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First Semester Final Review

Advanced Geometry. First Semester Final Review. What is a postulate?. A) A mathematical statement that can be proved. B) An unproved assumption States the meaning of a term or idea IDK. What is a theorem?. A) A mathematical statement that can be proved. B) An unproved assumption

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First Semester Final Review

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  1. Advanced Geometry First Semester Final Review

  2. What is a postulate? • A) A mathematical statement that can be proved. • B) An unproved assumption • States the meaning of a term or idea • IDK

  3. What is a theorem? • A) A mathematical statement that can be proved. • B) An unproved assumption • States the meaning of a term or idea • IDK

  4. What is a definition? • A) A mathematical statement that can be proved. • B) An unproved assumption • States the meaning of a term or idea • IDK

  5. Which of the following is always reversible? Remember! A GOOD definition is ALWAYS reversible! • A) Theorem • B) Postulate • Definition • IDK

  6. Write the following statement as a conditional, then converse, inverse, & contrapostive: “An angle whose measure is 60⁰ is acute” Conditional: If the measure of an angle is 60⁰, then it is acute. Converse: If an angle is acute, then its measure is 60⁰. Inverse: If the measure of an angle is not 60⁰, then it is not acute. Contrapostive: If an angle is not acute, then its measure is not 60⁰.

  7. Which of the following statements is true, given the conditional statement is true? The converse, inverse, or contrapostive: Conditional: If the measure of an angle is 60⁰, then it is acute. Converse: If an angle is acute, then its measure is 60⁰. Inverse: If the measure of an angle is not 60⁰, then it is not acute. Contrapostive: If an angle is not acute, then its measure is not 60⁰. Thm 3: If the conditional statement is TRUE, then the contrapositive is ALWAYS true! False! What if it is another acute angle, like 25° ? False! A 25° measure is not 60°, but it IS acute! TRUE! If an angle is not acute, it must be right, obtuse, or straight. . . (90°, 91°, 140°, 175° . . . 180°) None of these, among many others, are “NOT acute” measures of angles!

  8. Given points X and Y, write: Segment XY: XY Line XY: XY Ray XY: XY Length of XY: XY

  9. Triangle Classification By ANGLE: Remember! All triangles contain AT LEAST two acute angles! ACUTE: ALL angles are between 0⁰ and 90⁰ RIGHT: Contains one 90⁰ angle OBTUSE: Contains one angle greater than 90⁰

  10. Triangle Classification By SIDES: SCALENE: All sides have different measures ISOSCELES: At least two congruent sides EQUILATERAL: All sides have same measure

  11. Complementary vs. Supplementary Angles If two angles are, Complementary: these have a sum of 90⁰ Supplementary: these have a sum of 180⁰ C 90° S 180°

  12. Bisected Angle: M ∡ ABC = 60⁰ BD bisects ∡ABC, so m∡ABD = ? B 60⁰  2 = 30⁰ A C D

  13. Trisected Angle: M ∡ ABC = 60⁰ BD and BE trisect ∡ABC, so m∡ABD = ? B 60⁰  3 = 20⁰ A C D E

  14. Sides of an Angle: Given ∡ ABC, what are the sides of the angle? B BA and BC A C

  15. Adjacent Angles: Given ∡ ABC and ∡CBD are adjacent angles, what would it mean if they were also SUPP & congruent? Right Angles! C D A B

  16. ΔTRY ΔNOWName all congruent parts: Use CPCTC: Corresponding Parts of Congruent Triangles are Congruent Three Angle Pairs: ∡T  ∡N ∡R  ∡O ∡Y  ∡W Three Pairs of Sides: TR  NO RY  OW TY  NW T N W R Y O

  17. Find the measure: Quadrilateral ABCD is a rhombus and the measure of angle A is 30⁰. What is the measure of ∡B ? A B 30⁰ 150⁰ 180 - 30⁰ = D C

  18. Midpoint of a Segment Given endpoints: A (-2, 4) & B(10, 22) Find the midpoint of segment AB. -2 + 10 , 4 + 22 2 2 8 , 26 2 2 (4 , 13) Midpoint

  19. Quadrilaterals If a quadrilateral has one pair of opposite sides congruent and parallel, it is a: PARALLELOGRAM

  20. Quadrilaterals If a quadrilateral has opposite sides congruent and contains ONE RIGHT ANGLE, it is a: RECTANGLE

  21. Quadrilaterals If the diagonals of a parallelogram are perpendicular bisectors of each other, and CONGRUENT, then the parallelogram is a: SQUARE

  22. Quadrilaterals If the diagonals of a quadrilateral bisect each other, and it has two consecutive sides CONGRUENT, then it is a: RHOMBUS

  23. Name the methods used for proving triangles congruent. SSS Side-Side-Side SAS Side-Included Angle-Side ASA Angle-Included Side-Angle SSA Oops! Not this one! HL Hypotenuse-Leg (Right Triangles Only!)

  24. Parallel Lines Given: y = 3x – 5 Write an equation in slope-intercept form of a line that is PARALLEL: Sample Answer: y = 3x + 5 Write an equation in slope –intercept form of a line that is PERPENDICULAR. Sample Answer: y = - 1/3 x – 5 PARALLEL Same Slope Opposite Reciprocal Slope PERPENDICULAR - 1/3 You know it is the opposite reciprocal slope if when multiplied their product = -1 (3)(- 1/3) = -1!

  25. Find slope given two p0ints. Find the slope of the line containing the points (6, -9) and (10, -25) y – y ∆y m = or x – x ∆x -9 – (- 25) m = 6 – 10 16 – 4 – 4 = =

  26. Definitions of lines: Perpendicular Parallel Oblique Skew B C Intersecting at right angles A D Coplanar lines that do not intersect CUBE F Two intersecting lines that are not perpendicular G E Two lines that are not coplanar. These will never intersect . . . So that means these lines could never be perpendicular either!

  27. Isosceles Triangles If a triangle is isosceles, and the measure of the vertex angle is 80⁰, what is the measure of each base angle? 80⁰ 180⁰ – 80⁰ = 100⁰ 100⁰  2 = 50⁰ 50⁰ 50⁰

  28. Isosceles Triangles What does it mean about a triangle if one of its medians is also an altitude? Hint: The altitude⇒⏊ segments and the median divides the base into two congruent segments. Since the point at the vertex is ON the ⏊ bisector, it is equidistant from the endpoints of the base! Easy peasy 

  29. Exterior Angles of Triangles If the measure of one exterior angle of a triangle is 125⁰, and one of the remote interior angles is 25⁰, what is the measure of the other remote interior angle? Sum of angles in Δ = 180. ? 180 – (55 + 25) = 180 – (80) = 100 100⁰ 125⁰ 55⁰ 25⁰ 180 – 125 = 55

  30. Another  Bisector Problem BD is the perpendicular bisector of AC in Δ ABC. This means which angle is congruent to ∡ABD? B ∡CBD  ∡ABD A C D

  31. Intersections and Unions The UNION of segments WR, RT, and WT is _?_ R T W Δ WRT

  32. Intersections and Unions The INTERSECTION of ray TY with ray TR is _?_ R T Y Point T

  33. Intersections and Unions The UNION of ray TY with ray TR is _?_ R T Y ∡RTY

  34. A Making Triangles C B Which of the following measures could be used to form a triangle? REMEMBER! Triangle Inequality Rules: A + B > C B + C > A A + C > B 1, 2, 3 Is 1 + 2 > 3? Nope! 2, 4, 5 Is 2 + 4 > 5? Yes Is 4 + 5 > 2? Yes Is 2 + 5 > 4? Yes 3, 4, 5 Is 3 + 4 > 5? Yes Is 4 + 5 > 3? Yes Is 3 + 5 > 4? Yes 4, 4, 6 Is 4 + 4 > 6? Yes Is 4 + 6 > 4? Yes Is 4 + 6 > 4? Yes 2, 12, 21 Is 2 + 12 > 21? NO!

  35. Review Sheet Problems Now do problems 17 and 40 – 43 on your semester final review sheet, then we will go over them in class 

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