Parallel Lines and Angles

1. A B Parallel Lines and Angles Definition Two different given lines L1 and L2 on a plane are said to be parallel if they will never intersect each other no matter how far they are extended. Definition Two angles are called vertical angles if they are opposite to each other and are formed by a pair of intersecting lines. Theorem Any pair of vertical angles are always congruent.

2. L1 L2 T Parallel Lines and Angles Definition Given two line L1 and L2 (not necessarily parallel) on the plane, a third line T is called a transversal of L1 and L2 if it intersects these two lines.

3. a c L1   L2 T • Definitions • Let L1 and L2 be two lines (not necessarily parallel) on the plane, and T be a transversal. • a and form a pair of corresponding angles. • c and  form a pair of corresponding angles etc.

4. c d L1   L2 T • Definitions • Let L1 and L2 be two lines (not necessarily parallel) on the plane, and T be a transversal. • c and form a pair of alternate interior angles. • d and  form a pair of alternate interior angles.

5. a L1  L2 T • Definitions • Let L1 and L2 be two lines (not necessarily parallel) on the plane, and T be a transversal. • a and form a pair of alternate exterior angles. • b and  form a pair of alternate exterior angles.

6. Theorem • Let L1 and L2 be two lines on the plane, and T be a transversal. • If L1 and L2 are parallel, then • any pair of corresponding angles are congruent, • any pair of alternate interior angles are congruent, • any pair of alternate exterior angles are congruent. L1 L2 T

7. Theorem • Let L1 and L2 be two lines on the plane, and T be a transversal. • if there is a pair of congruent corresponding angles, then L1 and L2 are parallel. • if there is a pair of congruent alternate interior angles, then L1 and L2 are parallel. • if there is a pair of congruent alternate exterior angles, then L1 and L2 are parallel. L1 L2 T

8. Congruence of Triangles Definition Given two triangles ΔABC and ΔXYZ. If AB is congruent to XY, A is congruent to X , BC is congruent to YZ, B is congruent to Y , CA is congruent to ZX, C is congruent to Z then we say that ΔABC is congruent to ΔXYZ, and we write Y B Z X C A

9. Congruence of Triangles Side-Angle-Side Principle Given two triangles ΔABC and ΔXYZ. If AB is congruent to XY B is congruent to Y BC is congruent to YZ then ΔABC is congruent to ΔXYZ Z B Y C A X

10. Congruence of Triangles Angle-Side-Angle Principle Given two triangles ΔABC and ΔXYZ. If A is congruent to  X AC is congruent to XZ C is congruent to Z then ΔABC is congruent to ΔXYZ Z (( B Y (( C ( ) A X

11. Congruence of Triangles Side-Side-Side Principle Given two triangles ΔABC and ΔXYZ. If AB is congruent to XY BC is congruent to YZ CA is congruent to ZX then ΔABC is congruent to ΔXYZ Z B Y C A X

12. Theorem If ΔABC is congruent to ΔXYZ , then AB is congruent to XY BC is congruent to YZ CA is congruent to ZX and A is congruent to  X B is congruent to  Y C is congruent to  Z In short, corresponding parts of congruent triangles are congruent.

13. A D B E C Example 14.5 Show that the diagonals in a kite is perpendicular to each other. Recall that a kite is a quadrilateral with 2 pairs of congruent adjacent sides. In particular for the following figure, AB = AD and CB = CD.

14. 1 2 We first need to show that ΔADC and ΔABC are congruent. This is true because AD = ABDC = BC AC = AC and we have the SSS congruence principle. (b/c it is a kite) (b/c it is a kite) (b/c they are the same side) A D B Therefore, (click) C 1 is congruent to  2

15. A D B E C Now we only considerΔADE and ΔABE. They should be congruent becauseAD = AB  1 =  2AE = AE Hence SAS principle applies. 1 2  AED is congruent to  AEB, and they both add up to 180, hence each one is 90.

16. Similarity of Triangles Definition Given ΔABC and ΔXYZ. If A is congruent to  X  B is congruent to  Y C is congruent to Z and AB : XY = BC : YZ = CA : ZX then we say that ΔABC is similar to ΔXYZ, and the notation is ΔABC ~ ΔXYZ Y B X Z A C

17. Similarity of Triangles SSS similarity principle Given ΔABC and ΔXYZ. If AB : XY = BC : YZ = CA : ZX then ΔABC is similar to ΔXYZ. Y Z B A C X

18. Similarity of Triangles AAA similarity principle Given ΔABC and ΔXYZ. If A is congruent to  X B is congruent to  Y C is congruent to  Z then ΔABC is similar to ΔXYZ X B C A Z Y

19. Similarity of Triangles AA similarity principle Given ΔABC and ΔXYZ. If A is congruent to  X B is congruent to  Y then ΔABC is similar to ΔXYZ (because the angle sum of a triangle is always 180o) X B C A Z Y

20. Similarity of Triangles SAS similarity principle Given ΔABC and ΔXYZ. If AB : XY = BC : YZ and B is congruent to Y then ΔABC is similar to ΔXYZ Y B C A Z X

21. Indirect Measurements If the shadow of a tree is 37.5 m long, and the shadow of a 1.5m student is 2.5 m long. How tall is the tree? (assuming that they are all on level ground.)

22. A 47m B 40o 58m C Indirect Measurements What is the distance between the two points A and B on the rim of the pond?

23. Indirect Measurements How far is the boat from the point A on shore? B How do we measure angles? 72º 38º 36m C A land

24. Artillery Rangefinder

26. A large (3.5m) optical rangefinder mounted on the flying bridge of the USS Stewart (a Destroyer Escort)

27. A Transit is a surveying instrument to measure horizontal angles. Glass reticle on both models has stadia lines for measuring distance. Stadia ratio 1:100

29. Measure the height of Devils Tower in Wyoming

30. The first 'proper' ascent was in 1937 when some of America's best climbers took on the project. The Weissner Route was the result, a 5.7 (decent VS) classic on which he placed a single per runner on the crux pitch.

31. A bizarre incident took place only a couple of years later, in 1941 when local air ace Charles George Hopkins decided to parachute onto the top of the tower to advertise his aerial show. He came prepared with a length of rope, a block and tackle as well as a sharpened axle from a Model T Ford to act as an anchor for his planned escape. His parachute descent went OK but on arrival he found that his rope had disappeared over the edge and he was well and truly stuck! There was only one solution, the first ascensionists were called upon to drive halfway across America to repeat their great feat and bring down the hapless Hopkins who had spent a cold and lonely week on his island in the sky.

32. Example 1. Use similar triangles to find the value of x. x 7

33. Example 2. Use similar triangles to find the values of x and y. x y

34. R 4.5 x S Q Example 2. Use similar triangles to find the values of x and y. x y

35. R R 4.5 4.5 x x S S Q Q Example 2. Use similar triangles to find the values of x and y. x y

36. Example 2. Use similar triangles to find the values of x and y. R R 4.5 4.5 x x S S 8 8 x Q Q y

37. Example 3. Use similar triangles to find the values of x and y. y x

38. C 4 x D 9 A Example 3. Use similar triangles to find the values of x and y. y x

39. C C 4 4 x x D D 9 9 A A Example 3. Use similar triangles to find the values of x and y. y x

40. C C 4 4 x x D D 9 9 A A Example 3. Use similar triangles to find the values of x and y. y x

41. C x A B 12 3 D Example 4. Use similar triangles to find the value of x.

42. C C x x C D D B 3 12 A x A B 12 3 D

43. C C x x C D D B 3 12 A x A B 12 3 D

44. C C C x x x C D D D B 3 12 12 A A x A B 12 3 D

45. C C x x C D D B 3 12 A x A B 12 3 D

46. C C C x x x C D D D B B 3 3 12 A x A B 12 3 D

47. Constructions with Straight Edge and Compass Remark A straight edge is only used to draw line segments, and it should not have any marking on it. If a ruler is used, then all the markings should be ignored.

48. Reasons fornot using a Protractor or Ruler • We need to prepare for situations where some equipments are not available. • Small protractors are not accurate enough for large scale projects. • Many geometric constructions cannot be done by protractors but can be done by a compass and straight edge, such as finding the center of a circle.

49. What can the equipments do? • A straight edge is infinitely long and is only used to draw line segments connecting given points or extend an existing line segment. • A compass can be used to draw circles and circular arcs with radii that have already been constructed.

50. Basic Rules Any construction should consist of only a finite number of steps. Only one step can be carried out at a time. Each construction must be exact. Approximation is not counted as a solution. In particular, a construction should not require drawing a line tangent to an existing circular arc because that will not provide accurate results. A point must either be given in advance, or be constructed by the intersection of two line segments, or two arcs, or an arc and a line segment.