Construction Video
The video linked above is good. It's quick but goes through all the basics. Make sure you know how to do each of them as they provide the steps for the tougher ones (e.g. incenter, inscribed equilateral triangle, ...)
We'll go through those at the end of classes until Tuesday.
Wednesday, May 18, 2011
Saturday, March 12, 2011
Refreshers and Tips for 1-9
OK, y’all, I trust that you know the formulae for area and circumference of a circle. As I said in class, one way to keep them straight is to remember that area is always given in square units which means that a unit had to be squared. (cm x cm = cm2) So, area is the formula with the squared radius.
You have to complete section 1-9, largely on your own. Treat it like a proof packet which will have to be turned in. Much of it is review from middle school math but I’ll give you tips in case you’ve forgotten the details.
1-7) Basics (Draw a quick sketch for 6 and 7)
8-11.) Read directions. Don’t use a calculator.
12-15) I changed the directions to read “Find the circumference of the circle to 4 significant figures.” Refer to the sheet I gave you in class for help with significant figures. Use a calculator.
16-19.) Draw these out. For some sides, you’ll need the distance formula.
20-26.) Be careful with units. They must be the same. Give two answers, one in each unit given.
27-32.) Again, leave pi in your answer and give the units.
33-36.) Use the pi key on your calculator unless you happen to know pi to 9 decimal places. This time, give the area to 5 significant figures.
37-38.) What’s it going to be – addition or subtraction? You choose. One way is clearly easier for #38 but you can do it either way.
39.) Part c is good general knowledge to have. A square foot is a square with 1-foot sides.
40-46.) SKIP
47-49.) I know these are simple to do mentally but I’d like to see you create solvable equations from formulae. For each of these, follow this algorithm:
a. Top line – formula (No values substituted into the variables yet.)
b. Next line – fill in the parts that you know, making an equation
c. Solve, keeping your = signs lined up.
d. You will need to repeat this twice for 50c.
50.) Use good ole’ common sense for part a and b. For part c, this is an important skill to have. Pay attention to this… it says to “write a formula for the area of a square – “ (A) – “in terms of its perimeter. “- (P) This means that your formula will start with A= and can only have a P for its variable. The s variable will be the bridge from one to the other. Solve for s in the perimeter equation and then substitute what you get into the area equation.
51.) It helps to label each face of a solid with a number in order to draw its net.
52.) Use inches throughout the problem and show the units in each step. As mentioned above, units act just like variables. When you multiply inches by inches, you have inches2. If you divide inches2 by inches, you’ll have inches.
53-54.) SKIP (too easy)
55-56.) Just do it.
57.) Read the rounding part carefully. ( If I round a number to the nearest ten, that number must end with at least one zero. If I round a number to the nearest hundred, it will end with two zeros.)
58.) Good algebra practice. Show the expression that you used to find the answer.
59-62) SKIP
63.) Don’t forget that you’re first finding a diameter. Read carefully to decide whether or not to calculate pi.
64.) Refer to class notes on using your calculator to make tables of values. Specifically:
a.) Are you in FCN mode? You should be.
b.) In this mode, the variable will always be x. Your work will have b’s (for the base) but change them on your paper to x’s.
c.) Write an equation for the height of this corral in terms of the base (x). Hint: There’s 100 feet of fencing available so you will write an equation that looks like this: ______ + h = 100. Look at the picture. Solve this equation for h. (In other words, get h alone.)
d.) We enter equations into the calculator with the y= key. Enter your equation for h. (Clear whatever is on your y= screen.)
e.) Use TBLSET to start at 1 and to make steps of 1. Quit TBLSET. Show your table. (2ND GRAPH) Do you know what the columns represent?
f.) Now you want another column for the area of the corral. You will enter an area equation as y2. You will need to multiply every value of x to its corresponding y1 value. There is not a key for y variables but they’re in there. Find the VARS key. (VARS – YVARS – Y1)
g.) So your y2 equation will be y2 = xy1.
h.) When you press TBL now, you will have three columns. The first is the list of possible integral bases, the second is the corresponding heights if you’re using all 100 feet of fencing and the third is the corresponding areas. Isn’t that cool?!
i.) Find the greatest area and give its dimensions.
LOOKING AT THE TABLE, WRITE ALL THE VALUES ON YOUR CALCULATOR SCREEN WHEN THE HIGHEST AREA IS RIGHT IN THE MIDDLE.
j.) To see the graph, you have to adjust your window. We know that the x-values (the possible bases) range from 0 to 100. Well, not really but that’s a decent range. The area values range from 0 to something around 1400. (I’m not telling.) To set my window (the part of the graph that we can view), I enter the smallest and largest x’s for Xmin and Xmax. Then divide the difference by 10 to get XStep. Do the same for y’s. Reset your window to those numbers.
k.) We’re told to graph the base on the horizontal axis and the area on the vertical. We need to turn off the height equation so we don’t see it when we graph. Just go to the y= screen, highlight the = sign for y1, and press ENTER. This should un-highlight the = sign and turn off this graph.
DRAW A PICTURE OF YOUR GRAPH AND GIVE THE DIMENSIONS OF YOUR VIEWING WINDOW.
PHEW……….. Aren’t you glad that most of this assignment was so easy? This will seem easy to you once you’re halfway through advanced algebra next year.
65.) Awesome question. Easy to answer but harder to comprehend.
66-68.) I like this one too. Don’t be afraid to use variables as if they were numbers. They ARE numbers for us. Of course your answers will have variables in them. For #67, you could look back at your work for 50c and see if it worked.
69.) This is a cool problem. Now that you know the Pythagorean Theorem so well, you can find two squares with radicals in their sides whose areas add up to an 11 X 11 square. Find three squares with integral sides that add up to the 11 x 11 square. I just used guess and check. I didn’t find four squares but I did find more than four that added up to the right answer.
70.) Let’s use four significant digits for the answer. Think carefully about the radius of the two circular ends of the field. This problem isn’t bad, huh?!
Thursday, March 3, 2011
I was intending to post tips but I don't want to give too much because this is for extra credit - in other words, not required. Let's just cover the basics:
1. An equilateral triangle is made by starting a working line of random length and keeping the compass set to that radius the whole time. One X is all it takes.
2. An inscribed circle comes from finding the incenter, that is, finding the intersection of the angle bisectors. It is easier to do the circumcenter first though because you can use a perpendicular segment to find the perfect length for the inside circle. But, hey, wait a minute, isn't the triangle supposed to be equilateral for the first construction?! There's some connection between incenter and circumcenter for an equilateral triangle.
3. To construct a square, it might be easiest to construct two congruent diagonals that are perpendicular to each other. Faster than working with sides.
4. Rhombi (isn't that the plural of rhombus??) are easiest to draw with diagonals too. Remember the question on the quarter exam?
5. Remember the short cut for the circumcenter of a right triangle.
6. Warning: If you google "How to construct an equilateral triangle inside a circle", you might come to a schnazzy site that looks cool but goes the long way. You'd be better off starting at the center and lightly drawing a radius. Construct a 120 degree angle on it and.... You can divide the circle into thirds and connect the dots.
Hope this helps a little (and not any more!)
Mrs. Dant
1. An equilateral triangle is made by starting a working line of random length and keeping the compass set to that radius the whole time. One X is all it takes.
2. An inscribed circle comes from finding the incenter, that is, finding the intersection of the angle bisectors. It is easier to do the circumcenter first though because you can use a perpendicular segment to find the perfect length for the inside circle. But, hey, wait a minute, isn't the triangle supposed to be equilateral for the first construction?! There's some connection between incenter and circumcenter for an equilateral triangle.
3. To construct a square, it might be easiest to construct two congruent diagonals that are perpendicular to each other. Faster than working with sides.
4. Rhombi (isn't that the plural of rhombus??) are easiest to draw with diagonals too. Remember the question on the quarter exam?
5. Remember the short cut for the circumcenter of a right triangle.
6. Warning: If you google "How to construct an equilateral triangle inside a circle", you might come to a schnazzy site that looks cool but goes the long way. You'd be better off starting at the center and lightly drawing a radius. Construct a 120 degree angle on it and.... You can divide the circle into thirds and connect the dots.
Hope this helps a little (and not any more!)
Mrs. Dant
Thursday, January 27, 2011
Tips for Packet due Monday
Ok, here goes. I explained several of these in class so I will be brief on those.
#3. Do NOT set the expressions equal to each other. That would only happen in a rhombus.
#4. If you work with slopes first, you'll have to do two calculations. That will tell you parallelogram (?) and possibly rectangle. Regardless, you'll have to do a couple of distance calculations to see if you're dealing with a rhombus. Remember, rhombus + rectangle = square.
#5. You should have A = ....... and P = ......... In part c, the word evaluate means to find a value for A and P.
#6b. Kites?! What are they, I'd like to know. Do we take Wikipedia as a reliable source???? According to them, a kite may be a square. OK, so our textbook loses by Wiki rules.
#7. Focus on the angles that I numbered in class. Converse of AIA is useful.
#8. You already have a p-gram. To make it a rhombus, you only have to get two consecutive sides congruent. If you prove the overlapping triangles congruent, you've got CPCTC and you're almost done.
#9. Five steps, if you include the parallels in the Given. You could put Draw steps into the beginning of the proof which would add on two steps. It would be a professional geometer's way to go.
#10. Still in the A exercises... don't overthink it. I like an equidistance theorem to make this a slick proof.
#11. I have 9 steps because I just put in the fact that AE is NOT parallel to DC in the ending triple play. For a reason, just put Diagram. About that triple play, it includes one parallel statement, one non-parallel statement and one congruence statement. Wrap those up and you're done.
#12. Ah, one of my favorites. Here's a helpful hint... if I have to subtract two numbers from a third number but I only know the sum of those two numbers, can't I subtract the sum? In other words, I can subtract x and y from z even if I don't know what x and y each are. If I know their sum, it's all the same.
#13. No hints needed. Be observant.
#14. I worked on four slopes and two distances. You have other options. Keep in mind that slopes give a little more bang for the buck. They prove parallelograms and, if you're lucky, show perpendicular segments too. Show all work.
#15. Long proof. First focus on Triangle PQA. That will give you sides that you need in a transitive step. Then switch over to Triangle RAS. That will give you angles in a transitive step. Once you have the two angles at vertex S congruent, you're done. You'll need AIA and "If sides, then angles."
#16. Discussed in class. For the base, don't think horizontal.
#17. Draw the special quads and use their diagonals as mirrors.
#18. Took 10 steps for me. After I used the Given stating "Segments are drawn...", I followed up with simple parallel statements using the right letters. That makes the first Prove a breeze.
For the second part, do you remember which letter patterns we can use to describe alternate interior, corresponding, same-side interiors, ...? For example, SSI's take on a U-shape. Looking at letter patterns might help you. It is fine to say BC is parallel to MP, even though BC is an extension of the parallelogram. I figured that as a Restate (or Given would be fine).
#19. Discussed in class. Don't forget to finish up with the Perimeter.
#20. How can you figure out (x, y). One is already on the diagram, the other comes from the fact that the point is on a line.
#21. Area = pi r squared
#22. Answers will vary. But we all know what shape will result.
#23. After our circumcenter/centroid work, this is child's play, right? Midpoints are averages of x's and y's.
#24. Reflection of a point over an axis only changes the sign of one coordinate. So label D appropriately. Then set up an algebra equation by setting the slopes equal. Once you set up that equation, the answer might jump off your paper (or you can cross multiply).
#25. Probability of two things happening = (probability of first) X (probability of second) When you figure out the probability of choosing one true statement, assume SUCCESS. This means that there is one less true statement left in the pot. The first probability will have a denominator of 4. The second will have a denominator of 3.
#26. See #24 for reflection tips. I did two slopes and counted two distances.
#27. I proved two overlapping triangles congruent to get angles of one isosceles triangle. Then a quick subtraction property closes it up.
#28. Same old stuff.
#29. Imagine the whole figure shaded. What's its area? Subtract out the white part. And speaking of that white part, you know that x must be less than some quantity. So x squared will be less than that quantity squared. (And of course you might throw in there that x must be greater than 0.)
FEEL FREE TO LEAVE A COMMENT SO THAT I KNOW WHO WAS HERE.
#3. Do NOT set the expressions equal to each other. That would only happen in a rhombus.
#4. If you work with slopes first, you'll have to do two calculations. That will tell you parallelogram (?) and possibly rectangle. Regardless, you'll have to do a couple of distance calculations to see if you're dealing with a rhombus. Remember, rhombus + rectangle = square.
#5. You should have A = ....... and P = ......... In part c, the word evaluate means to find a value for A and P.
#6b. Kites?! What are they, I'd like to know. Do we take Wikipedia as a reliable source???? According to them, a kite may be a square. OK, so our textbook loses by Wiki rules.
#7. Focus on the angles that I numbered in class. Converse of AIA is useful.
#8. You already have a p-gram. To make it a rhombus, you only have to get two consecutive sides congruent. If you prove the overlapping triangles congruent, you've got CPCTC and you're almost done.
#9. Five steps, if you include the parallels in the Given. You could put Draw steps into the beginning of the proof which would add on two steps. It would be a professional geometer's way to go.
#10. Still in the A exercises... don't overthink it. I like an equidistance theorem to make this a slick proof.
#11. I have 9 steps because I just put in the fact that AE is NOT parallel to DC in the ending triple play. For a reason, just put Diagram. About that triple play, it includes one parallel statement, one non-parallel statement and one congruence statement. Wrap those up and you're done.
#12. Ah, one of my favorites. Here's a helpful hint... if I have to subtract two numbers from a third number but I only know the sum of those two numbers, can't I subtract the sum? In other words, I can subtract x and y from z even if I don't know what x and y each are. If I know their sum, it's all the same.
#13. No hints needed. Be observant.
#14. I worked on four slopes and two distances. You have other options. Keep in mind that slopes give a little more bang for the buck. They prove parallelograms and, if you're lucky, show perpendicular segments too. Show all work.
#15. Long proof. First focus on Triangle PQA. That will give you sides that you need in a transitive step. Then switch over to Triangle RAS. That will give you angles in a transitive step. Once you have the two angles at vertex S congruent, you're done. You'll need AIA and "If sides, then angles."
#16. Discussed in class. For the base, don't think horizontal.
#17. Draw the special quads and use their diagonals as mirrors.
#18. Took 10 steps for me. After I used the Given stating "Segments are drawn...", I followed up with simple parallel statements using the right letters. That makes the first Prove a breeze.
For the second part, do you remember which letter patterns we can use to describe alternate interior, corresponding, same-side interiors, ...? For example, SSI's take on a U-shape. Looking at letter patterns might help you. It is fine to say BC is parallel to MP, even though BC is an extension of the parallelogram. I figured that as a Restate (or Given would be fine).
#19. Discussed in class. Don't forget to finish up with the Perimeter.
#20. How can you figure out (x, y). One is already on the diagram, the other comes from the fact that the point is on a line.
#21. Area = pi r squared
#22. Answers will vary. But we all know what shape will result.
#23. After our circumcenter/centroid work, this is child's play, right? Midpoints are averages of x's and y's.
#24. Reflection of a point over an axis only changes the sign of one coordinate. So label D appropriately. Then set up an algebra equation by setting the slopes equal. Once you set up that equation, the answer might jump off your paper (or you can cross multiply).
#25. Probability of two things happening = (probability of first) X (probability of second) When you figure out the probability of choosing one true statement, assume SUCCESS. This means that there is one less true statement left in the pot. The first probability will have a denominator of 4. The second will have a denominator of 3.
#26. See #24 for reflection tips. I did two slopes and counted two distances.
#27. I proved two overlapping triangles congruent to get angles of one isosceles triangle. Then a quick subtraction property closes it up.
#28. Same old stuff.
#29. Imagine the whole figure shaded. What's its area? Subtract out the white part. And speaking of that white part, you know that x must be less than some quantity. So x squared will be less than that quantity squared. (And of course you might throw in there that x must be greater than 0.)
FEEL FREE TO LEAVE A COMMENT SO THAT I KNOW WHO WAS HERE.
Saturday, December 18, 2010
Final Exam Tips
1. Know the postulates and theorems on pp. 770-2.
2. Go over the Chapter Reviews in the text.
3. Go through your quizzes.
4. There are extra practice problems starting on p.716. I would go over the following problems:
Chapter 1 (p.716): #18 - 37
#42 - 47
Chapter 2 : #1 - 28
Chapter 3: #1 - 13
#19 - 41
Chapter 4: #1 - 16
# 21 - 28
# 31-33
Chapter 5: #1 -9
# 13 - 20
# 33 - 42
Of course, you don't have to answer all of those but you can glance at each one and determine whether or not you know how to do it. If you're unsure, try the problem. Odd answers start on p.850.
I would say that most students had a little trouble with finding equations of lines. Make sure you know that material. Know how to find the slope of a line given two points, how to write the equation once you have a slope and how to tell whether lines are parallel or perpendicular.
If you have any questions, post them here or email me. I'll check in regularly.
Study!
Mrs. Dant
2. Go over the Chapter Reviews in the text.
3. Go through your quizzes.
4. There are extra practice problems starting on p.716. I would go over the following problems:
Chapter 1 (p.716): #18 - 37
#42 - 47
Chapter 2 : #1 - 28
Chapter 3: #1 - 13
#19 - 41
Chapter 4: #1 - 16
# 21 - 28
# 31-33
Chapter 5: #1 -9
# 13 - 20
# 33 - 42
Of course, you don't have to answer all of those but you can glance at each one and determine whether or not you know how to do it. If you're unsure, try the problem. Odd answers start on p.850.
I would say that most students had a little trouble with finding equations of lines. Make sure you know that material. Know how to find the slope of a line given two points, how to write the equation once you have a slope and how to tell whether lines are parallel or perpendicular.
If you have any questions, post them here or email me. I'll check in regularly.
Study!
Mrs. Dant
Tuesday, December 7, 2010
Tips for New Proofs and Problems
You will have to use theorems from your notes for these problems. Problem #11 tells you to use Theorem 24, which is not in your book. Here it is:
Thm 24 - If two points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of that segment.
Another useful theorem:
Thm 25 - If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of that segment.
TIPS:
4.) In order for PQ to be the perp. bis. of RS, P and Q must be equidistant from R and S. That means, PR = PS and QR = QS. You don't have to go from congruent segments to equal distances. If you have that PR congruent to PS, that means equidistant. Tip - draw radii. (4 step proof)
5.) You do not have to state that E is on the perpendicular bisector. Let's all agree that it is! Use Thm 25. (5 step proof)
6.) Use subtraction property. (3 step proof, with semicolons)
7.) Just answer it.
8.) Same.
9.) Again.
10.) a.) Ignore angle EBC. You can set up an equation with the two expressions. Make sure you use the right equation. Don't assume that segment AB is perpendicular to CD when you write the eq'n.
b.) Use the solution for x from part a to answer this.
11.) Given: PQ = PR (actually congruent but I don't have that key)
SQ bisects <PQR
______________ (you fill in this one)
Prove: PS perp QR
I used the Division property here, followed by another common theorem. You'll also use Thm. 24. (about 6 steps)
12.) There's a hint already there. To tell which segment is the perp. bisector, consider one point at a time. Once you find two points that are each equidistant to the endpoints of a segment, "connect them" to form the perp. bisector.
13.) Don't neglect to use the transitive property so that you can get all four segments congruent.
14.) Once you have the segments perpendicular, choose one angle to be right and then you can conclude the Prove.
15.) Number four acute angles. (Not the right angles.) You'll have to use Subtraction and 'If angles, then sides.' Theorem 24 is handy too.
16.) Make sure you DO NOT use congruent triangles. To prove something is an altitude, the step before that must include perpendicular segments. (4 step proof)
17.) Ok, on this one, you can prove one set of triangles to be congruent. Then use Thm 24.
18.) Again, what's the definition of altitude? Remember that the area of a triangle is 1/2 (base)(height). But, in a right triangle, the base and the height are the legs.
19.) a.) Figure out the coordinates of C and T.
b.) When you rotate point E 90 degrees clockwise, its coordinates are easier to figure out. (Only 1, 12, -1, or -12) are possible. Use that for a simpler sample and follow the pattern.
Thm 24 - If two points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of that segment.
Another useful theorem:
Thm 25 - If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of that segment.
TIPS:
4.) In order for PQ to be the perp. bis. of RS, P and Q must be equidistant from R and S. That means, PR = PS and QR = QS. You don't have to go from congruent segments to equal distances. If you have that PR congruent to PS, that means equidistant. Tip - draw radii. (4 step proof)
5.) You do not have to state that E is on the perpendicular bisector. Let's all agree that it is! Use Thm 25. (5 step proof)
6.) Use subtraction property. (3 step proof, with semicolons)
7.) Just answer it.
8.) Same.
9.) Again.
10.) a.) Ignore angle EBC. You can set up an equation with the two expressions. Make sure you use the right equation. Don't assume that segment AB is perpendicular to CD when you write the eq'n.
b.) Use the solution for x from part a to answer this.
11.) Given: PQ = PR (actually congruent but I don't have that key)
SQ bisects <PQR
______________ (you fill in this one)
Prove: PS perp QR
I used the Division property here, followed by another common theorem. You'll also use Thm. 24. (about 6 steps)
12.) There's a hint already there. To tell which segment is the perp. bisector, consider one point at a time. Once you find two points that are each equidistant to the endpoints of a segment, "connect them" to form the perp. bisector.
13.) Don't neglect to use the transitive property so that you can get all four segments congruent.
14.) Once you have the segments perpendicular, choose one angle to be right and then you can conclude the Prove.
15.) Number four acute angles. (Not the right angles.) You'll have to use Subtraction and 'If angles, then sides.' Theorem 24 is handy too.
16.) Make sure you DO NOT use congruent triangles. To prove something is an altitude, the step before that must include perpendicular segments. (4 step proof)
17.) Ok, on this one, you can prove one set of triangles to be congruent. Then use Thm 24.
18.) Again, what's the definition of altitude? Remember that the area of a triangle is 1/2 (base)(height). But, in a right triangle, the base and the height are the legs.
19.) a.) Figure out the coordinates of C and T.
b.) When you rotate point E 90 degrees clockwise, its coordinates are easier to figure out. (Only 1, 12, -1, or -12) are possible. Use that for a simpler sample and follow the pattern.
Thursday, November 25, 2010
Five Challenging Proofs
OK guys, these are tough, particularly #17 and #18. Once you get them, however, they might become your favorite proofs of the year! I don't want to give away too much because I'll be robbing you of the joy that comes from figuring them out. (Yes, I did say 'joy'.)
OK, here goes... For each of these, you'll have to prove at least two sets of triangles congruent. Use the Givens and the diagrams to get the first sets. Then, carefully consider each part of those triangles until you find the parts that can be used ~ with CPCTC ~ to get the next set of triangles. Always keep your eye on which parts you're aiming for.
USE COLORED PENCILS TO OUTLINE TRIANGLES ONCE YOU HAVE THEM CONGRUENT.
14.) Subtraction property is quicker than the Congruent Complements Theorem. Start with the small triangles.
15.) Draw. I used SSS and ASA.
16.) This proof took me 11 steps and I did not combine Givens. From the Division Property, you'll need to get two conclusions. You can list them both right away and then use Restates if/when you need to. 'If angles, then sides' and its converse will be very useful.
17.) OK, now comes the real fun!! The first time through, this took me 19 steps but I was convinced I could be more efficient. Sure enough, I had neglected to use the obvious vertical angles. Fixing that saved me four steps! I used THREE sets of triangles on this one so that means I have six triangles listed in my proof. Two of them are RPU and SUQ (which are not congruent to each other, of course). I won't spoil the ending.
Some of my reasons include Subtraction property, Reflexive property, CPCTC, SAS and Vertical angles are congruent. (**Note: For the reflexive property, you can call the same angle by two different names. I did this for the first set of triangles.)
You might want to use backward braces for those statements which give you more than one important conclusion. For example, the congruence statement for triangle RPU gave me two parts that I wanted to use so I used a backward brace there.
18.) As soon as I used a colored pencil, I figured this one out. It's tough because there are so many triangles that you could try. This one took me 14 steps and two sets of triangles. The first set of triangles is easy because the Givens are all you need. Use a colored pencil to outline those.
Then, you move on to the next set of triangles. I chose a set that included angle DAC. You'll be able to use SAS for those. (Number the small angles by vertex D.)
Do not use any triangles that do not have a labeled point at each vertex.
Lastly, 'if sides, then angles' will come in handy with segments AD and BD. A little subtraction is all you'll need to close it up.
TO RECAP THE GRADING:
If you seriously attempt all five, you will be eligible for an A. If you try 3 or 4, you can earn a B. If you try 1 or 2, you can earn a C. I will be grading for content from there.
Have fun with these... every bit as enjoyable as a riddle, brain teaser or Sudoku puzzle. = + )
Happy Thanksgiving and go Cadets!!!!
OK, here goes... For each of these, you'll have to prove at least two sets of triangles congruent. Use the Givens and the diagrams to get the first sets. Then, carefully consider each part of those triangles until you find the parts that can be used ~ with CPCTC ~ to get the next set of triangles. Always keep your eye on which parts you're aiming for.
USE COLORED PENCILS TO OUTLINE TRIANGLES ONCE YOU HAVE THEM CONGRUENT.
14.) Subtraction property is quicker than the Congruent Complements Theorem. Start with the small triangles.
15.) Draw. I used SSS and ASA.
16.) This proof took me 11 steps and I did not combine Givens. From the Division Property, you'll need to get two conclusions. You can list them both right away and then use Restates if/when you need to. 'If angles, then sides' and its converse will be very useful.
17.) OK, now comes the real fun!! The first time through, this took me 19 steps but I was convinced I could be more efficient. Sure enough, I had neglected to use the obvious vertical angles. Fixing that saved me four steps! I used THREE sets of triangles on this one so that means I have six triangles listed in my proof. Two of them are RPU and SUQ (which are not congruent to each other, of course). I won't spoil the ending.
Some of my reasons include Subtraction property, Reflexive property, CPCTC, SAS and Vertical angles are congruent. (**Note: For the reflexive property, you can call the same angle by two different names. I did this for the first set of triangles.)
You might want to use backward braces for those statements which give you more than one important conclusion. For example, the congruence statement for triangle RPU gave me two parts that I wanted to use so I used a backward brace there.
18.) As soon as I used a colored pencil, I figured this one out. It's tough because there are so many triangles that you could try. This one took me 14 steps and two sets of triangles. The first set of triangles is easy because the Givens are all you need. Use a colored pencil to outline those.
Then, you move on to the next set of triangles. I chose a set that included angle DAC. You'll be able to use SAS for those. (Number the small angles by vertex D.)
Do not use any triangles that do not have a labeled point at each vertex.
Lastly, 'if sides, then angles' will come in handy with segments AD and BD. A little subtraction is all you'll need to close it up.
TO RECAP THE GRADING:
If you seriously attempt all five, you will be eligible for an A. If you try 3 or 4, you can earn a B. If you try 1 or 2, you can earn a C. I will be grading for content from there.
Have fun with these... every bit as enjoyable as a riddle, brain teaser or Sudoku puzzle. = + )
Happy Thanksgiving and go Cadets!!!!
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