RPDP Geometry Seminar Quarter 1 Handouts

Transcription

1 RPDP Geometry Seminar Quarter 1 Handouts

2 Geometry lassifying Triangles: State Standard Syllabus Objectives: 5.11, 6.1, 6.4, 6.5 enchmarks: 2 nd Quarter - November Find the distance between: (1, 1) and (6, 4). Plot the points. onstruct a right triangle. Use the Pythagorean Theorem. Use (x 1, y 1 ) & (x 2, y 2 ) to find the Distance formula: Find the midpoint of (-9, 4) and (-, -2). Plot the points and find the point in the middle. Midpoint formula: 19. Use the triangle below: 20. Use the triangle below: a) Find the value of x. a) Find the value of y b) lassify the triangle. b) lassify the triangle. 77 x y

3 11. Use the triangle below: 12. Use the triangle below: a) Find the value of x. a) Find the value of y. 7x- y x+51 6x+6 b) lassify the triangle. b) lassify the triangle. 11. Use the triangle: 12. Use the triangle: a) Find the value of x. x+8 a) Find the value of y. y 147 b) lassify the triangle. x+41 7x+1 b) lassify the triangle. 15. Find the measures of the other 2 angles: 16. Find the measures of the other 2 angles: a 18 1 b a b 17. True or False: is isosceles. Why? 18. Find the lengths of the sides if the perimeter is 19 cm x x + 2

4 10. Find the values of x and y: 11. Find the measures of the base angles: x a y b 12. If two sides of a triangle are 5 inches and 4 inches, what length is not possible for the third side? a) 9 inches b) 6 inches c) 5 inches d) 4 inches e) none of these 14. XY,YZ and ZX are midsegments. 15. XY is a midsegment. The perimeter of = 50. If a = 14, then b = a = x + 2 a X Y b = 2x + 2 If b = 11, then a = c = 4x b c a b Y x = Z X 16. The m = 7x, m = 6x + 5 and m = 24x 10. a) Find x, the measure of the angles and the type of triangle. b) List the sides from shortest to longest: 7. If a triangle has congruent sides, then it is equiangular. Write the statement. Is it true or false? onverse: 9. If two sides of a triangle are 11 inches and 2 inches, what length is possible for the third side? a) 9 inches b) 1 inches c) 11 inches d) 2 inches e) none of these 12. Give the point of concurrency of the 1. Name each segment from VXZ: altitudes () and of the perpendicular bisectors () of the sides of XYZ. median X (0, 0), Y (4, 0) altitude Z (0, -6) bisector V U Z Y W X

5 14. XY,YZ and ZX are midsegments. 15. Find the perimeter and the area of. The perimeter of XYZ = 57. = 6x + 1 = 2x + 5 = 4x x = X b Z a c Y 7m 0m 1m D 12m 18. a) Solve for the vertex angle if the base angles are 7. b) Solve for the base anglesif the vertex angle is If a triangle has congruent sides, then it is equiangular. Write the statement. Is it true or false? ontrapositive: 6. If two sides of a triangle are 5 inches and 9 inches, what length is not possible for the third side? a) 8 inches b) 7 inches c) 6 inches d) 5 inches e) none of these 7. Name each segment from : 8. Find the perimeter and the area of. median altitude E F 10 6 bisector D Give the point of concurrency of the altitudes () and of the perpendicular bisectors () of the sides of XYZ. X (6, -4), Y (6, 0) and Z (0, -4)

6 Geometry onstructions: State Standard Syllabus Objectives: 5.1, 6.6 to 6.9 enchmarks: 2 nd Quarter - November Match the diagrams with the proper construction: a) construct a segment congruent to a given segment d) construct an angle bisector b) construct a perpendicular bisector of a segment e) construct the midpoint of a segment c) construct an angle congruent to a given angle I. II. III. IV.

7 1. On your half sheet of paper draw a triangle any triangle you choose. Please use the majority of the page. ut out your triangle follow the lines!!! 2. Number the angles of your triangle.. Label the separate, full sheet of paper Exterior ngle Theorem. 4. Use a ruler to draw a line on your full sheet of paper. Mark a point on the line approximately 2/ of the length of the line. 5. Line up one of the sides of the triangle, making sure that one of the vertices of the triangle is on the point, on the line you drew. Outline your triangle on the full sheet of paper. arefully number the angles of the triangle onto your tracing. You should have some line left over and not used. 6. Tear off the two remaining angles do not use scissors to cut them off!!! Put these aside for use later. 7. Glue the body of your triangle into the traced triangle. You should be able to see the outline of the two angles that were torn off. 8. Take the two angles that you set aside and carefully place them so that the vertices are also on the point, they do not overlap each other, and they completely fill in the space between the triangle and the line. Glue these on your page. Write a conjecture about the external angle and the two remote interior angles on the bottom of the page.

8 Geometry Lines: State Standard , & Syllabus Objectives: 2.2, , enchmarks: 1 st Quarter - October Slope: incline, slant, steepness, rate of change (+ up, - down). rise run = m = y-y 1 2 x-x 1 2 = y -y x -x = change in y change in x = y x = dy dx verage velocity: change in position change in time or va s = t Ex.1 What is the average velocity of a jet between 5pm and 7 pm if it travels 2642 miles? Ex.2 Find y so that m = 1 4 between (5, 8) and (, y).

9 18. a) What is the slope of a horizontal line? b) Find the slope between (-, 0) & (4, -5) Graph: y = x+ 2, using a t-table: 20. Graph 2x y = 6, using the intercepts: y-intercept: slope: x-intercept: y-intercept: x y 11. Graph using a t-table: y = x 12. Graph 4x y = 12, using the intercepts: 2 y-intercept: slope: x-intercept: y-intercept: x y 16. Which lines are parallel? 17. Which lines are perpendicular? ) 2y = -4x ) y = x ) 16x 4y = 6 D) x = 17 E) y = 2x + 17 F) 8y + 16x = 6 G) y = -2x + 17 H) y = 17

10 Use the following equations: ) y = x ) y = x + 1 ) y = D) x = 0 E) 16y 40x = F) 16x 40y = G) -5y = 2x + H) y = x 14. Which lines are parallel? 15. Which lines are perpendicular? a) only & a) only & D b) only & H b) only & H c) only F & G c) only & H d) E, F & G d) & D and E & G e) & H and F & G e) E & F and & Use the following equations: 1 ) y = x ) y = x + 1 ) y = x 1 D) y = -x 14. Which lines are parallel? & 15. Which lines are perpendicular? & 18. The lines in #19 & #20 are: a) parallel b) perpendicular c) skew d) horizontal e) none Graph using a t-table: y = x Graph 2x - 5y = 10, using the intercepts: 2 y-intercept: slope: x-intercept: y-intercept: x y

11 12. These lines y = 2x and y = -2x are: a) parallel b) perpendicular c) skew d) horizontal e) none 1. Find the slope and then the equation of the line through (2, -1) & (4, -7). 16. List all of the indicated angles: a) orresponding: b) Same side interior: a) If the m 4 = 12, then m 8 = b) Solve for x if m = x + 67 & m 6 = x Use for #s a) If the m 4 = 12, then m 6 = b) Solve for the measures of the angles if m = 2x + 7 & m 5 = 5x List all of the indicated angles: 1. Solve for x: m = 5x - & m 4 = x a) orresponding: b) lternate interior: Solve for x: m 8 = 9x + 2 & m 5 = x Write the Theorem that allows you to solve the problem: 9. Solve for x: m 4 = 7x 7 & m = 5x Solve for the measures of the angles if 7 2 m 5 = 7x 4 & m 8 = 9x Write the Theorem that allows you to solve the problem: 9. Solve for x: m 4 = 2x + 26 & m 8 = x Solve for the measures of the angles: 7 2 m 4 = 5x + 5 & m 6 = 11x

12 Geometry Review (F04) Graphing Lines with Slope Intercept Form Name: Date: Per: 2 y + y = mx + b Example: = x 1 1. = x + 7, y = x & y = x 9 #1) Plot the y-intercept (b) on the y-axis. #2) Find the rise (numerator) of the slope (m). #) Find the run (denominator) of the slope. Go up 2 and over and plot all of the points. Lines with negative slopes go down (L to R) y 2. y = x + 8, y = 5x & y = x 7. y = x + 6, y = x & y = x y = x + 7, y = x & y = x 6 5. = x + 4, y = x & y = x 5 1 y 6. y = 2x + 6, y = x & y = 2x 4 2

13 7 12 Write an equation of a line with the given slope and y-intercept. 7. m =, b = 2 8. m = 1, b = - 9. m = -4, b = m =, b = m = 4.5, b = m = Find the slope of the line passing through each pair of points. 1. (2, 7), (1, 4) 14. (4, ), (-2, ) 15. (-5, 5), (-1, 7) Write the equation of the line: y = x - 2 slope = 20. y = -x + 1 slope = 21. y = -2x + slope = y int = y int = y - int =

14 In the following figure m n and cut by transversal t. Place a sheet of patty paper over the figure below and trace the figure use a ruler for accuracy! t m n Number the figure on the patty paper the same as the figure above. Line your patty paper up over the figure above to determine the following. Note: you may need to rotate or flip the patty paper. Name all the angles congruent to: 1 = 2 = = 4 = 5 = 6 = 7 = 8 = Name all angles supplementary to: 1 = 2 = = 4 = 5 = 6 = 7 = 8 =

15 1 & 5 are in the same position and are called corresponding angles. Name all pairs of corresponding angles. orresponding angles are always: congruent supplementary circle the correct answer 4 & 5 are inside the parallel lines, but on opposite sides of the transversal. These angles are called alternate interior angles. Name all pairs of alternate interior angles. lternate interior angles are always: congruent supplementary circle the correct answer

16 1 & 8 are outside the parallel lines and on opposite sides of the transversal. These angles are called alternate exterior angles. Name all pairs of alternate exterior angles. lternate exterior angles are always: congruent supplementary circle the correct answer & 5 are inside the parallel lines and on the same side of the transversal. These angles are called same side interior angles. Name all pairs of same side interior angles. Same side interior angles are always: congruent supplementary circle the correct answer

17 Proofs Given : a b Find : x and y and provide reasons (properties) for each step. Given : = E Prove : E bisects D a b Given : = + Proof : is a right triangle

18 D O Given : D D Prove : O is isosceles E D Given : Rectangle D E is the midpoint of Prove : E = ED F E G H D Given : Rectangle D E, F, G, H are the midpoints of the sides of the rectangle Prove : EFGH is a rhombus

19 Geometry Quadrilaterals: State Standard , , Syllabus Objectives: enchmarks: 2 nd Quarter December 18. True or false? 19. Define or draw & label a rhombus & a kite: ) Every rectangle is a quadrilateral. ) No parallelogram is a trapezoid. ) No rectangle is a square. 6. IRLE the true statements: a) ll rhombuses are squares. b) Some squares are rectangles. c) No kites are parallelograms. d) ll isosceles trapezoids are trapezoids. e) Some rhombuses are kites. Use the graph for Graph quadrilateral RJZT and then determine the most precise name for the figure. R(, 5), J(2, 5), Z(1, 2) & T(5, 2)

20 11. Identify each pair of statements that forms a contradiction. a) I. D is a rectangle. b) I. D is a quadrilateral II. D is a quadrilateral. III. m + m = 170 II. D is a parallelogram III. D is a isosceles trapezoid 16. Find the perimeter of the parallelogram. 17. Find the area of the non-shaded region Find the area of the parallelogram enclosed by: 20. If the width of a rectangle is 10 inches and it s perimeter is 28 inches, then find y = 2x + 1 the length and the area. y = -1 y = 2x -7 y = GRPH the points/lines and then solve. 1. Find the area of the trapezoid between the lines: 2 + y = -2x - 8, x = -, y = x 5 and x = Find the perimeter and area of rectangle D: (6, 2), (6, -5), (9, -5) and D(9, 2).. Find the perimeter and area of the regular triangle. 6 m m

21 14-15 Determine the values of the variables if the figures are parallelograms: a) b) D = x 2, = 2x 12 c) E = y and E = y 8 x y 5y 18 2x 27 z E 40 y 4 D 16. m D = 1, = 2x 4, 17. Find the measures of 1-5 if m = 91. D= 17 x & ED = 4 2 = 1 5 E = E 4 E = D 67 D m D = m = Determine the values of the variables if the figures are parallelograms: a) b) c) 5y y x 7 46 z x 90 y + 24 D E D = x 2, = x 12 E = 1 and E = y Which of these properties apply to LL rhombi, rectangles and squares? (circle all that apply) a) ll sides are congruent. b) ll angles are right angles. c) Opposite sides are congruent. d) Opposite sides are parallel. e) Opposite angles are congruent. f) Diagonals bisect each other. g) Diagonals are congruent. h) Diagonals are perpendicular.

22 Geometry Polygons: State Standard Syllabus Objectives: enchmarks: 2 nd Quarter - December 15. How many sides does each polygon have? 16. Is the polygon (circle): convex or concave? Find the missing interior angle: a) Dodecagon b) Heptagon c) Nonagon d) Pentagon e) 57-gon Find the measure of an exterior & interior angle of a regular hexagon. 15. Use the polygon to answer these questions: 1. Use the polygon to answer these questions: a) lassify the regular polygon. a) lassify the regular polygon. b) Find the sum of the interior angles. b) Find the sum of the interior angles. c) Find the measure of an exterior angle. c) Find the measure of an exterior angle.

23 Geometry ongruent Triangles: State Standard 4.8.2, Syllabus Objectives: 7.1 to 7.8 enchmarks: 2 nd Quarter - November 19. If DEF, = 18, m = 41 and the m F = 5, which statements are false? a) D b) FD = 18 c) = EF d) m = Write the congruence and state the postulate or write not possible. a) b) c) Postulate: Postulate: Postulate: H H F L K E G I D 17. Given: X is the midpoint of G & NR Prove: NX GRX J M Statements Reasons Given 2. X GX X is the midpoint of NR NX RX 5. N X 1 2 R G 6. 6.

24 Write the congruence and state the postulate or write not possible. a) b) c) Postulate: Postulate: Postulate: T T Y I N O F R N W D I 15. Given: D and D is the midpoint of Prove: Statements Reasons Given 2. D D 2. lines form ( ) right angles. D is the midpoint of. 4. D D D D D D