nerice
Uploaded by
19 SLIDES
336 VUES
190LIKES

Understanding Parallelism in Geometry: Lines, Angles, and Quadrilaterals

DESCRIPTION

This chapter explores the concept of parallelism in geometry, focusing on key terms and theorems related to lines and angles. It discusses skew lines, parallel lines, and transversals, as well as properties of triangles and quadrilaterals, including parallelograms, trapezoids, and rhombuses. Key theorems such as the Alternate Interior Angles Theorem and the properties of bisected diagonals in quadrilaterals are explained with proofs and reasons. Understanding these principles is essential for solving geometric problems effectively.

1 / 19

Télécharger la présentation

Understanding Parallelism in Geometry: Lines, Angles, and Quadrilaterals

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

Playing audio...

  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          

More Related