Rectangles, Rhombuses, and Squares

1. Kelompok 3 Rectangles, Rhombuses, and Squares Annisa Luthfi Fadhilah Ma’ruf ; Rosyida Khikmawati ; Rizqi Dwi Maharani ; Nadiatul Khikmah

2. Theorem 8-9 • A parallelogramis a rectangle if an only if its diagonal are congruent • Jajargenjangadalahsebuahpersegipanjangjikadanhanyajikadigonalnyakongruen

3. We must prove two things 1. if the diagonals of a parallogram are congruent, then the paralellogram is rectangle 2. if a parallelogram is a rectangle, then the diagonals are congruent

6. Jadi, segitiga ABD dan segitiga BAC kongruen (SSS postulate)

7. Theorem 8-10 • A parallelogramis a rhombus if an only if its diagonal are perpendicular to each other • Jajargenjangadalahbelahketupatjikadanhanyajikadiagonalnyategaklurusdengan diagonal yang lainnya

8. 2 1 1 1 2 2 2 1

11. Theorem 8-11 A parallelogram is a rhombus if and only if each diagonal bisects a pair of opposite angles.

12. PROOF Given: Prove that parallelogram ABCD would be rhombuses C D 30 60 60 30 90 O 60 30 30 60 A B

13. Answer: ˂O = 90 ˂ DAO˂ BAO ˂ BCO ˂ DCO ˂CDO ˂ADO ˂ ABO ˂ CBO So, the parallelogram would be rhombuses C D 30 60 60 30 90 O 60 30 30 60 A B

14. Theorem 8-12 The segment joining the midpoints of the two nonparallel sides of a trapezoid is parallel to the two bases and has a length equal to one half the sum of the lengths of the bases.

15. PROOF Given: ABCD is a trapezoid with DC AB E the midpoint of AD and F the midpoint of BC Prove: EF AB, EF DC and EF= ½ (AB+CD)

16. Extend AB and DF to meet at G. Then prove that F is midpoint of DG and use the Midsegment Theorem C D F E A B G

17. It show that EF AB, EF DC and EF= ½ ( AB+CD)

18. An isosceles trapezoid is a trapezoid with congruent nonparallel sides C D B A AD and BC are nonparallel sides. <A and <B are called base angles. <C and <D together are another pair of base angles.

19. Theorem 8-13 In an isosceles trapezoid base angles are congruent and the diagonals are congruent

20. 13. Given : ABCD is an isosceles trapezoid with AB ││CDProve : AC BD A

21. ∆ Plan : Draw perpendiculars from D to AB intersecting at E and from C to AB intersecting at F and prove ∆ DEA ∆ CFB

22. ˂

23. 15. Given : ABCD is an isosceles trapezoid with AB││ CD Prove : <A <B Plan : Draw DE ││CB with E on AB

25. Theorem 8-14 The sum of the measure of the angles of a convex polygon of n sides is (n-2) 180

26. Masih ingat dengan rumus yang satu ini?? • α1+ α2+ α3+… + αn = (n-2) 180 • dengan  α1,α2,α3,… ,αn adalah besar sudut-sudut dalam dari suatu bangun datar segi-n.

27. Segitiga Gambarlah sebuah segitiga, kemudian bagi segitiga tersebut menjadi segitiga (tidak harus sama besar) dan berikan label pada sudut-sudut yang terbentuk seperti gambar berikut

28. Dari gambardiataskitaperolehtigapersamanberikut : • Θ1 = 180-(ß1+ ß2) • Θ2 = 180-(ß3+ ß4) • Θ3 = 180-(ß5+ ß6) • Jumlahkanketigapersamaantersebutsehinggadiperoleh: Θ1+ Θ2+ Θ3 = 3x180 – (ß1+ ß2+ ß3+ ß4+ ß5+ ß6) 360 = 3x180 – (ß1+ ß2+ ß3+ ß4+ ß5+ ß6) ß1+ ß2+ ß3+ ß4+ ß5+ ß6 = (3-2)x180

29. Segiempat Gambarlah sebuah segiempat, kemudian bagi segiempat tersebut menjadi 4 segitiga (tidak harus sama besar) dan berikan label pada sudut-sudut yang terbentuk seperti gambar berikut

30. Dari gambar di atas kita peroleh tiga persaman berikut : Θ1 = 180-(ß1+ ß2) Θ2 = 180-(ß3+ ß4) Θ3 = 180-(ß5+ ß6) Θ4 = 180-(ß7+ ß8)

31. Jumlahkankeempatpersamaantersebutsehinggadiperoleh: Θ1+ Θ2+ Θ3+ Θ4 = 4x180 – (ß1+ ß2+ ß3+ ß4+ ß5+ ß6 + ß7+ ß8) 360 = 4x180 – (ß1+ ß2+ ß3+ ß4+ ß5+ ß6 + ß7+ ß8) ß1+ ß2+ ß3+ ß4+ ß5+ ß6 + ß7+ ß8 = (4-2)x180

32. Theorem 8-15

33. The sum of the measures of the exterior angles a polygon one at each vertex, is 360 Theorem 8-16 Bukti Jumlah n buah sudut dalam & sudut luar= n.180° Jumlah n buah sudut dalam =(n-2)180°- Jumlah n buah sudut luar =2.180°

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