introductory exercises about congruence theorems

1 Section 4.3.2: 10) Here are some introductory exercises about congruence theorems. a) What does it mean for two triangles to be congruent ? b) You may assume that SSS , SAS , and ASA are all valid congruence theorems. Discuss and draw a picture illustrating the meaning of each of these theorems. c) Is A AS a valid triangle congruence theorem? If so, briefly show how it follows from the congruence theorems in part (b). If not, produce a counterexample. d) Repeat part (c) for AAA and SSA . 11) Here we connect the Law of Cosines with SSS . a. Does the value of cos γ uniquely determine an angle γ satisfying 0 ≤ γ ≤ π ? Why? b. Use the Law of Cosines to show that if we know all three sides of a triangle (say, of lengths A, B, and C), then we can uniquely determine all three angles. How does part (a) play a role in this process? c. Construct a large triangle. Measure the three side lengths. Then, use the Law of Cosines to compute the cosine of each of the angles in the triangle. d. Using your figure from part (c), measure the angles and compute the cosines of the angle measures. How well do the cosines you just computed agree with the cosines from part (c)? 2 Section 4.6.1: 1) Recall that sine is an odd function. a. Show that 𝐬𝐢𝐧 𝜽 𝜽 is an even function. b. Argue that 𝐥𝐢𝐦− 𝜽→𝟎 𝐬𝐢𝐧 𝜽 𝜽 = 𝟏, given that 𝐥𝐢𝐦+ 𝜽→𝟎 𝐬𝐢𝐧 𝜽 𝜽 = 𝟏. Section: 5.1.4 𝒙𝟐 𝒚𝟐 4) Here we study the hyperbolas given by Equation 5.7(Equation 5.7: 𝒓𝟐 − 𝒇𝟐 −𝒓𝟐 = 𝟏), (which you can read more about this equation in page 147 of book). a) What happens to the asymptotes if r approaches 0? Sketch a typical hyperbola for which r is very small. b) What happens to the asymptotes if r approaches f ? Sketch a typical hyperbola for which r is almost as large as f. 3 Section 5.3.2: 1) Is cosh an even function, an odd function, or neither? Is sinh an even function, an odd function, or neither? Explain. Compare to the analogous circular trigonometric functions. 3) Using a calculator, make a table of values for cosh θ and sinh θ for θ = 0, ± 0.5, ± 1, ± 1.5, ± 2, ± 2.5, and ± 3. Use these to give rough graphs of cosh θ and sinh θ. Then, plot the ordered pairs (cosh θ, sinh θ ) along the hyperbola 𝒙𝟐 − 𝒚𝟐 = 𝟏.
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