700 likes | 822 Views
18/02/2014 CH.6.5 Conditions for Special Parallelogram. Classwork. Homework. Homework Booklet Chapter: 6.5. (Pages 434 to 436 ) Exercises 1, 4, 5, 6, 9 to 16, 17, 18, 20, 22, 24 to 26, 27, 28, 33(a, b, c), 34, 39, 40, 41, 43. Warm Up Solve for x . 1. 16 x – 3 = 12 x + 13
E N D
Classwork Homework Homework Booklet Chapter: 6.5 • (Pages 434 to 436 ) Exercises 1, 4, 5, 6, 9 to 16, 17, 18, 20, 22, 24 to 26, 27, 28, 33(a, b, c), 34, 39, 40, 41, 43.
Warm Up Solve for x. 1.16x – 3 = 12x + 13 2. 2x – 4 = 90 ABCD is a parallelogram. Find each measure. 3.CD4. mC 4 47 104° 14
opp. s Warm Up 1.Find AB for A (–3, 5) and B (1, 2). 2. Find the slope of JK for J(–4, 4) and K(3, –3). ABCD is a parallelogram. Justify each statement. 3.ABC CDA 4. AEB CED 5 –1 Vert. s Thm.
Objective Prove that a given quadrilateral is a rectangle, rhombus, or square.
When you are given a parallelogram with certain properties, you can use the theorems below to determine whether the parallelogram is a rectangle.
A manufacture builds a mold for a desktop so that , , and mABC = 90°. Why must ABCD be a rectangle? Both pairs of opposites sides of ABCD are congruent, so ABCD is a . Since mABC = 90°, one angle ABCD is a right angle. ABCD is a rectangle by Theorem 6-5-1. Example 1: Carpentry Application
Check It Out! Example 1 A carpenter’s square can be used to test that an angle is a right angle. How could the contractor use a carpenter’s square to check that the frame is a rectangle? Both pairs of opp. sides of WXYZ are , so WXYZ is a parallelogram. The contractor can use the carpenter’s square to see if one of WXYZ is a right . If one angle is a right , then by Theorem 6-5-1 the frame is a rectangle.
Below are some conditions you can use to determine whether a parallelogram is a rhombus.
Caution In order to apply Theorems 6-5-1 through 6-5-5, the quadrilateral must be a parallelogram. To prove that a given quadrilateral is a square, it is sufficient to show that the figure is both a rectangle and a rhombus. You will explain why this is true in Exercise 43.
Remember! You can also prove that a given quadrilateral is a rectangle, rhombus, or square by using the definitions of the special quadrilaterals.
Example 2A: Applying Conditions for Special Parallelograms Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. Given: Conclusion: EFGH is a rhombus. The conclusion is not valid. By Theorem 6-5-3, if one pair of consecutive sides of a parallelogram are congruent, then the parallelogram is a rhombus. By Theorem 6-5-4, if the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. To apply either theorem, you must first know that ABCD is a parallelogram.
Quad. with diags. bisecting each other Example 2B: Applying Conditions for Special Parallelograms Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. Given: Conclusion: EFGH is a square. Step 1 Determine if EFGH is a parallelogram. Given EFGH is a parallelogram.
with diags. rect. with one pair of cons. sides rhombus Example 2B Continued Step 2 Determine if EFGH is a rectangle. Given. EFGH is a rectangle. Step 3 Determine if EFGH is a rhombus. EFGH is a rhombus.
Example 2B Continued Step 4 Determine is EFGH is a square. Since EFGH is a rectangle and a rhombus, it has four right angles and four congruent sides. So EFGH is a square by definition. The conclusion is valid.
Check It Out! Example 2 Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. Given:ABC is a right angle. Conclusion:ABCD is a rectangle. The conclusion is not valid. By Theorem 6-5-1, if one angle of a parallelogram is a right angle, then the parallelogram is a rectangle. To apply this theorem, you need to know that ABCD is a parallelogram .
Example 3A: Identifying Special Parallelograms in the Coordinate Plane Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. P(–1, 4), Q(2, 6), R(4, 3), S(1, 1)
Example 3A Continued Step 1 Graph PQRS.
Since , the diagonals are congruent. PQRS is a rectangle. Example 3A Continued Step 2 Find PR and QS to determine if PQRS is a rectangle.
Since , PQRS is a rhombus. Example 3A Continued Step 3 Determine if PQRS is a rhombus. Step 4 Determine if PQRS is a square. Since PQRS is a rectangle and a rhombus, it has four right angles and four congruent sides. So PQRS is a square by definition.
Example 3B: Identifying Special Parallelograms in the Coordinate Plane Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. W(0, 1), X(4, 2), Y(3, –2), Z(–1, –3) Step 1 Graph WXYZ.
Since , WXYZ is not a rectangle. Example 3B Continued Step 2 Find WY and XZ to determine if WXYZ is a rectangle. Thus WXYZ is not a square.
Since (–1)(1) = –1, , WXYZ is a rhombus. Example 3B Continued Step 3 Determine if WXYZ is a rhombus.
Check It Out! Example 3A Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. K(–5, –1), L(–2, 4), M(3, 1), N(0, –4)
Check It Out! Example 3A Continued Step 1 Graph KLMN.
Since , KMLN is a rectangle. Check It Out! Example 3A Continued Step 2 Find KM and LN to determine if KLMN is a rectangle.
Check It Out! Example 3A Continued Step 3 Determine if KLMN is a rhombus. Since the product of the slopes is –1, the two lines are perpendicular. KLMN is a rhombus.
Check It Out! Example 3A Continued Step 4 Determine if KLMN is a square. Since KLMN is a rectangle and a rhombus, it has four right angles and four congruent sides. So KLMN is a square by definition.
Check It Out! Example 3B Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply. P(–4, 6) , Q(2, 5) , R(3, –1) , S(–3, 0)
Check It Out! Example 3B Continued Step 1 Graph PQRS.
Since , PQRS is not a rectangle. Thus PQRS is not a square. Check It Out! Example 3B Continued Step 2 Find PR and QS to determine if PQRS is a rectangle.
Since (–1)(1) = –1, are perpendicular and congruent. PQRS is a rhombus. Check It Out! Example 3B Continued Step 3 Determine if PQRS is a rhombus.
Lesson Quiz: Part I 1. Given that AB = BC = CD = DA, what additional information is needed to conclude that ABCD is a square?
Lesson Quiz: Part II 2. Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid. Given:PQRS and PQNM are parallelograms. Conclusion:MNRS is a rhombus. valid
AC ≠ BD, so ABCD is not a rect. or a square. The slope of AC = –1, and the slope of BD = 1, so AC BD. ABCD is a rhombus. Lesson Quiz: Part III 3. Use the diagonals to determine whether a parallelogram with vertices A(2, 7), B(7, 9), C(5, 4), and D(0, 2) is a rectangle, rhombus, or square. Give all the names that apply.
Objectives Prove and apply properties of rectangles, rhombuses, and squares. Use properties of rectangles, rhombuses, and squares to solve problems.
Vocabulary rectangle rhombus square
A second type of special quadrilateral is a rectangle. A rectangleis a quadrilateral with four right angles.
Since a rectangle is a parallelogram by Theorem 6-4-1, a rectangle “inherits” all the properties of parallelograms that you learned in Lesson 6-2.
diags. bisect each other Example 1: Craft Application A woodworker constructs a rectangular picture frame so that JK = 50 cm and JL = 86 cm. Find HM. Rect. diags. KM = JL = 86 Def. of segs. Substitute and simplify.
Check It Out! Example 1a Carpentry The rectangular gate has diagonal braces. Find HJ. Rect. diags. HJ = GK = 48 Def. of segs.
Check It Out! Example 1b Carpentry The rectangular gate has diagonal braces. Find HK. Rect. diags. Rect. diagonals bisect each other JL = LG Def. of segs. JG = 2JL = 2(30.8) = 61.6 Substitute and simplify.
A rhombus is another special quadrilateral. A rhombusis a quadrilateral with four congruent sides.
Like a rectangle, a rhombus is a parallelogram. So you can apply the properties of parallelograms to rhombuses.
Example 2A: Using Properties of Rhombuses to Find Measures TVWX is a rhombus. Find TV. WV = XT Def. of rhombus 13b – 9=3b + 4 Substitute given values. 10b =13 Subtract 3b from both sides and add 9 to both sides. b =1.3 Divide both sides by 10.
Example 2A Continued TV = XT Def. of rhombus Substitute 3b + 4 for XT. TV =3b + 4 TV =3(1.3)+ 4 = 7.9 Substitute 1.3 for b and simplify.
Example 2B: Using Properties of Rhombuses to Find Measures TVWX is a rhombus. Find mVTZ. mVZT =90° Rhombus diag. Substitute 14a + 20 for mVTZ. 14a + 20=90° Subtract 20 from both sides and divide both sides by 14. a=5
Example 2B Continued Rhombus each diag. bisects opp. s mVTZ =mZTX mVTZ =(5a – 5)° Substitute 5a – 5 for mVTZ. mVTZ =[5(5) – 5)]° = 20° Substitute 5 for a and simplify.
Check It Out! Example 2a CDFG is a rhombus. Find CD. CG = GF Def. of rhombus 5a =3a + 17 Substitute a =8.5 Simplify GF = 3a + 17=42.5 Substitute CD = GF Def. of rhombus CD = 42.5 Substitute