1. Geometry and Trig

2. Geometry
Goals
• Define a ray as originating at an endpoint, A, along a half-line containing a point B, as
• Represent an angle, , between two rays, with a common endpoint, A, by the counterclockwise rotation of ray onto ray
Big Idea
A line segment is a part of a line. A line is denoted as
A ray is a half-line and originates at a point.
Intersecting rays form an angle and the size of the angle is the measure of the angle and denoted

3. Geometry
Goals
• Define acute, obtuse, right, and straight angles.

4. Geometry
Goals
• Represent the complement of an angle with a geometric figure and calculate the complement.
• Represent the supplement of an angle with a geometric figure and calculate the supplement.
For adjacent angles

5. Geometry
Goals
• Prove and apply the fact that the congruence of vertical angles formed by intersecting lines.
Parallel lines
These vertical angles are congruent.
Why?
Intersecting lines

6. Geometry
Goals
• Prove and apply the fact that the congruence of vertical angles formed by intersecting lines.
These vertical angles are congruent.
Intersecting lines

7. Geometry transversal We won’t do an informal proof:
When parallel lines are cut by a transversal the alternate interior angles have equal measure.

8. Geometry
Goals
• Prove and apply the fact that the interior angles on opposite sides of a transversal of parallel lines are congruent.
• Prove and apply the fact that the corresponding angles formed by the transversal of parallel lines are congruent.
The measure of this angle is equal to
Why?

9. Geometry
Goals
• Prove and apply the fact that the interior angles on the same side of a transversal of parallel lines are supplements.
Find the measures of all other angles.
HW due Tuesday 1/4
250.3, 5, 6, 7, 9, 255.4, 6, 16, 27, 30, 38 and

10. Geometry
Goals
• Classify triangles as acute, equiangular, right, and obtuse.

11. Geometry
Goals
• Differentiate the legs and hypotenuse of a right triangle.

12. Geometry
Goals
• Classify triangles as scalene, isosceles, and equilateral.

13. Geometry
Goals
• Calculate one interior angle of a triangle if two are known.

14. Geometry First construction: Congruent sides
• Draw two rays that intersect
• Use the compass to locate equal distances from S

15. Geometry Second construction: Congruent angles
• Draw intersecting rays with endpoint S
• Draw a ray with endpoint M
• Use the compass to locate equal distances from S, A and B
• Adjust the compass with point at B to intersect at A
• With compass at L locate the point N

16. Geometry Third construction: Perpendicular bisector
• Draw intersecting ray containing points A and B
• Use the compass to construct a circle centered on A
• Construct a circle with the same diameter centered on B
• Connect the intersections of the circles

17. Geometry Third construction: Angle bisector
• Draw rays containing points A and B intersecting at O
• Use the compass to construct a circle centered on A
• Construct a circle with the same diameter centered on B
• Connect the intersections of the circles

18. Geometry
Goals
• Calculate the interior angle of a quadrilateral if three interior angles are known.

19. Geometry
Goals
• Calculate the interior angle of an n-sided polygon if n-1 interior angles are known.
Within an n-sided polygon (n-2) triangles can be constructed and the sum of the angles in each is 180 so
The sum of the angles in an n-sided polygon is (n-2)180°

20. Geometry
Goals
• Apply mathematical reasoning to construct the area of an irregular polygon by decomposition into simpler geometric figures where possible.
A triangle can be inscribed in a rectangle. In this case there are two.
Each has an area that is half of a rectangular area.

21. Geometry
Goals
• Apply mathematical reasoning to construct the area of an irregular polygon by decomposition into simpler geometric figures where possible.
A piece of land is bounded by two parallel roads and two roads that are not parallel form a trapezoid. Along the parallel roads the land measures 1.3 miles and 1.7 miles. The distance between the parallel roads, measured perpendicular to the roads, is 282 miles.
Find the area of the land. Express the area to the nearest tenth of a mile.
b. An acre is a unit of area often used to measure land. There are 640 acres in a square mole. Express to the nearest hundred acres, the area of land.

22. Geometry
A piece of land is bounded by two parallel roads and two roads that are not parallel form a trapezoid. Along the parallel roads the land measures 1.3 miles and 1.7 miles. The distance between the parallel roads, measured perpendicular to the roads, is 282 miles.
Find the area of the land.
First draw a picture.

23. Geometry Find the area of the land. First draw a picture.
Then work out a symbolic representation of the area.
Finally, evaluate the representation.

24. Geometry
Goals
• Apply mathematical reasoning to construct the area of an irregular polygon by decomposition into simpler geometric figures where possible.
ABCD is a quadrilateral. The diagonals are perpendicular at E.
Find the area of ABCD.

25. Geometry
Goals
• Calculate the surface area and volumes of rectangular and cylindrical solids, and spheres.
Some important areas and volumes are memorized formulas.

26. Geometry
Goals
• Calculate the surface area and volumes of rectangular and cylindrical solids, and spheres.
Some areas and volumes are constructed from these formulas.
What is the area of the region that is
shaded black?
First you find the area of the regular
hexagon and then you subtract it from
the area of the circle.

27. Geometry
Goals
• Calculate the surface area and volumes of rectangular and cylindrical solids, and spheres.
Some areas and volumes are constructed from these formulas.
What is the volume of the prism?

28. Geometry
Goals
• Calculate the relative error in the measurement of a length, surface area or volume.
HW due Wednesday 1/5
268.13, 14, 27, 31, , , 281.3, 4, 5, 8, , , 17,

29. Geometry Homework Quiz #1 Regents Exam question:
Homework problem: The complement of an angle is 14 times as large as the angle. Find the complement.

30. Geometry Homework Quiz #2 Regents Exam question:
Homework problem: Do an informal proof that the interior angles when parallel lines are cut by a transversal are equal.

31. Geometry
Homework Quiz #3

32. Trigonometry Goals • Prove and apply the Pythagorean theorem.
• Calculate the length of a hypotenuse from the lengths of the legs.
If the lengths of two sides of a right triangle are known then
the length of the third side can be determined.

33. Trigonometry Goals • Prove and apply the Pythagorean theorem.
• Calculate the length of a hypotenuse from the lengths of the legs.
The negative root is rejected since the length is positive.

34. Trigonometry
Goals
• Calculate the length of a hypotenuse from the lengths of the legs.
If the hypotenuse = 25 and one leg = 20, what
Is the length of the other leg?
but so
The negative root is rejected since the length is positive.

35. Trigonometry
Goals
• Represent the tangent of the acute angle of a right triangle as the ratio of the lengths of the opposite and adjacent sides.

36. Trigonometry
Goals
• Represent the tangent of the acute angle of a right triangle as the ratio of the lengths of the opposite and adjacent sides.
The leg OA is adjacent to A
The leg OB is adjacent to B

37. Trigonometry
Goals
• Represent the tangent of the acute angle of a right triangle as the ratio of the lengths of the opposite and adjacent sides.
The ratio of the adjacent
and opposite sides remains
constant for these similar
triangles.
The tangent of the angle a is the ratio of the length of the
opposite side to the length of the adjacent side.

38. Trigonometry
Goals
• Represent the tangent of the acute angle of a right triangle as the ratio of the lengths of the opposite and adjacent sides.
The tangent of the angle a, tan(a), is the ratio of the length of the opposite side to the length of the adjacent side.

39. Trigonometry
Goals
• Calculate the tangent of the acute angle of a right triangle.
311.3 In a right triangle with sides of length 3 and 3 find the tangents of both acute angles.
Calculate the tangent of 1°
Calculate the tangent of 67°
Calculate tan(20°) and tan(40°) to the nearest ten-thousandth. Does tan(a) double when the angle doubles?
HW due Thursday 1/6
305.1, 2, 9, 11, 20, 28, 31, 34, 36, 311.3, 6, 13, 16, 17, 22, , 31, 35

40. Trigonometry
Goals
•Calculate the ratio of the opposite and adjacent sides of a right triangle using the inverse tangent of the acute angle.
The inverse tangent tan-1(a), is the answer to the question: if the ratio is known what is the angle?

41. Trigonometry
Goals
• Calculate the ratio of the opposite and adjacent sides of a right triangle using the inverse tangent of the acute angle.
Calculate the measure of the angle A if tan(A)=0.0875
Calculate the measure of the angle A if tan(A)=3.0777
in a right triangle with legs of length 6 and 6:
a. find tangent of the acute angles.
b. find the acute angle
HW due Thursday 1/6
305.1, 2, 9, 11, 20, 28, 31, 34, 36, 311.3, 6, 13, 16, 17, 22, , 31, 35

42. Trigonometry
HW Quiz #2

43. Trigonometry
HW Quiz #2

44. Trigonometry
HW Quiz #2 C
6x+20=10x
4x=20
x=5 A
6x+20 E
10x B D

45. Trigonometry Applications of the tangent
• if the opposite side and an angle are known the adjacent side can be determined
A stake is to be driven into the ground away from the base of a 50-foot pole. A wire from the stake to the top of pole is to make a 52° angle with the ground. Where is the stake?

46. Trigonometry Applications of the tangent
• if the opposite side and an angle are known the adjacent side can be determined
A stake is to be driven into the ground away from the base of a 50-foot pole. A wire from the stake to the top of pole is to make a 52° angle with the ground. Where is the stake?

47. Trigonometry Applications of the tangent
• if the adjacent side and the angle of elevation is known then the length of the opposite side can be determined.
A tree casts a 25-foot shadow on a sunny day. If the angle formed by the tip of the tree and the tip of the shadow is 32° what is the height of the tree?

48. Trigonometry Applications of the tangent
• if the adjacent side and the angle of elevation is known then the length of the opposite side can be determined.
A tree casts a 25-foot shadow on a sunny day. If the angle formed by the tip of the tree and the tip of the shadow is 32° what is the height of the tree?

49. Trigonometry Applications of the tangent
• if the adjacent side and the angle of elevation is known then the length of the opposite side can be determined.
A tree casts a 25-foot shadow on a sunny day. If the angle formed by the tip of the tree and the tip of the shadow is 32° what is the height of the tree?

50. Trigonometry
In some problems you’re interested in making measurements of depth rather height. Then the “angle of elevation” is replaced with an “angle of depression.”
HW due Thursday 1/10
316.2, 3, 5, 9, 12, 13, , 21, 320.1, 3, 4, 5, 6, 8, 14, 20, 31, 32, 39, , 47, 50

51. Trigonometry
Goals
• Represent the sine of the acute angle of a right triangle as the ratio of the lengths of the opposite side and the hypotenuse.
• Calculate the sine of the acute angle of a right triangle.
A second constant ratio
is called the sine of a,
sin(a).

52. Trigonometry
Goals
• Represent the sine of the acute angle of a right triangle as the ratio of the lengths of the opposite side and the hypotenuse.
The sine of the angle a, sin(a), is the ratio of the length of the opposite side to the length of the hypotenuse.

53. Trigonometry
Goals
•Calculate the ratio of the opposite side and the hypotenuse of a right triangle using the inverse sine of the acute angle.
The inverse sine sin-1(a), is the answer to the question: if the ratio is known what is the angle?

54. Trigonometry
Goals
• Represent the cosine of the acute angle of a right triangle as the ratio of the lengths of the adjacent side and the hypotenuse.
• Calculate the cosine of the acute angle of a right triangle.
A third constant ratio
is called the sine of a,
cos(a).

55. Trigonometry
Goals
• Represent the cosine of the acute angle of a right triangle as the ratio of the lengths of the adjacent side and the hypotenuse.
The cosine of the angle a, cos(a), is the ratio of the length of the adjacent side to the length of the hypotenuse.

56. Trigonometry
Goals
•Calculate the ratio of the adjacent side and the hypotenuse of a right triangle using the inverse cosine of the acute angle.
The inverse sine cos-1(a), is the answer to the question: if the ratio is known what is the angle?

57. Trigonometry
Goals
• Calculate the sine of the acute angle of a right triangle.
• Calculate the cosine of the acute angle of a right triangle.
sin(42°) to the nearest ten-thousandth
cos(88°) to the nearest ten-thousandth
Calculate sin(25°) and sin(50°). Is the sine of the angle doubled if the angle is doubled?
HW due Thursday 1/10
316.2, 3, 5, 9, 12, 13, , 21, 320.1, 3, 4, 5, 6, 8, 14, 20, 31, 32, 39, , 47, 50

58. Trigonometry Applications of the sine and cosine
• if the hypotenuse and the angle of elevation are known then the length of the opposite side can be determined.
A hot-air balloon is tied to the ground with two taut (straight) ropes. One rope is directly under the balloon and makes a right angle with the ground. The other rope forms an angle of 50° with the ground.

59. Trigonometry Applications of the sine and cosine
• if the hypotenuse and the angle of elevation are known then the length of the opposite side can be determined.
A hot-air balloon is tied to the ground with two taut (straight) ropes. One rope is directly under the balloon and makes a right angle with the ground. The other rope forms an angle of 50° with the ground. What is the balloon’s height?

60. Trigonometry Applications of the sine and cosine
• if the hypotenuse and the angle of elevation are known then the length of the adjacent side can be determined.
A hot-air balloon is tied to the ground with two taut (straight) ropes. One rope is directly under the balloon and makes a right angle with the ground. The other rope forms an angle of 50° with the ground. What is the distance along the ground between the two ropes?

61. Trigonometry Applications of the sine and cosine
• if the hypotenuse and the angle of elevation are known then the length of the opposite side can be determined.
A hot-air balloon is tied to the ground with two taut (straight) ropes. One rope is directly under the balloon and makes a right angle with the ground. The other rope forms an angle of 50° with the ground. What is the distance along the ground between the two ropes?

62. Trigonometry
Goals
• Solve narrative problems by applying trigonometry of a right triangle.
HW due Tuesday 1/11
325.6, 7, 9, 11, 14, 16, , 23, 26, , 30

63. Trigonometry
Goals
• Solve narrative problems by applying trigonometry of a right triangle.
HW due Wednesday 1/12
329.13, 18, 19, , 24, 28, 29, 32