congruent - Wikispaces · PDF fileWest Springfield High School Geometry 2009-10 12 1. Graph the lines 2x – y = 5 and x + 2y = -10 on the same Cartesian coordinate plane. a) Using

West Springfield High School Geometry 2009-10

10

1. Some terminology: Figures that have exactly the same shape and size are called congruent (symbol: ≅). Dissect the region shown at right into two congruent parts. How many different ways of doing this can you find? 2. Le t A = ( 2 , 4 ) , B =( 4 , 5 ) , C = ( 6 , 1 ) , T = (7 , 3 ) , U = (9 , 4) , and V = (11,0). Triangles ABC and TUV are specially related to each other. Make calculations to verify this statement. Write a few words to describe what you discover. Note: A triangle that has two sides of equal length is called isosceles. 3. Make up an example of an isosceles triangle, one of whose vertices is (3, 5). a) If you can, find an isosceles triangle that does not have any horizontal or vertical sides. b) What does the perpendicular bisector have to do with this problem? 4. Let A = (1,5) and B = (3, −1). Verify that P = (8,4) is equidistant from A and B. Find at least two more points that are equidistant from A and B. Describe all such points. 5. Find two points on the y-axis that are 9 units from (7,5). (Hint: Your solution will not be lattice points). Note: A lattice point is a point whose coordinates are integers. For example, (2, 3) is a lattice point, but (2.5, 3) is not. 6. a) Find two lattice points that are 5 units apart, but do not form horizontal or vertical lines. b) Find two lattice points that are exactly 13 units apart. c) Is it possible to find lattice points that are 15 units apart? If so, give an example. If

not, explain. d) Is it possible to form a square whose area is 18 by connecting four lattice points?

Explain. 7. Two complementary angles have degree measures (5x-5) and (x+11). Find x. 8. Two angles are supplementary. a) Do they have to be a linear pair? b) If one of the two angles measures (5x-5) and the other (x+11), what is x? 9. T/F: a) A Linear pair is made up of supplementary angles. b) Supplementary angles form a linear pair. c) Perpendicular lines intersect at right angles. d) Lines that intersect at right angles are perpendicular. e) Two points determine a line. f) Three points determine a plane. g) Three non-collinear points determine a plane. h) Two angles that form a vertical pair are congruent. i ) Two congruent angles form a vertical pair.


11

1. Solve for x +1 = 2x - 3 . (Hint: you can square both sides to eliminate the radical.) [Review Properties of Equality] 2. a) Write a formula for the distance from A = (-1, 5) to P = (x, y). b) Write another formula for the distance from P = (x, y) to B = (5, 2). c) Write an equation that says that P is equidistant from A and B. c) Simplify your equation to linear form (slope-intercept form). Note: You should take this as far as you can. Don’t give up. It involves a lot of algebra. The algebra is lengthy, but not difficult. 3. (Continuation) The line you just found is called the perpendicular bisector of AB. Verify this as follows: Determine the slope of this line. Calculate the slope of line AB. Find the mid-point of segment AB. Plot these on a coordinate plane. 4. Plot each set of points. Find the slope of the line through a) (4,4) and (-2, -2) b) (4,-4) and (-2, 2) c) (-5, 9) and (12, -7) d) (3, 1) and (3 + 4t, 1 + 3t) Note: You may recall me saying, “Two points determine a line.” The statement, “A unique line exists through any two points.” In one form or another, this statement is a fundamental postulate of Euclidean geometry – a statement accepted as true, without proof. 5. Taking this for granted, what can be said about three non-collinear points? Why must you include the words “non-collinear” in your statement? Note: The three angles of a triangle fit together to form a straight angle. This is another postulate. We call it the triangle sum postulate. 6. Taking this for granted, tell what can be said about the two non-right angles in a right triangle by completing this sentence, “The two acute angles of a right triangle are _____” Your statement, if correct, is a corollary, a statement readily deduced from a theorem. 7. Let P = (a, b), Q = (0,0), and R = (−b, a), where a and b are positive numbers. Show that angle PQR is right, by introducing two congruent right triangles into your diagram. Verify that the slope of segment QP is the negative reciprocal of the slope of segment QR. (Hint: I suggest you draw point P on a coordinate plane such that the x-coordinate is farther from the vertical axis, and the y-coordinate is closer to the vertical axis. Use the location you drew for (a,b) to determine where you place ( - b, a). ) 8. Find an example of an equilateral hexagon whose sides are all 13 units long. a) Give coordinates for all six points. b) Your figure is equilateral. Is it also equiangular? Explain. 11.Two iron rails, each 50 feet long, are laid end to end with no space between them. During the summer, the heat causes each rail to increase in length by 0.04 percent. Although this is a small increase, the lack of space at the joint makes the joint buckle upward. What distance upward will the joint be forced to rise? Assume that each rail remains straight, and that the other ends of the rails are anchored. (Hint: draw and label a figure.)


12

1. Graph the lines 2x – y = 5 and x + 2y = -10 on the same Cartesian coordinate plane. a) Using your protractor, measure the angle of intersection. b) What are the slopes of the two lines. c) How are the slopes related? 2. Given that 2x – 3y = 17 and 4x + 3y = 7, without using paper, pencil or calculator, find the value of x. 3. Blair is walking along the edge of her room toward a wall where a lady bug is crawling along the crown molding (top edge of the wall). Assuming the bug does not change direction, will their paths ever cross? Are their paths parallel? Note: If two lines are in the same plane, they must either intersect, or be parallel. Skew lines are two lines not in the same plane. Note: Midpoint is like an average. Two students were talking, “I scored 80 and 100 on my last two tests.” “What is your midpoint?” “You mean average? It was 90.” Note: Attempt these next few problems numerically, algebraically and graphically. 4. The point on segment AB that is equidistant from A and B is called the midpoint of AB. For each of the following, find coordinates for the midpoint of AB. Explain your solutions. a) A = (-3,-3) and B = (5, 5) b) A = (−1, 5) and B = (5,−7) c) A = (m, n) and B = (k, l) Problems (c) and (d) show the “general case” d) A = (x1, y1) and B = (x2, y2) 5. If A is an endpoint and B is the midpoint, find the other endpoint. a) A = (4, 0) and B = (4, 12) b) A = (-1, 0) and B = (13, 0) c) A = (-1, -4) and B = (4, 1) d) A = (2, -3) and B = (11,5) 6. Find the slope of the line through a) A = (0,0) and B = (17, 18) b) A = (-3, 1) and B = (1, -7) c) A = (-14, -5) and B = (16, 27) 7. Find two points not on segment AB, that are equidistant from both points A and B: a) A = (4, 0) and B = (4, 12) b) A = (-1, 0) and B = (13, 0) c) A = (-1, -4) and B = (4, 1) d) A = (2, -3) and B = (11,5) 8. What values of a, b, and c in the equation of a line: ax + by = c will give: a) y = x b) y = 3 c) x = 5 9. a) Is it possible for a line to lack a y-intercept? b) … an x intercept? Explain.


13

1. For each of the following questions, fill in the blank with always true (A), never true (N), or sometimes true (S). Write a few sentences for each, explaining your choice. a) Two parallel lines are ________ coplanar. b) Two lines that are not coplanar _______ intersect. c) Two lines parallel to the same plane are ________ parallel to each other. d) Two lines parallel to a third line are ________ parallel to each other. e) Two lines perpendicular to a third line are ________ perpendicular to each other. 2. The dimensions of rectangular piece of paper ABCD are AB = 10 and BC = 9. It is folded so that corner D is matched with a point F on edge BC. a) Given that length DE = 6, find EF , EC, and FC. b) The area of triangle EFC is a function of the length DE. Write a formula for this function, using the letter x to stand for DE. c) Find the value of x that maximizes the area of triangle EFC. Explain how you made your determination. 3. Rewrite the equation 3x-5y=30 in the form y = m x + b. 4. Rewrite the equation 3x−5y = 30 in the form ax + by = 1. Are there lines whose equations cannot be rewritten in this form? If so, give an example. If not, explain. 5. What is the equation of the line that passes through (3,0) and is vertical? Can this equation be written in the form y = mx + b? Explain. What is m? … b? … x and y? 6. a) Find a and b so that ax + by = 1 has x-intercept 5 and y-intercept 8. b) Write the equation for this line in slope-intercept form (y=mx+b). 7. Consider the linear equation y = 3.5 (x - 1.3) + 2. a) What is the value of x when y = 2? b) What is the value of y when x = 1.3? c) What is the slope of this line? d) This equation is written in “point-slope” form: y = m (x – x1) + y1. Explain the terminology. e) Find an equation for the line through (4.2, -2.5) that is parallel to the given line. Leave your answer in point-slope form. f) Rewrite the equation from (f) in slope-intercept form. g) Describe how you would write the equation of a line that has slope -2 and that goes

through the point (-7, 3). h) Describe how you would graph the equation of a line that has slope -2 and that goes

through the point (-7, 3). 8. In the linear equation that relates distance to rate and time, d = r t, distance is directly proportional to rate and to time. What is the relationship between rate and time? 9. Golf balls cost $0.90 each at Jerzy’s Club, which has an annual $25 membership fee. At Rick & Tom’s sporting-goods store, the price is $1.35 per ball for the same brand. Where you buy your golf balls depends on how many you wish to buy. Explain, and illustrate your reasoning by drawing a graph. 10. A slope can be considered to be a rate. Explain this interpretation.


14

1. Given the points A = (−2, 7) and B = (3, 3), a) Find two points P1 and P2 that are on the perpendicular bisector of AB. b) In each case, what can be said about the triangle PAB? 2. Explain the difference between a line with an undefined slope and one with a slope of zero. 3. Find a way to show that points A = (−4, −1), B = (4, 3), and C = (8, 5) are collinear. Can you come up with more than one way to show this? 4. Must the slope of a line always be either positive or negative? Explain. 5. Some questions regarding linear equations: a) Write the slope-intercept form for the equation of a line. b) What does m represent? c) How do you determine the slope? d) Where on a graph is the y-intercept? What is the x-coordinate of the y-intercept? e) Where on a graph is the x-intercept? What is the y-coordinate of the x-intercept? f) In the slope-intercept form of an equation of a line, which terms are coefficients? g) Which terms are variables? h) What is the difference between a coefficient and a variable? 6. Use the drawing to the right for this problem. a) Find the point of intersection of the lines: 3 x + 2 y = 1 and –x + y = –2 b) Find the point of intersection of the lines: 3 x + 2 y = 1 and –4 x + 9y = 22. c) Find the point of intersection of the lines: –x + y = –2 and –4x + 9y = 22. d) The sides of the triangle at right are formed by the graphs of

3x + 2y = 1, y = x−2, and −4x + 9y = 22. Is the triangle isosceles? Verify your answer (in other words, show your calculations!) 7. What is the relation between the lines described by equations: –20 + 12y = 36 and –35x + 21y = 63? Find a third equation in the form ax + by = 90 that fits this pattern. 8. Given the line y = (3/4) (x + 3) – 2 and the point (9,2). a) Using point-slope form, (y = m(x – x1) + y1, write equations for the lines parallel and

perpendicular to this line through the given point. b) Explain why it is easier to answer this question using this than the slope-intercept form. 9. A clock takes 3 seconds to chime at 3 pm, how long does it take to chime at 6 pm? (Hint: There are pauses between the chimes. Draw a number line showing the chimes.) 10. Using both sides of your ruler, draw a pair of parallel lines like those shown. Draw a line that crosses both of the parallel lines. a) Measure all the angles formed between your line and both of the parallel lines. Write the angle measures in the angles you form. b) Do this same thing a second time, but use a different slope to draw the transversal (a line that crosses two or more lines). c) Describe what you notice.


15

z y

xw

c

ba

117°

1. A transversal crosses parallel lines to form the angels as shown below left. Without using a protractor, determine the values of a, b, c, w, x, y and z.

2. Given that both pair of lines are parallel as shown above right, and the angle measures are as shown in terms of x and y, find x and y. Explain your work. 3. Given that both pair of opposite lines in the drawing below left are parallel. Find x. Explain your work. (Hint: Draw auxillary lines – e.g. extend the lines shown, then look for angle relationships)

4. In the diagram above right, l||m and the angles are as marked. Find x.

5. Can you find x in the diagram above left? 6. In the figure above right, show that –B @ –D and –A @ –C . 7. One angle in a triangle is twice another angle, and the third angle is 54°. What is the measure of the smallest angle in the triangle?

3x-15y

x

3y+15

x

3x

140

m x

130°

l

C

B

A

3x-y

3x+y

CD

A B


16

1. In the diagram below left, m||n and the angles are all as marked. Find x.

n

m

x°

35°

40°

B

A

C

2. In the diagram above right, k||l and m||n. If the angles are as marked, find x and y. 3. Three triangle are shown below. a) Use your protractor to measure the three angles in each. b) Can you guess a statement that is always true about the sum of the measures of the angles in a triangle?

A B D E G H

C

F

I

4. Given that l m! in the figure at right, find BDC– and BCD– . 5. Prove that the measures of the angles in a triangle always sum to 180° (Triangle Sum Postulate) as follows: a) Draw a triangle, ∆ABC and a line, k, through point A such that k BC

!""#$ .

b) Find angles in you diagram that are equal to B– and C– . Use little arcs to mark pairs of congruent angles. c) Prove that 180m CAB m ABC m BCA– + – + – = . 6. In the diagram m n! , AB m^ , m–CEB = 50! ,

3m ADC m BCE x– = – = ,and m BCD x– = . Find x. 7. Two angles in a triangle have measures 30 degrees and 57 degrees. What is the measure of the third angle?

k

l

m n

x+y

2x-2y

x-y

l

m

50°

70°

DC

B

n

m50°

x

3x

3x

A

B E

CD


17

1. The angles in a triangle are in the ratio 1:2:3. What are the measure of the angles? 2. In the diagram at right, find m–ZYP . 3. One of the angles in a triangle is a right angle. Show that the other two angles are complementary. 4. Below left, solve for x. (Hint: Draw auxiliary lines. Use what you know about triangle)

5. Given the angles as shown above right, find x. 6. Prove that m X m Y m XZP– + – = – in the diagram at right. 7. In the diagram below left, AB DE

!""# !""#$ , 2 20m BAC x– = - ,

30m ACB– = , and 55m DEF x– = + as shown. Find m CED– . Explain your work.

x+55°30°

2x-20E

DB

A

C

F

8. Above right, find x+y+z . Why this might be called the “Walk-around” Theorem? 9. Find x in the figure below left. 10. Find m∠C in the figure above right.

29°

52°

58°

Z

Y

X P

m x

130°

l

C

B

A

x°79°

35°

C

A

B D

Y

X

Z P

z

yx

A

B

C

x°

39°

43°

110°

38°C

A

B


18

–5

–4

–3

–2

–1

1. Find m∠Y in the figure at right. 2. Must an exterior angle of a triangle always be greater than 90°? 3. Using what you know about exterior angles of a triangle, and the triangle sum postulate, determine the sum of the measures of angles 1 through 5 below without using a protractor. 4. a) In the diagram below left, what can we say about lines k and m? Why? b) Explain what’s wrong with ∆PQR. c) Explain why in the second figure the two lines to the right of PQ

!""# cannot meet as

shown.

m

k

40°

40°

40°

40°Q

P

Q R

P

Note: The Corresponding Angles Postulate states that if a transversal crosses parallel lines, then the corresponding angles are congruent. The Converse Corresponding Angles Theorem states that if two corresponding angles, formed by a transversal crossing two lines are congruent, then the two lines are parallel. Do you see the difference between the two statements? 5. Given that the angles have the measures indicated in the diagram, prove that AB CD

!""# !""#$ and.BC AD

!""# !""#$ .

123°

97°

X

Y

Z

70° 80°

30°

80°


19

1. Let A = (4, 2), B = (11, 6), C = (7, 13), and D = (0, 9). Show that ABCD is a square: a) Determine the lengths of the sides of the square. b) Calculate the slopes of the sides of the square. c) What significant patterns do you notice? 2. (Contd.) Three squares are placed next to each other as shown. The vertices A, B, and C are collinear. Find the dimension n. (Be careful. It’s trickier than you may think.) 3. (Contd.) Replace the lengths 4 and 7 by m and k, respectively. Express k in terms of m and n. 4. A five-foot Terrier (a short freshman) casts a shadow that is 40 feet long while standing 200 feet from a streetlight. How high above the ground is the lamp? 5. (Contd.) How far from the streetlight should the Terrier stand, in order to cast a shadow that is exactly as long as the Terrier is tall? 6. Find as many ways as you can to dissect each figure at right into two congruent parts. 7. One leg of a right triangle is 12 units long. The other leg is b units long and the hypotenuse c units long, where b and c are both integers. Find all possible values of b and c. (Hint: both sides of the equation c2 – b2 = 144 can be factored.) This problem is much more challenging than it looks at first glance. Take some time with it. 8. Show that a 9-by-16 rectangle can be transformed into a square by dissection. In other words, the rectangle can be cut into pieces that can be reassembled to form the square. Do it with as few pieces as possible. 9. Write the converse of each of the following statements, then identify whether or not that converse is true. a) If two teams are playing in the World Cup Finals, then the teams must be playing soccer. b) If two of the angles of a triangle sum to 80°, then one angle of the triangle must be 100°. c) If a river is the longest river in the world, then it must be the Nile. d) If an animal is a duck, then it must be a bird. e) If two angles form a linear pair, then they must be supplementary. f) If two angles are complementary, then their measures must sum

to 90. 10. What Is wrong with the figure shown at right? 11. Suppose that numbers a, b, and c fit the equation a2 + b2 = c2, with a = b. a) Express c in terms of a. b) Draw a good picture of such a triangle. c) What can be said about its angles? d) How would you classify this triangle?


20

4x-10

x+10

x

5x

3x

1. In the diagram below left, we accept without further proof that if k l! then x = y (Corresponding Angles Postulate). Prove that if k l! , then y = z (Alternate Interior Angles Theorem).

2. In the figure at right, find x. 3. How many seconds does it take the second hand of a clock to rotate through an angle of 72°? 4. Two angles of a triangle are 30° and 70°. What is the third angle? 5. Given !ABC . m–B = 60! . If an exterior angle at A is 190°, what is m–C ? 6. Below left, find x. 7. Below right, find x. 8. Given PQ RS

!""# !""#$ , and TV PQ^

!""# !""#, if TV!""#

intersects PQ!""#

at X and RS!""#

at Y, find m∠RYX. 10. Three angles, ∠A, ∠B, and ∠C have the property that ∠A is complementary to ∠B, ∠B is complementary to ∠C and ∠C is complementary to ∠A. Determine the measure of ∠A. 11. The interior angles of ∆ABC are in the ratio 2:3:4. What are the measures of the angles? 12. Is it possible for two exterior angles of a triangle to be supplementary? Explain. 13. One angle of a triangle is 20°. If the largest angle of the triangle has six times the measure of the smallest, what are the angles of the triangle? 14. The three angles of a triangle have measures ∠A = x – 2y, ∠B = 3x+5y, and ∠C=5x-3y. a) Find x. b) Is it possible to determine the value of y? c) If you are told one of the angles is 10°, what are all the possible positive values of y?

l

k

m

z

y

x

x

30°

30°

122°


21

2x

4x3x

A E

C

B

D

1. Three lines intersect as shown below left. –DOE @ –DOC . m–EOD :m–DOB = 2 : 7 . Find m–FOB .

2. Is it possible for the angles in the diagram below right to have the measure indicated? If so determine x and y. If not, explain why not.

3. Below left, find x. 4. Below right, find x.

5. Show that if a transversal cuts two lines such that same side interior angles are supplementary, the lines are parallel. 6. In the diagram below left, if Θ=28°, find φ. 7. Below right, find x.

8. What is the number of degrees formed by the minutes and hour hands at 11:10 pm? 9. Below left, Find m∠A + m∠B + m∠C + m∠D in the diagram below left. 10. Below right, find x.

2y

x+y

xC

O

A

DB

EF

j

q x

50°50°

60°

50°

x85°

45°

x

2x

2x

C

D

A E

B

A

B

C

D


22

zy

x

w

1. Find the values of x, y, and z if the degree measures of the angles are as shown. (NCTM Calendar 1998) 2. What is the measure of the acute angle formed by the hands of a clock when the clock shows that the time is 12:15? (NCTM Calendar 2008) 3. Find the measure of the angles in the diagram below left, given that the two horizontal segments are parallel. (NCTM Calendar 2008) 4. Above right, If m∠AOB=m∠COD=90°, find m∠COB+m∠AOD. (NCTM Calendar 199903) 5. One angle of a triangle equals the sum of the other two. a) Show that the sum of two exterior angles of the triangles is 180° greater than the third. b) If one of the angles of the triangle equals 40°, what are the other two angles’ measures? 6. a) It is possible for the interior angles of a triangle to be of a ratio 1:2:6. b) Is it possible for the exterior angles of a triangle to be of a ratio 1:2:6? Explain. 7. Find w + x + y + z. 8. Point Z is on PR of ∆PRQ such that PZ=ZQ, and m–PQR - m–PRQ = 42! . Find m–RQZ .

135°

145°


23

By Desiree Spingler 11/2005

36°

56°

26°

1. Find a, b, c, and d. 2. Find m, n, p, q, r, s, t, and u. 3. Find all the angle measures in Desi’s drawing.

Documents

congruent - Wikispaces · PDF fileWest Springfield High School Geometry 2009-10 12 1. Graph the lines 2x – y = 5 and x + 2y = -10 on the same Cartesian coordinate plane. a) Using