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Unit 4: Triangles

Name: ___________

Teacher: __________

Period: ______

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AIM: SWBAT determine whether three leg measurements could be the sides of a triangle

(The Triangle Inequality)

“DO NOW” Given two side lengths of 5 and 6 inches. What information could you give about the third side? 5, 6, ? Notes The Triangle Inequality states that the sum of any 2 sides of a triangle must be greater than the third side AND any the difference of any two sides must be less than the third side. Therefore, where a, b, and c are sides: (a – b) < c < (a + b)

Example 1: Could 7, 8, & 5 be the three sides of a triangle?

7 + 8 > 5 and 8 – 7 < 5

7 + 5 > 8 and 7 – 5 < 8

8 + 5 > 7 and 8 – 5 < 7

Yes, 7, 8, and 5 could be the three sides of a triangle.

Example 2: Could 4, 2, & 6 be the three sides of a triangle?

4 + 2 > 6 and 4 – 2 < 6

4 + 6 > 2 and 6 – 4 < 2

2 + 6 > 4 and 6 – 2 < 4

No, since 4 + 2 is not greater than 6 and 6 – 4 is not less than 2; 4, 2, & 6 could not be the three sides of a triangle. The Sum of the 2 smallest sides must be larger than the third AND the Difference between the 2 largest must be smaller than the 3rd.

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The Sum of the 2 smallest sides must be larger than the third AND the Difference between the 2 largest must be smaller than the 3rd. For each of the following, state whether the three leg lengths could be the three sides of a triangle. Show work to support your answer. 1) 5, 7, 4 2) 8, 3, 4 3) 6, 2, 6 4) 12, 3, 9 The lengths of two sides of a triangle are given. What can you say about the length of the third side? 5) 3 ft and 5 ft 6) 9 in. and 11 in. 7) 16 cm and 20 cm The 3rd side is greater than ____ ft and less than ___ ft. 8) You have moved to a new city, and are told that your apartment is 5 mi from your school and 6 mi from the restaurant where you have a part-time job. These three places do not lie on a straight line. What can you say about the distance between your school and the restaurant? The distance between the school and the restaurant is greater than _________ mile and less than _________ miles.

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Homework: Triangle Inequality For each of the following, state whether the three leg lengths could be the three sides of a triangle. Show work to support your answer. 1) 3, 4, 6 2) 4, 10, 6

3) 11, 7, 5 4) 161

, 41

, 83

The lengths of two sides of a triangle are given. What can you say about the length of the third side? 5) 3 m and 8 m 6) 30 m and 45 m 7) 10 ft and 21 ft 8) 100 cm and 225 cm 9) You are at an amusement park and meeting a friend at the Giant Wheel. Your friend said that it takes 5 min to walk from Gemini to the Corkscrew and 2 min to walk from the Corkscrew to the Giant Wheel. How long does it take to walk from Gemini to the Giant Wheel? Show work to support your answer.

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Aim: SWBAT find the measures of missing angles of a triangle.

The sum of the measure of the angles of a triangle is equal to ___________ degrees.

m ∠ 1 + m ∠ 2 + m ∠ 3 = _______

Find the measure of each angle algebraically. Classify the triangle by its angles and sides. x+2 x+3 x-5 In the triangle below, find the measure of the missing angles ALGEBRAICALLY. 40° x° 70° y°

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Find the measure of each angle ALGEBRAICALLY.

1)

2)

3)

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Homework Find the missing angles ALGEBRAICALLY. 1) (x+20) x˚ yo

2) 72˚ 54o xo yo

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3) 122° xo 61o yo

4) In ∆ABC the m∠A is x˚, m∠B is (x+16)˚, find the m∠C, (2x)˚.

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Aim: SWBAT identify corresponding parts of similar triangles and determine that

corresponding parts are proportional.

“DO NOW”

1) Find the m∠CBA ________

2) Find the m∠FED ________

Similar Triangles have the same shape but not necessarily the same size.

If two triangles are similar, then the angles of one triangle are congruent to the corresponding

angles of the other triangle.

If two triangles are similar, then their corresponding sides are proportional.

1) ∆ABC~ ∆DEF List the corresponding sides List the corresponding angles

_____________________ _____________________

_____________________ _____________________

_____________________ _____________________

2) ∆DOG~ ∆CAT List the corresponding sides List the corresponding angles

_____________________ _____________________

_____________________ _____________________

_____________________ _____________________

3) Is ∆DEC~ ∆ABC ? _________

If so list the corresponding sides and the corresponding angles?

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Determine whether the following triangles are similar and if so list the corresponding sides and

angles.

4) Is ∆RNT~ ∆SMT ? _________ 5) Is ∆AEB~ ∆ADC ? _________

6) Is ∆ABC~ ∆ADE ? _________ 7) Show how ∆PQR~ ∆LMK

______________________________________

______________________________________

______________________________________

______________________________________

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Homework : Similar Triangles

1) Determine whether the following triangles are similar, support your answer.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

2) Show that ∆CBD ∆SUT and list the corresponding parts.

List the corresponding sides List the corresponding angles

_____________________ _____________________

_____________________ _____________________

_____________________ _____________________

Why is ∆CBD ∆SUT ? _____________________________________________________

3) Show that ∆DCE ∆DBA and list the corresponding parts.

List the corresponding sides List the corresponding angles

_____________________ _____________________

_____________________ _____________________

_____________________ _____________________

Why is ∆DCE ∆DBA ? _____________________________________________________

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4) Determine which triangles are similar.

A) ∆_____ ∆_____ B) ∆_____ ∆_____ C) ∆_____ ∆_____

5) Is ∆DOG ∆CAT ? _____

Explain why or why not.______________________________________________________

_______________________________________________________________________

_______________________________________________________________________

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Aim: SWBAT identify corresponding parts of similar triangles and solve for missing sides of

similar triangles.

“DO NOW” Solve each proportion algebraically.

1) 104 =

x6 2)

x + 916 =

x4

If two triangles are similar, then their corresponding sides are proportional.

1) Identify corresponding sides of the similar triangles

2) Using the corresponding sides set up a proportion

3) Solve the proportion algebraically

The following triangles are similar, use your knowledge of corresponding sides to solve for the

missing side.

1) 2)

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3) 4)

5) 6)

7) Solve for x and y:

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Homework: Solving for missing sides of similar triangles, SHOW ALL WORK!!!

1) Solve for x and y. 2) Solve for m and n.

3) Solve for x and find the missing side. 4) Solve for x and find the missing side.

5) Solve for x: 6) Solve for x:

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NOTE: The phrase “at the same time” is

important when dealing with problems

involving shadows because the angle of the

sun changes throughout the day.

Aim: SWBAT Solve word problems using similar triangles.

Do Now: Find the value of x:

x

35mm

18mm 25mm

45mm

Solving Word Problems Using Similar Triangles

When solving a word problem involving similar triangles, it is helpful to draw a picture and label the corresponding parts of the triangles.

� Use a let statement to define your variable. � Write a proportion using the corresponding sides, but be sure to be CONSISTENT! � Your final answer should be a sentence.

Example 1: Find the width of the river to the nearest meter.

Example 2: A yardstick casts a 6 foot shadow at the same time a tree casts a shadow of 24 feet. How tall is the tree?

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Example 3:

A girl 160 cm tall, stands 360 cm from a lamp post at night. Her shadow from the light is 90 cm long. How high is the lamp post?

160cm

90cm 360cm Example 4: Sam built a ramp to a loading dock. The ramp has a vertical support 2m from the base of the loading dock and 3m from the base of the ramp. If the vertical support is 1.2m in height, what is the height of the loading dock?

Example 5: A 40-foot flagpole casts a 25-foot shadow. Find the shadow cast by a nearby building 200 feet tall.

Example 6: The lengths of the sides of a ΔABC are 5m, 6m, and 7m. Triangle RST is similar to ΔABC. The longest side of ΔRST is 21m. What is the length of the shortest side of ΔRST?

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HW Math 8: Word Problems Using Similar Triangles

Draw and label a diagram to represent each problem. Remember to use a let statement to define your variable, write an equation and solve. Your final answer should be a complete sentence.

1) A person 6 feet tall casts a shadow 15 feet long. At the same time, a nearby tower casts shadow 100 feet long. What is the height of the tower? 2) What is the height of a vertical pole that casts a shadow 8 feet long at the same time that another vertical pole 12 feet high casts a shadow 3 feet long? 3) The heights of two flagpoles are 20 feet and 30 feet. If the shorter pole casts a shadow of 8 feet, how long is the taller pole’s shadow?

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4) Mark wants to cut a triangular patch to make an emblem. The pattern for the emblem is a triangle with sides of 8, 8, and 10 cm. If Mark wants to make the longest side of the emblem 25 cm., how long should the other sides be? 5) On a map, the length from Cleveland to New York is 7cm, from Cleveland to Atlanta is 10cm, and from New York to Atlanta is 13cm. If on a larger map the length from Cleveland to New York is 17.5cm, what is the distance from Cleveland to Atlanta?

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Triangle Review

Classify each triangle according to the length of its sides.

1) 2) 3)

_______________ _______________ _______________

Classify each triangle according to the measure of its angles.

4) 5) 6)

_______________ _______________ _______________

Find the measure of the third angle ALGEBRAICALLY. Classify each triangle by its sides and angles.

7) 8) 9)

Sides: _______________ _______________ _______________

Angles:_______________ _______________ _______________

10) Solve for x and find the measure of each given angle.

11) Solve for x and solve for the missing angles.

?

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12) Solve for x and solve for the missing angles.

13) Could 48°, 37°, 111° be the angle measures of a triangle? Show work to support your answer.

14) Could 52°, 27°, 101° be the angle measures of a triangle? Justify your answer.

15) The measure of an angle in a right triangle is 32°. Find the measure of the missing angle algebraically.

16) State whether 8cm, 6cm, 9cm could be the three sides of a triangle. Show work to support your answer.

17) The lengths of two sides of a triangle are 5 in. and 7 in. What can you say about the length of the third side.

If 2 sides of a triangle are 12 inches and 18 inches, state whether the following could be the measure of the third side:

18) 30 inches 19) 18 inches 20) 6 inches 21) 12 inches 22) 24 inches

23) From questions 18 – 22, which measurements would make the triangle an isosceles triangle?

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24) State how you can determine that ∆ ABC is similar (∼) to ∆ EDC AND Solve for x.

8 in

x

12 in

3 in

25) Find the missing side.

26) Use the triangle above to answer the following questions

A) Name the side of the triangle that corresponds to NR _______

B) Name the side of the triangle that corresponds to TM _______

C) Name the angle that corresponds to ∠TRN _______

D) Name the angle that corresponds to ∠SMT _______

27) A 6-foot man casts a 7-foot shadow. Find how tall a nearby building is if it casts a 21-foot shadow at the same time?