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Chapter 9 Project

Chapter 9 Project. Parallelism  Triangles  Quadrilaterals. Parallelism. Key Terms. skew lines : non-coplanar lines that do not intersect (lines AB and EF are skew lines) parallel lines : non-intersecting coplanar lines (lines AD and EF are parallel lines). C. B. A. D. E. F.

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Chapter 9 Project

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  1. Chapter 9 Project Parallelism  Triangles  Quadrilaterals

  2. Parallelism Key Terms skew lines: non-coplanar lines that do not intersect (lines AB and EF are skew lines) parallel lines: non-intersecting coplanar lines (lines AD and EF are parallel lines) C B A D E F

  3. transversal: line that intersects two coplanar lines (line AB is a transversal) alternate interior angles: angles that lie inside two lines and on opposite sides of the transversal (r and s are alternate interior angles) corresponding angles: if r and s are alternate interior angles and q is a vertical angle to r, thenq and s are corresponding angles. B q r s A

  4. Theorems AIP Theorem If you are given two lines intersected by a transversal, and a pair of alternate interior angles are congruent, then the lines are parallel. Restatement: Given line AB and line CD cut by transversal EF. If x  y, then line AB is parallel to line CD. A B F x C y D E

  5. Given: line segment GH and line segment JK bisect each other at F. Prove: line segment GK and line segment JH are parallel. J H Statements Reasons 1. GH and JK bisect at F 2. GF = FH and JF = FK 3. JFH  KFG 4. ∆JFH  ∆KFG 5. HJF  GKF 6. JH is parallel to GK 1.Given 2. Def. of bisector 3. VAT 4. SAS 5. CPCTC 6. AIP F G K

  6. PCA Corollary Corresponding angles are congruent if you are given two parallel lines cut by a transversal. Restatement: v  w if line AB and line CD are parallel and are cut by transversal EF. F A v B C w D E

  7. Given: the figure with CDE  A and line LF  line AB. Prove: line LF  line DE. Statements Reasons C L 1. CDE  A, LF  AB 2. DE is parallel to AB 3. GFA  LGD 4. GFA is a R.A. 5. m GFA = 90˚ 6. m LGD = 90˚ 7. LGD = R.A. 8. LF  DE 1. Given 2. CAP 3. PCA 4. Def. of perp. 5. Def. of R.A. 6. Def. of congruence 7. Def. of R.A. 8. Def. of perp. G E D H A B

  8. Triangles Key Terms concurrent lines: two or more lines that all share a common point (lines AB, CD, and EF are concurrent.) point of concurrency: the common point shared by concurrent lines. (point G is the point of concurrency.) E D A G C B F

  9. Theorems Theorem 9-13 The measures of all the angles in a triangle add up to 180. Restatement: Given ABC. mA + m B +m C = 180. B C A

  10. Given: ABC, BA  AC and mB = 65. Prove: mC = 155. Statements Reasons 1. BA  AC, m B = 65 2. A is a R.A. 3. mA = 90 4. mA + mB +mC = 180 5. 90 + 65 + mC = 180 6. mC = 155 1. Given 2. Def. of perp. 3. Def. of R.A. 4. ms in  = 180 5. Sub. 6. SPE B 65˚ C A

  11. Theorem 9-28 If one side of a right triangle is half the length of the hypotenuse, then the measure of the opposite angle is 30. Restatement: Given right triangle ABC. If AB = 1/2BC, then mBCA is 30. B C A

  12. Given: DEF is a right triangle. D = 90 and DE = 1/2EF. Prove: mE = 60 Statements Reasons E 1. DEF is a R.T. DE = 1/2EF 2. mF = 30 3. mD = 90 4. mD + mE + mF = 180 5. 90 + mE + 30 = 180 6. mE = 60 1. Given 2.  opp. side 1/2 as long as hyp. = 30 3. Def. of R.T. 4. s add up to 180 5. Sub. 6. SPE F D

  13. Quadrilaterals Key Terms diagonal: a line segment connecting two nonconsecutive angles in a quadrilateral. (segment AC is a diagonal) parallelogram: quadrilateral that has opposite parallel lines. (ABCD is a parallelogram) trapezoid: quadrilateral that has one pair of opposite parallel lines and one pair of nonparallel lines. (JKLM is a trapezoid.) B C K L J M A D

  14. rhombus: parallelogram that has 4 congruent sides. (QRST is a rhombus) rectangle: parallelogram with 4 right angles. (ABCD is a rectangle) square: rectangle that has 4 congruent sides. (FGHJ is a square) H R G B C Q S F T J A D

  15. Theorems Theorem 9-21 If the diagonals in a quadrilateral bisect each other, that quadrilateral is a parallelogram. Restatement: Given ABCD. If AC and BD bisect each other at E, then ABCD is a parallelogram. B C E A D

  16. Given: WXYZ WT = TY and XT = TZ Prove: WXYZ is a parallelogram. Statements Reasons X Y 1. WT = TY, XT = TZ 2. WY and XZ bisect each other. 3. WXYZ is a parallelogram. 1. Given 2. Def. of bisectors 3. If diagonals bisect each other, = parallelogram T W Z

  17. Theorem 9-24 A rhombus’ diagonals are perpendicular to each other. Restatement: Given ABCD is a rhombus, then AC  BD. B A C D

  18. Given: FGHK is a rhombus. Prove: GJF  GJH Statements Reasons 1. FGHK is a rhombus 2. GK  FH 3. GJK and GJH are R.A. 4. GJK  GJH 1. Given 2. Diagonals are  in rhombus 3. Def. of perp. 4. R.A.  G J H F K

  19. Happy Holidays          

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