5.5/5.6 Theorems
Opposite Sides & Angles Theorem
Questions:
On a lined sheet of paper, record your responses to the following:
1. If the red angle is the largest angle then the _______ side is the longest side (you can use your mouse to change the shape of the triangle).
2. If the green angle is the largest angle then the _______ side is the longest side.
3. If the blue angle is the smallest angle then the _______ side is the shortest side.
4. If the red side is the longest side then the _______ angle is the largest angle.
5. If the green side is the shortest side, then the ______ angle is the smallest angle.
6. In conclusion, the ______ side will always be _______ from the ______ angle and the ______ side will always be _______ from the ______ angle. This is called the Opposite Sides & Angles Theorem. It is on page 328 ( your book broke it into two theorems). Now would be a good time to add it to your Truth Bank.
7. In ΔXYZ, ∠Y is the smallest angle, so _____ must be the shortest side.
8. In ΔQRS, QR is the longest side. What can you conclude?
9.In ΔTUV,∠U is the largest angle. What can you conclude?
10. In ΔHKJ, m∠K =125º, m∠H =11º andHJ = 8 in.
a) What is m∠J?
b) Is KJ =, <, or > 8 in.?
c) Is HK =, <, or > 8 in?
11. In ΔDEF, m∠D =80º, m∠E =60º ,DE = 500 ft., and EF = 800 ft.
a) What is m∠F?
b) List the angles in ΔDEF in order from smallest to largest.
c) List the sides of ΔDEF in order from shortest to longest.
d) What is the shortest that DF could be? What is the longest it could be?
12. In ΔPQR, PQ = 18m, QR = 26m, and PR = 18.5m. List the angles of ΔPQR in order from smallest to largest.
13. In ΔLMN, m∠L =º, and m∠N =10.75º.
a) List the angles of ΔLMN in order from smallest to largest.
b) List the sides of ΔLMN in order from shortest to longest.
Check your answers before moving on to the next part.
Triangle Inequality Theorem
Questions
It turns out that not every group of 3 segments can make a triangle. The goal of this next part is to discover what the pattern is.
14. Use the applet above. With AB=10, AC=4, and BC=7, is it possible to move the C points together to create a triangle?
15. Now use AB=10, AC=4, and BC=5.5. You should notice that it is impossible to create a triangle with these three segment lengths. Why is this? What could you change to make it work?
16. Now use AB=8, AC=3, and BC=1. These segments don't make a triangle either. Leave AB=8 and AC=3. How long do you have to make BC so that it can form a triangle? BC must be at least larger than _______. How does this number relate to the lengths of AB and AC?
17. Leave AB=8, and AC=3, but make BC = 15. This can't make a triangle either. Is BC too short or too long? In order to form a triangle, BC must be at least shorter than ________. How does this number relate to the lengths of AB and AC?
18. Now use AB=12 and AC = 5. In order to form a triangle, BC must be longer than _____ but shorter than ______. In other words: _____ < BC < ______.
19. Now use AB=3 and AC=6. In order to form a triangle, _____ < BC < _____.
20. Suppose AB=500 and AC=200. In order to form a triangle, _____ < BC < _____.
21. Let's write a rule that works for any AB and AC. In order to form a triangle with AB and AC,
____________ < BC < ____________. This is called the Triangle Inequality Theorem. Now would be a good time to add it to your truth bank. Don't forget a picture!
Triangle Inequality Theorem again
Questions
Just incase you didn't get it with the last applet, here's another applet that demonstrates the same thing.
Notice that when you change AC and AB, the possibility for CB changes too.
22. If AC=3 and AB=10, then _____ < CB < _____.
23. If AC=12 and AB=3, then _____ < CB < _____.
24. If AC=7 and AB=7, then _____ < CB < _____.
25. If AC=482 and AB=900, then _____ < CB < _____.
26. If AC= and AB=, then _____ < CB < _____.
27. If AC= and AB=, then _____ < CB < _____.
The Hinge Theorem.
Questions
Notice that in the above applet, the blue segments are congruent and the black segments are congruent. The blue and black segments also do not change length.
28. Notice that when you lengthen BE, what happens to ∠A?
29. If BE > DF, then what is true about angle A and angle C?
30. If m∠C > m∠A, then what is true about BE and DF?
31. Set BE=5. If DF > 5, then m∠C > _____.
32. Set m∠A=70. If m∠C < 70, then DF < _____.
33. In triangle QRS and triangle XYZ, QR=XY and RS=YZ. If QS > XZ, then what can you say about ∠R and ∠Y? (Drawing a picture might help).
34. This is called the Hinge Theorem. It's in the green box on page 335 of your textbook (they broke it into two theorems). Now would be a good time to add this to your truth bank.
THE LAST SET OF QUESTIONS ARE FROM THE TEXTBOOK:
p331: 8,11-13,17-23
p338: 3-9