadrilaterals meet this condition?

  19. ANSWER It is a kite as kite is a quadrilateral that has two pair of consecutive congruent sides, but opposite sides are not congruent. for Examples 1, 2 and 3 GUIDED PRACTICE Give the most specific name for the quadrilateral.Explain your reasoning.

  20. ANSWER VWXY is a trapezoid. one pair of opposite sides are parallel, and the diagonals do not bisect each other, therefore it is a trapezoid. for Examples 1, 2 and 3 GUIDED PRACTICE Give the most specific name for the quadrilateral.Explain your reasoning.

  21. ANSWER Quadrilateral; there is not enough information to be more specific. for Examples 1, 2 and 3 GUIDED PRACTICE Give the most specific name for the quadrilateral.Explain your reasoning.

  22. ANSWER MNPQ could be a rectangle or a square since you do not know the relationship between MQ and NP. There is not enough information to conclude it is an isosceles trapezoid. 5. Error AnalysisA student knows the following information about quadrilateral MNPQ:MN PQ ,MP NQ , and P Q. The student concludes that MNPQis an isosceles trapezoid. Explain why the student cannot make this conclusion. for Examples 1, 2 and 3 GUIDED PRACTICE

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