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Analytic Geometry

Analytic Geometry. EOCT Review. Proofs. Which item can be given as a statement in a proof? A. Given B. Def. of congruent segments C. m<1 + m< 2= 180 D. Trans. Prop. of Equality. Proofs. Identify the property that justifies the statement. m = n, so n = m KL = KL

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Analytic Geometry

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  1. Analytic Geometry EOCT Review

  2. Proofs • Which item can be given as a statement in a proof? • A. GivenB. Def. of congruent segmentsC. m<1 + m< 2= 180 D. Trans. Prop. of Equality

  3. Proofs • Identify the property that justifies the statement. • m = n, so n = m • KL = KL • p = q and q = -1, so p = -1

  4. Proofs • Algebraic Proof • Solve the equation below. Write a justification for each step. • 1/5(a + 10) = -3

  5. Parallelograms • Properties of Parallelograms • - Opposite sides are parallel and congruent • - Opposite angles are congruent • - Consecutive angles are supplementary • - Diagonals bisect each other

  6. Parallelograms • WXYZ is a parallelogram. • Find the measure of angle W. • Find the value of x.

  7. Parallelograms • In parallelogram JKLM, what is the value of <K?

  8. Parallelograms • ABCD is a parallelogram. Find AB and BX.

  9. Parallelograms • In parallelogram DEFG, what is EG?

  10. Angles formed by Lines and Transversals • Corresponding Angles are congruent • Alternate Interior Angles are congruent • Alternate Exterior Angles are congruent • Same Side Interior Angles are supplementary

  11. Angles formed by Lines and Transversals • Find each angle measure.

  12. Angles formed by Lines and Transversals • Find x.

  13. Congruence • 5 Triangle Congruence Theorems • Side-Side-Side • Side-Angle-Side • Angle-Angle-Side • Angle-Side-Angle • Hypotenuse Leg (right triangles only) • Angle-Side-Side is NOT a theorem

  14. Congruence • If ΔKLM ≅ΔRST, find the value of x.

  15. Congruence • What is the measure of angle U?

  16. Congruence • ΔJKL≅ΔMNP. KL = 21x - 2, NP = 20x, LJ = 15x and PM = 13x + 4. Find LJ.

  17. Congruence

  18. Similarity • 3 Triangle Similarity Theorems • Side-Side-Side • Side-Angle-Side • Angle-Angle

  19. Similarity • What theorem proves the triangles are similar?

  20. Similarity • What theorem proves the triangles are similar?

  21. Similarity • What is the length of AC?

  22. Similarity • Find SP.

  23. Similarity

  24. Similarity

  25. Similarity • A drawing of a garden uses a scale of 1 in : 3 ft. Find the length of the garden if the length on the drawing is 13 inches.

  26. Exterior Angles Theorem • Find measure of <RST.

  27. Midsegment Theorem • Find QR. What type of segment is QR?

  28. Midsegment Theorem

  29. Triangles • What is the length of the longest side of the triangle?

  30. Angle relationships in Triangles • What is the value of x if the acute angles of a right triangle measure 8x° and 12x°? • The angles of a triangle measure 4°, 86°, and 90°. Which classification of the triangle is correct? • One angle of an equilateral triangle measures (4x - 20). What is the value of x?

  31. Special Right Triangles • There are 2 types of special right triangles: • 1. 45-45-90 • In a 45-45-90 triangle, the legs have equal length and the hypotenuse is the length of one of the legs multiplied by √2. • 2. 30-60-90 • In a 30-60-90 triangle, the hypotenuse is the length of the shorter leg multiplied by 2, and the longer leg is the length of the shorter leg multiplied by √3.

  32. Special Right Triangles • 45-45-90 • Find the value of x.

  33. Special Right Triangles • 30-60-90 • Find the value of x.

  34. Trigonometry • SOHCAHTOA • Sin = opp/hyp • Cos = adj/hyp • Tan = opp/adj

  35. Trigonometry • 1. Find tan K. • 2. Find cos M. • 3. Find sin K. • 4. To the nearest degree, what is the measure of <M?

  36. Trigonometry • A 24-foot ladder forms a 76° angle with the ground. The top of the ladder rests against a building. To the nearest inch, how high up the building does the ladder reach? • One acute angle of a right triangle measures 28°. To the nearest tenth, what is the length of the side opposite that angle if the hypotenuse measures 16 meters? • A skateboard ramp makes a 22° angle with the ground. To the nearest foot, how high is the ramp?

  37. Trigonometry • 1. Find sin (1.54). • 2. If sin A = 8/17, find the measure of angle A.

  38. Trigonometry • Use the figure below to find each of the following: • 1. m<A. • 2. length of AB • 3. m<B.

  39. Lines that Intersect Circles • Use the figure below to find each of the following: • Chord • Secant • Tangent • Diameter • Radius

  40. Lines that Intersect Circles • To the nearest tenth, what is the length of MN?

  41. Central and Inscribed Angles • A central angle is EQUAL to the measure of its intercepted arc. • An inscribed angle is HALF the measure of its intercepted arc. • An angle inscribed in a semicircle is ALWAYS a right angle. • If two inscribed angles intercept the same arc, the angles are congruent.

  42. Central and Inscribed Angles • Find the measure of arc JK. Then, find the measure of arc JIL.

  43. Central and Inscribed Angles

  44. Central and Inscribed Angles

  45. Central and Inscribed Angles

  46. Central and Inscribed Angles

  47. Inscribed Quadrilaterals • Opposite angles in an inscribed quadrilateral are supplementary.

  48. Arc Length • Find the measures of arcs MN and XY. • Formula is not on the sheet

  49. Sector Area • Find the areas of sectors BAC and QPR. • Formula is not on the sheet

  50. Spheres • Volume and Surface Area formulas are on the sheet • Find the volume and surface area of the sphere.

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