Unit 4: Triangles Booklet #1 of 2

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Name: Geometry Period Unit 4: Triangles Booklet #1 of 2 In this unit you must bring the following materials with you to class every day: Please note: Calculator Pencil This Booklet A device

1 Name: Geometry Period Unit 4: Triangles Booklet #1 of 2 In this unit you must bring the following materials with you to class every day: Please note: Calculator Pencil This Booklet A device Headphones You may have random material checks in class Some days you will have additional handouts to support your understanding of the learning goals in that lesson. Keep these in a folder and bring to class every day. All homework for part one of this unit is in this booklet. Answer keys will be posted as usual for each daily lesson.

2 4-1 Notes Today s Learning Goal: What is the Side-Angle Relationship in triangles? Angle-Side Relationships in Triangle Consider ABC as shown below with given side lengths and angle measures: Turn and Talk: With your elbow partner, answer the three questions below 1) List the angles (by name) in order from smallest to biggest: 2) List the sides (by name) of the triangle in order from smallest to biggest: 3) Write a conjecture that describes a relationship between the measure of angles and the measure of sides: BE PREPARED TO SHARE OUT! Together! Angle - Side Relationship The side is across from the angle. The side is across from the angle. The side is across from the angle.

3 Let s Try It! Example 1) InΔ ABC, m A = 30 and m B = 50. What is the name the longest side of the triangle? Example 2)

4 Angle Side Relationship Partner Playtime Choose who will be partner peanut butter and who will be partner jelly. Work on your column s questions. Each problem is different, but your multiple choice answers should match. If there is a discrepancy between answers, figure out who is right and help your partner correct their mistake. Work on the last 2 problems together Partner Peanut Butter 1 In scalene triangle ABC, and. What is the order of the sides in length, from longest to shortest? Partner Jelly In, and = 70. The lengths of the sides of in order from smallest to largest are 1),, 2),, 3),, 4),, 2 In,m C < m A < m B. Which inequality is true? 1),, 2),, 3),, 4),, In,. Which statement is false? 1) 2) 3) 4) 1) 2) 3) 4)

5 3 For which measures of the sides of is angle B the largest angle of the triangle? For which measures of the sides of smallest angle of the triangle? is angle B the 1) 2) 3) 4) 1) AC = 12, AB = 10, BC =14 2) AC = 8, AB = 9, BC = 6 3) AC = 8, AB = 9, BC = 10 4) AC = 4, AB = 3, BC = 8 Peanut Butter Jelly Time! Try these as a team now 4) The degree measures of the angles of are represented by x, 3x, and. Find the value of x. Classify this triangle by its angles. 5) As shown in the diagram of below, B is a point on and is drawn. If,, and, what is the longest side of?justify your answer.

6 6) In,,, and B is a point on side, such that. Which line segment is shortest? 1) 2) 3) 4) 7)

7 4-1 Homework 1. In,, and. Which expression correctly relates the lengths of the sides of this triangle? 1) 2) 3) 4) 2) Determine and state the largest angle of RST, and justify how you obtained your answer. 3) In triangle RST, which is a possible side length for ST? (Select any that apply)

8 4) the accompanying diagram, is a straight line, and angle E in triangle BEC is a right angle. What does equal? 1) 135 3) 180 2) 160 4) 270 5) In the diagram below of with side extended through D, and. Which side of is the longest side? Justify your answer. 6) In, and is an obtuse angle. Which statement is true? 1) and is the longest side. 2) and is the longest side. 3) and is the shortest side. 4) and is the shortest side.

4-2 Notes The Triangle Inequality Theorem Learning Goals: What is the Triangle Inequality Theorem? What does the Triangle Inequality Theorem help us to conclude? Will any three sides make a triangle?

3. Measure the size lengths of the straws used. 4. Record the information on the table provided below. (Make sure to place the small/medium/large sizes in the correct order) 5. Repeat seven times. 6.

9 4-2 Notes The Triangle Inequality Theorem Learning Goals: What is the Triangle Inequality Theorem? What does the Triangle Inequality Theorem help us to conclude? Will any three sides make a triangle? Pair Exploration Guide Materials 1. Baggie of Straws and connectors. 2. Ruler 3. Pencil 4. Cell Phone/Device 5. Thinking Caps. Directions 1. Pick any three length straws. 2. Attempt to make a triangle. 3. Measure the size lengths of the straws used. 4. Record the information on the table provided below. (Make sure to place the small/medium/large sizes in the correct order) 5. Repeat seven times. 6. Get Ready to PLAY! Find your partners, find a baggie with sticks and try to make triangles! Shortest Side Length Medium Side Length Longest Side Length We were able to make a triangle? (Yes/No) As you work with your partners see if you can find any patterns you might want to share to the class! Discuss with your partners! What is special about the sides that make a triangle? How about those that don t? STOP HERE AND WAIT FOR DIRECTIONS!

10 Let s come back together and discuss! Triangle Inequality Theorem: Used to determine. If the of ANY are than the third, then those sides form a triangle. Test the two sides to see if it s bigger than the third. Let s try this! Determine if the following side lengths would form a triangle. Show work to verify your answer. 1) 7, 5, 4 2) 3, 6, 2 3) 8, 2, 8 Same theorem- different type of question! Determine a range for possible side lengths, when given two other sides. Show work to verify your answer. 4) 9,5

11 4-2 Practice 1. Can you make a triangle with the following given side lengths? Explain your reasoning. a) 8m, 9m, 15m b) 5cm, 7cm, 12cm c) 4ft, 108in, 10ft ***CAREFUL! 2. A triangle has one side of length 6 and another side of length 10. State the possible integer lengths of the third side.

12 3. Consider the triangle below. Describe the possible lengths of the third side using an inequality. How did you come to this answer? (Be careful! This is a multi-step question!) 4. Modeling With Mathematics You travel from Fort Peck Lake to Glacier National Park and from Glacier National Park to Granite Peak. a) Write an inequality to represent the possible distances from Granite Peak back to Fort Peck Lake. b) How is your answer to part a affected if you know that m 2 < m 1 and m 2 < m 3? 5. Can you make a triangle with any side lengths? Explain why or why not.

13 4-2 Homework 1. Is it possible to construct a triangle with the given side lengths? Show work that supports your answer. a) 3, 12, 17 b) 5, 21, 16 c) 8, 5, 7 d) 10, 3, A triangle has two sides with lengths 5 inches and 13 inches. State the possible integer lengths the third side of the triangle can be.

14 3. Which of the statement/s about ΔTUV is false? (a) UV > TU (b) UV + TV > TU (c) UV < TV (d) m U > m V 4. List the sides of ΔDFG in order from longest to shortest. 5. Your house in on the corner of Hill Street and Eighth Street. The library is on the corner of View Street and Seventh Street. What is the shortest route to get from your house to the library? Explain why it works.

We will use it so find the missing side of a right triangle.

15 The Pythagorean Theorem Learning Goals: What is the Pythagorean Theorem? For which type of triangle does the Pythagorean Theorem apply? How do we determine if a triangle is acute or obtuse based on the sides? Part 1: The Pythagorean Theorem We can use the Pythagorean Theorem with triangles. We will use it so find the missing side of a right triangle. Guided Example: Find the missing side of the triangle below Example 4-3 Notes 1) a = 5, b = 12, c = x (c is always the hypotenuse) 2) c 2 = a 2 + b 2 x 2 = x 2 = x 2 = 169 x 2 = 169 x = 13 c 2 = a 2 + b 2 Remember! c is always the hypotenuse, which is opposite the right angle! Steps 1) Identify the measure of each side length and classify them as a, b, or c. 2) Use the Pythagorean Theorem and properties of Algebra to find the missing side. 3) Put your answer in simplest radical form, if applicable. Try one! Find the missing side. Show all of your work! Keep your answer as a radical. PART 2: Classifying Triangles Using the Pythagorean Inequality Theorem, we classify triangles as acute, obtuse, or right based on their side lengths. Acute Triangle Obtuse Triangle Right Triangle c 2 a 2 + b 2 c 2 a 2 + b 2 c 2 a 2 + b 2 *Let s remember, c is always the largest side of the given triangle and we should put it on the left of our inequality when classifying triangles!

and classify them as a, b, or c. 2) Use the Pythagorean Inequality to compare if c 2 is greater than, less than, or equal to a 2 + b 2. 3) Make a conclusion and justify. Try one!

16 Guided Example for CLASSIFYING TRIANGLES: A triangle has side lengths 10, 11, 14. Is the triangle acute, right, or obtuse? 1) a = 10, b = 11, c = 14 (c is always the largest side) 2) < 221 3) The triangle is acute since c 2 < a 2 + b 2 Steps: 1) Identify the measure of each side length and classify them as a, b, or c. 2) Use the Pythagorean Inequality to compare if c 2 is greater than, less than, or equal to a 2 + b 2. 3) Make a conclusion and justify. Try one! A triangle has side lengths 6, 4, 9. Is the triangle acute, right, or obtuse? Let s Refresh! Put the following in simplest radical form: 32 PART 3: Partner Exploration! Complete the following questions and check in with the key when you are done before you proceed! 1. Solve for the length of side BC? Leave your answer in simplest radical form. 2. Given the following segments: 12, 16, 20 a. Determine if the segments form a triangle: b. Is the triangle acute, right or obtuse? 3. What is 96 is simplest radical form?

17 Mixed Practice 1) Determine if the following triangles are acute, obtuse, or right. Show all of your work! 2) Determine whether the given triangle is a right triangle. Explain how you arrived at your answer. 3) Write the following in simplest radical form. Show all of your work! a. 12 b. 75x 2 4) Determine if the following triangles are acute, obtuse, or right. Show all of your work! Side 1 = 9, Side 2 = 12, Side 3 = 5 5) Determine if the following triangles are acute, obtuse, or right. Show all of your work! 5, 12, 13

18 6) In baseball, the lengths of the paths between consecutive bases are 90ft, and the paths form right angles. The player on first base tries to steal second base. How far, to the nearest tenth of a foot, does the ball need to travel from home plate to second base to the get the player out? (Draw a picture!!) 7) A triangle is isosceles and right angled. Which of the following statements must be false? A) The triangle has angles 45, 45 and 90 B) The triangle has two of its sides equal C) The triangle has one line of symmetry D) The triangle has sides of lengths 3, 4 and 5 8) Find the error(s) in the student s work, then solve for x correctly. 9) Quadrilateral ABCD is shown (with coordinates labeled). Solve for the length of BC in simplest radical form:

19 1. Determine if the triangle is a right triangle: 4-3 Homework 2. Given the following segments: 12, 16, 20 Is the triangle acute, right or obtuse? 3. Parallelogram ABCD has points,,, and. Calculate the distance for the following, leaving answers in simplest form: a) AB b) BC 4. If (x, 40, 41) are the sides of a right triangle, where 41 is the largest side, what is the value of x?

20 Pick one question from each column for each number. Show all of your work!

21 4-4 Notes 4-4 Isosceles Triangle Theorem Today s Learning Goal: What is the isosceles triangle theorem? Reactivate your knowledge! What's an isosceles triangle? Write a Conjecture! Consider each of the triangles below to answer the following questions with a neighbor: a) Classifying by sides, what type of triangle is each of the triangles? Justify your answer. b) What do you observe about the angles in each of these triangles? c) Write a conjecture about the angle measures of an isosceles triangle.

22 Isosceles Triangle Theorem Works two ways! If a triangle has two congruent sides, then it has two congruent If a triangle has two congruent, then it has two congruent, therefore making the triangle Isosceles. Key Features: Isosceles Triangle( AC BC) Base Angles: Vertex angle: Let's try some together! Applying the Theorems 1) Solve for x: Our Thinking In the diagram we are given that two Therefore, by Isosceles Triangle Theorem 2) Find the measure of angle y: Our Thinking In the diagram we are given that two Therefore, by Isosceles Triangle Theorem

23 You can figure it out! IN math, not knowing isn t failing, not trying is failing. Your Task: With your partner, determine and state the measure of <1. (Assume all segments are straight) Hint!! Try covering up triangle BCD with your hand to help you break down the diagram a little bit! Practice Putting it All Together! Don t forget to write about your thinking! 1) In the accompanying diagram, the roof of a house is in the shape of an isosceles triangle. The vertex angle formed at the peak of the roof is 84 degrees. What is the measure of angle x?

24 2) In triangle XYZ, side XZ is extended to point p outside of the triangle. Explain why the m<b = 136. Push Yourself. Achieve great things Select 2 problems you want to challenge yourself on in this section. Get as far as you can go, don t settle by giving up on a challenge 3) In the diagram of parallelogram FRED shown below, is extended to A, and is drawn such that. Hint: Use Linear pairs! If m <FDE =124, what is the largest side of triangle AFD?

more than three times m B. Find m C. Hint: Draw a picture!

25 4) Solve for angle k Hint: Redraw the isosceles triangles you see! 5) Vertex angle A of isosceles triangle ABC measures 20 more than three times m B. Find m C. Hint: Draw a picture! Use details from the problem for accuracy! 6) Hint: Think about linear pairs

2) Tina wants to sew a piece of fabric into a scarf in the shape of an isosceles

26 4-4 Homework Directions: Answer the following questions to the best of your ability. Show all work. 1) In, and. Find. 2) Tina wants to sew a piece of fabric into a scarf in the shape of an isosceles triangle, as shown in the accompanying diagram. What are the values of x and y? 1) 2) x = 69 and y = 42 3) 4) 3) Solve for angles 1,2,3 and 4:

27 Mixed Practice 4) Which of these lengths could be the sides of a triangle? 5) Find the value of x 6) Find the values of x and y. Highlight the SHORTEST side of the triangle. Check the key online!

28 4-5 Notes Exterior Angle Theorem Learning Goals: What are exterior angles? What is the Exterior Angle Theorem? I. Exterior Angles of a Triangle Fact Check: An exterior angle is an angle that is created when the side of a triangle is extended outside the triangle a) Which angle in the diagram to the right is the exterior angle? b) What is the relationship between the exterior angle and the adjacent interior angle? How do you know? II. Pair Discovery- The Exterior Angle Theorem 1. The sum of the interior angles of a triangle is 2. Given CUP, identify (by angle number): a) an exterior angle: Adjacent next to; share a ray and vertex b) an interior angle adjacent to the exterior angle: c) TWO remote interior angles (meaning the two angles not adjacent to the exterior): and 3. Given that <1 = 60 0, <2 = 40 0, <3 = 80 0 and <4 = fill those values in the diagram! 4. What do you notice about the exterior angle and two other interior angles? 5. Write a conjecture (a hypothesis) about the relationship you see:

why can t we just use the sum of interior angles and linear pairs to solve for the exterior angle?

29 Exterior Angle Theorem: The SUM of the TWO REMOTE INTERIOR ANGLES is equal to the EXTERIOR angle. Let s Try it! Example 1) Determine the measure of b But.why can t we just use the sum of interior angles and linear pairs to solve for the exterior angle? Let s take a look at the next example! Example 2) Find the m JKM Example 3) Sketch it! In ΔPQR side QR is extended to a point M, outside of the triangle. Sketch what is being described:

30 4-5 Homework Complete each of the following problems, and make sure you check your answer in a different color when you are done! 1. What is the measure of the angle marked D in the triangle below? 2. In the diagram below, is shown with extended through point D. If,, and, what is the value of x? 3. Determine the measure of the unknown angle, a. Give reasons for your calculations. 4. In ΔABC, m<a = 48, and m<c = 24. What type of triangle is triangle ABC? What s the largest side in ΔABC?

31 5. Find the measures of the numbered angles in the diagram below: a. <1 e. <5 b. <2 f. <6 c. <3 g. <7 d. <4 h. <8 6. Find the values of x and d. 7. All isosceles triangles are acute triangles. True or False? Explain.

Type of Triangle Definition Drawing. Name the triangles below, and list the # of congruent sides and angles:

Name: Triangles Test Type of Triangle Definition Drawing Right Obtuse Acute Scalene Isosceles Equilateral Number of congruent angles = Congruent sides are of the congruent angles Name the triangles below,