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Unit 2 Notes / Secondary 2 Honors Day 1: 3.1/3.2 Triangle Angle Theorems and Inequalities 3.1: Pg 212 Triangle Sum Activity: The teacher has a large colored triangle on the board. A student is going to tear off each corner of the triangle to make a discovery about the sum of the angles in a triangle. Write a sentence describing the discovery made in the activity: The Triangle Sum Theorem says: 1. a. b. c. d. e. f. Write a description for each of the following triangles: Acute: Obtuse: Right: Scalene: Isosceles: Equilateral: 2. Pg 213 Use a straight edge to draw a large scalene triangle in the space below. Label the sides of the triangle S, M and L for small, medium and large. Use a protractor to measure and record the size of each interior angle of the triangle and label the angles S, M and L. Compare your results with your partner and the class. What conclusion can we draw about the relationship between the lengths of the sides of a triangle and the measure of the interior angles? 3. Pg 217 List the sides from shortest to longest. Complete the problems below, then compare with your partner. Pg 219 The remote interior angles of a triangle are the two angles that are non-adjacent to the specified angle. 4. Pg 220. Prove the Exterior Angle Theorem. Work with a partner. Be prepared to explain your reasoning to the class. The Exterior Angle Theorem says: The measure of the exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles of the triangle. Given: Triangle ABC with exterior angle โ ๐ด๐ถ๐ท Prove: ๐โ ๐ด + ๐โ ๐ต = ๐โ ๐ด๐ถ๐ท STATEMENTS REASONS 2. triangle sum theorem 3. linear pairs are supplementary 5.Subtraction property 5. Pg 221 #14 Solve for x and give the angle measures. Complete problem b if you have time complete d. 6. The Exterior Angle Inequality Theorem says: an exterior angle must be larger than either remote interior angles. Use the diagram below to discuss this theorem as a class: 3.2. Pg 230 Pasta Activity Sarah thinks any three lengths can represent the lengths of the sides of a triangle. Sam does not agree. Letโs explore. Take your piece of pasta and break it a two random points so the strand is divided into three pieces. Measure each of your three pieces in centimeters to the tenths place. Try to form a triangle from your three pieces of pasta. List your three lengths below and state whether or not the lengths could form a triangle. __________________________________________________________________________________________ Random sample of class measurements: Piece 1 (cm) Piece 2 (cm) Piece 3 (cm) Forms a triangle? (yes/no) 7. With your partner write a hypothesis for what must be true for the 3 lengths to be able to form a triangle. Be prepared to share your statement with the class. Is it possible to form a triangle using segments with the following measurements? Sketch a diagram and explain your answers. a. 1.9 cm, 5.2 cm, 2.9 cm b. 152 cm, 73 cm, 79 cm Pg 233: The Triangle Inequality Theorem states: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Example: If a triangle has two sides measuring 4 cm and 7 cm, what are the possible lengths for the third side? Day 2: 3.3/3.4 Special Triangles 1. What is the Pythagorean Theorem? (Write out the formula using the words โlegโ and โhypotenuseโ) Does it work for all triangles? Explain. 2. a. Use a ruler to measure the two legs of the triangle to the right. What kind of triangle is the triangle to the right? b. Use a protractor to measure the two acute angles in the triangle. What is another name for the triangle? c. Use your ruler to measure the length of the hypotenuse. Do you get an integer answer? Write down the decimal answer, to the nearest tenth of a centimeter, for the length of the hypotenuse. Compare your estimate for the length of the hypotenuse with your neighbor. Are your answers exactly the same? Would we expect all the member of the class to have exactly the same estimate? How could we get an exact length for the hypotenuse? (Hint: look at step #1 above) Use this method to find an exact/reduced root length for the length of the hypotenuse. Record this length on your diagram above. 3. Use Pythagorean Theorem to find the length of the hypotenuse for a 45°-45°-90° triangle with 6 inch legs. What pattern do you notice in the answers to #2 and #3? Sketch a diagram of a 45°-45°-90° triangle with side lengths ๐ long. Record the length of the hypotenuse on the diagram. 4. Use Pythagorean Theorem to find the missing side lengths of the triangles. Answer in reduced root form. x x 8 5 15 5 b = 2โ3 , c = 6 5. Are any of the triangles above a 45°-45°-90° triangle? Explain. 6. Find all the missing side lengths for the triangles below. x 6 2 45๏ฐ x x 6 6 7. An equilateral triangle is shown below. One angle has been bisected with a perpendicular bisector. Record the angle measures of the angles in the 2 triangles formed. a. What are the 3 angle measurements of the newly formed triangles? ฬ ฬ ฬ ฬ ? b. What is the length of ๐ด๐ท ฬ ฬ ฬ ฬ ? What is the length of ๐ต๐ท c. Use Pythagorean Theorem to find the length of ฬ ฬ ฬ ฬ ๐ถ๐ท . What pattern do you notice in the side lengths of the 30°-60°-90° triangles? Compare your answer with your partner. Sketch a diagram of a 30°-60°-90° triangle with a Short Leg ๐ long. Record the length of the Long Leg and the Hypotenuse on the diagram. 8. Use the above investigation to find the missing side lengths of following triangle. First label the side lengths: Short Leg, Long Leg and Hypotenuse then find the missing lengths. x 3 30 ๏ฐ y 9. What is a rule for finding the side lengths of a 30°-60°-90° triangle in relationship to the short leg: Hypotenuse = _____________________ Long Leg = ________________________ 10. Find all missing side lengths in the following diagrams: 8 y 60 ๏ฐ y 30๏ฐ 60๏ฐ x x a) 6 b) 11. Why are the 45°-45°-90° triangle and the 30°-60°-90° triangle โspecialโ? Day 3: 4.1/4.2 Similar Triangles 1) Dilation vocabulary to know: Scale Factor Point of dilation Corresponding parts Proportional Larger or Smaller โ Carnegie p266 Problem 3 Consider โ๐ฎ๐ฏ๐ฑ shown on the coordinate plane. You will dilate the triangle by using the origin as the center and by using a scale factor of 2. 2) How will the distance from the center of dilation to a point on the image of โ๐บโฒ๐ปโฒ๐ฝโฒ compare to the distance from the center of dilation to a corresponding point on โ๐บ๐ป๐ฝ? Explain your reasoning. 3) How do the coordinates of the image compare to the coordinates of the pre-image? 1 4) โ๐๐๐ต, with vertices ๐(0, โ3), ๐(โ12,6) and ๐ต(5,4), has been dilated by a factor of , using the origin as 3 the center of dilation. What are the new vertices? Making a Shadow โ Carnegie p262 Problem 2 Consider โ๐จ๐ฉ๐ช, โ๐ซ๐ฌ๐ญ, and point ๐. Imagine that point ๐ is the flashlight and โ๐ซ๐ฌ๐ญ is the shadow of โ๐จ๐ฉ๐ช. This creates Similar Triangles. 5) Use your measurements to express the following ratios as fractions then as decimals. ๐ท๐ธ = 7.1 ๐ด๐ต 4.8 ๐ธ๐น = ๐ต๐ถ 9 ๐ท๐น = ๐ด๐ถ 6) Use a protractor to measure the corresponding angles in the triangles. What can you conclude? 2.8 1.9 3.6 PROVING TRIANGLES ARE SIMILAR What do we always know about similar triangles? *The corresponding angles are ______________________. The corresponding sides are ________________. So, if โ๐ถ๐๐~โ๐๐ผ๐บ, what do we know? AA Similarity Theorem SAS Similarity Theorem SSS Similarity Theorem Determine whether the given triangles are similar. If so, use symbols to write a similarity statement AND state which theorem you used. Carnegie p279 #8 Carnegie p278 #6 Two triangles with side lengths that can be written: ๐ด๐ต ๐ต๐ถ ๐ด๐ถ = = ๐๐ ๐๐ ๐๐ Carnegie p276 #6 Carnegie p276 #7 GIVEN: โก๐ด๐ต โฅ โก๐ธ๐ท PROVE: โ๐ด๐ต๐ถ~โ๐ท๐ธ๐ถ Day 4: 4.3/4.4 Triangle Proportionality Theorems 4.3: Pg 328 Applying the Angle Bisector/Proportional Side Theorem 1. The Angle Bisector/Proportional Side Theorem says: A bisector of an angle in a triangle divides the opposite side into two segments whose lengths are in the same ratio as the lengths of the sides adjacent to the angle. Path E bisects the angle formed by path A and path B. Path A is 143 feet long. Path C is 65 feet long. Path D is 55 feet long. ฬ ฬ ฬ ฬ bisects โ ๐ถ. Solve for DB. ๐ถ๐ท Find the length of path B. Pg 328 Applying the Triangle Proportionality Theorem 2. The Triangle Proportionality Theorem says: If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. Are the triangles similar? Justify Now solve for FG using similar triangles. GE โฅ HD, DE = 30, EF = 45, GH = 25, FG = ? Solve using the theorem above. Pg 329. Write the CONVERSE of the Triangle Proportionality Theorem. 3. The converse of the triangle proportionality theorem can be used to test whether two lines segments are parallel. Given: DE = 33, EF = 11, GH = 66, FG = 22. ฬ ฬ ฬ ฬ โฅ ฬ ฬ ฬ ฬ Is ๐ท๐ป ๐ธ๐บ ? Justify using proportions. Pg 329 Applying the Proportional Segments Theorem. 4. The Proportional Segments Theorem says: If three parallel lines intersect two transversals, then they divide the transversals proportionally. a) Given: ๐ฟ1 โฅ ๐ฟ2 โฅ ๐ฟ3 , AB = 52, BC = 26, DE = 40, find EF. b) Given: ๐ฟ1 โฅ ๐ฟ2 โฅ ๐ฟ3 , AB = 90, EF=15, DE = 75, find BC. Pg 330 Applying the Triangle Midsegment Theorem. 5. The Triangle Midsegment Theorem says: The midsegment of a triangle is parallel to the third side of the triangle and half the measure of the third side of the triangle. a. Given E is the midpoint of FD, G is the midpoint of FH, and DH = 15, find the measure of EG. b. What do you know about โ ๐ธ and โ ๐ท? Why? Pg 330. 6. Using the Right Triangle/Altitude Theorem. If an altitude is drawn from the vertex of a right angle to the hypotenuse, then three similar right triangles are formed! Use proportions to solve for the missing lengths. Can you see the three similar triangles? Hint: Label the sides of the triangles S, M and L, then set up the proportion. Solve for all variables. Day 5: 4.5/4.6 Similar Triangle Applications What is the converse of the Pythagorean Theorem? 1) If a triangle has sides that measure, 12, 16, and 20, would it be a right triangle? Justify your answer with calculations. How Tall Is That Oak Tree? Carnegie p 320 Problem 2 #1, #3 and How Wide Is That Creek? Carnegie p322 Problem 3 #1 1) You go to the park and place a mirror on the ground so you can see the top of a tree. You then gather enough information to calculate the height of one of the oak trees. The figure shows your measurements. a. Are the triangles similar? How do you know? b. Calculate the height of the tree. 2) Stacey notices that another tree casts a shadow and suggests that you could also use shadows to calculate the height of the tree. She lines herself up with the treeโs shadow so that the tip of her shadow and the tip of the treeโs shadow meet. She then asks you to measure the distance from the tip of her shadows to her, and then measure the distance from her to the tree. Finally, you draw a diagram of the situation as shown to the right. Calculate the height of the tree. Explain your reasoning. a. Are the triangles similar? How do you know? b. Calculate the height of the tree. 3) You stand on one side of the creek and your friend stands directly across the creek from you on the other side as shown in the figure. Your friend is standing 5 feet from the creek and you are standing 5 feet from the creek. You and your friend walk away from each other in opposite parallel directions. Your friend walks 50 feet and you walk 12 feet. a. How do you know the triangles formed are similar? b. Calculate the distance from your friendโs starting point to your side of the creek. c. What is the width of the creek? Explain your reasoning.