Unit 7:Basic elements ofgeometry. Sexagesimalsystem.

In this lesson you will learn about:

Points, lines, planes and their possiblepositions.

Angles, their measuremetes and how to workwith them.

Time and how to work with it.

Vocabulary you need in this lesson:You are going to use some vocabulary related to this lessonand the problems you will work with. The same words appe-ar in the crossword, in the word scramble and in the wordsearch.

Some theoryKinds of angles:

Supplementary angles Complementary angles

Alternate Interior angles Alternate Exterior angles

Corresponding angles

Adjacent angles

Degree: is the measurement of any of the ninety equal parts you can divide aright angle into.

Complex form: measurement espressed using different units.

Incomplex form: measurement expressed using only one unit.

Examples:

Incomplex form: 3456º 567sg

Complex form: 45º 33´ 52´´ 3h 24min 43sg

Now it´s time to practiseExercises

1) Express in minutes:

a) 43º b) 34,5º c) 1260´ d) 23º 32´ e) 4567´´ f) 0.7º

2) Express in seconds:

a) 15h 16min 17sg b) 55h 78min c) 87min d) 8.2h e) 8h 8min8sg

3) Express using seconds, minutes and degrees:

a) 9876´´ b) 58,17º c) 6789,5´ d) 23,18º e) 89º 567´

4) Work outa)23º 90´32´´+ 52º 72´ 33´´´b)42º 25´44´´ - 37º 30´52´´c)3 (19º 92´72´´)d)58º 28´38´´ : 2

5) Calculate the measurement of:

a)The complementary and the supplementary angles of: 45º 15´33´´b)The complementary and the supplementary angles of: 51º 12´51´´

6) Given the angles: A = 15º 23´52´´ and B = 34º 28´59´´ calculate:

a) 4A +2B b) 5A -3B c) A:2 d) B: 3

7) a) How many planes contain a certain point?b) How many planes contain two given points?c) How many planes contain three given points?

8) The twelve edges of this box are numbered. Check that 1 and 2 have one point incommon, 2 and 4 are parallel and 1 and 3 miss do not intersect. Find out other sideswith one point in common, other two sides that are paralell and other two that do notintersect.

9) In this regular, hexagonal prism every face is contained in a plane. Lateral faces arenumerated from 1 to 6 and the two bases are A and B.

A) State whether the following planes have points in common or are parallel: 1 and2, 1 and 3, 2 and 5, 2 and B, A and B.

B) Calculate the angle formed by 1 and 2 and 2 and A.C) Calculate the angle formed by 1 and 3

Problems

1. Five friends (Aly, Beth, CJ, David, and Emily) decide to run the Chicago Marathontogether. Aly completes the marathon in 16,400 seconds. Beth finishes in 274 min-utes and 45 seconds, CJ in 5 hours and 1000 seconds, David in 2 hours 157 minutesand 125 seconds. Emily finishes in 45 minutes 12,845 seconds.

a. Can you find all of the friends’ times in seconds?b. Calculate the times in minutes.

2. After the marathon…a. Aly lives in Boston. Her flight leaves Chicago at 3 pm and she needs to get to the

airport 1.5 hours early. Her hotel is 15 minutes from the airport by cab. From thefinish line to the hotel it takes 35 minutes by train. Does she have time to take a 30minute shower and still catch her flight?b. Beth is supposed to meet her mom at the Park Grill in Millennium Park for an earlydinner. She must go back to her apartment in Lakeview to shower and get dressed.From the finish line to her apartment it takes 46 minutes and 89 seconds by train.She takes a long shower that lasts 34 minutes and 118 seconds. Afterwards she isstill exhausted and she decides to take a nap. She sleeps for 1 hour 67 minutes and1200 seconds. She needs 43 minutes and 33 seconds to get dressed and put on hermake up. If she goes by taxi to the restaurant it takes 39 minutes. If she goes bytrain it takes 1 hour and 16 minutes. Beth’s mom has asked the waiter Mike to havethe salads on the table when Beth arrives.

- What time should Mike bring the salad if Beth takes a cab to the restaurant?- If she takes the train?

c. CJ and David have plans to meet after the race in Wrigleyville for Mexican food.David lives above Caesar’s Mexican Restaurant. He can be at his table at the restau-rant 57 seconds after leaving his apartment. He arrives home 37 minutes and 12 sec-onds after he finishes the race. He takes a shower, takes a nap, and gets ready to goto dinner. All together this takes him 3 hours 23 minutes and 26 seconds. Before hegoes to dinner he must walk his dog. He walks his dog for 17 minutes and brings itback to the apartment. CJ lives 5 minutes and 20 seconds from the finish line. Hewalks home and takes a shower. 47 minutes and 987 seconds later he is in bed takinga nap. He wakes up after sleeping for 7345 seconds. It takes him 44 minutes and 68seconds to get to the restaurant by train.- Who gets to the table first CJ or David?- At what time does the second person arrive?d. Who won the Marathon? What time did the winner cross the finish line?

Solutions:

Race TimesAly: 16400’’/273’20’’/4h33’20’’Beth: 16485’’/274’45’’/4h34’/45’’CJ: 19000’’/316’’40’/5h16’40’’David: 16745’’/279’5’’/4h39’5’’Emily 15545’’/259’’5’/4h19’’5’

After the marathonNo, Aly finishes the marathon at 12:33:20pm; she must leave for the

airport at 1:15pm.Cab: 5:14:45pm/Train: 5:51:45pmDavid: 4:56:40pm/CJ: 4:33:40pmEmily won the marathon at 12:24 pm.

3) A windmill has a six sails. What is the angle between two con-secutive sails? What will be the angle if the number of sailswere eight?

4) John goes every week to see his friends Paul, Mary and Anne. It takes him 30min 23sec to go to Paul's house, 1h 15min to go to Mary's house and 45min40sec to go to Anne's house. He visits Paul three times a week, Mary twice aweek and Anne once a week. How long does it take John to visit his friends fortwo weeks?

5) A telephone company charges you 2.35 dollars for using the connection and 1.25dollars for every minute. What will be the price of a phone call whose length is8 min and 24 sec?

6) A truck has been travelling at a speed of 85km/h for 6 h and 24 min. How manykm has it travelled?

Some interesting websites!

If you want to take a test about angles and their measurements go to:

http://www.thatquiz.org/tq/practice.html?angles

To listen how to check relative positions and practice with someexamples go to:

http://my.hrw.com/math06_07/nsmedia/homework_help/geo/geo_ch03_01_homeworkhelp.html

To read a bit more about measuring angles go to:

http://www.mathsisfun.com/geometry/degrees.html

To see some examples about the different types of angles go to:

http://www.mathsisfun.com/angles.html

To practice a little with a test about angles click here:

http://www.glencoe.com/sec/math/studytools/cgi-bin/msgQuiz.php4?isbn=0-02-834817-6&chapter=3&lesson=2

If you want to read about the sexagesimal system, its origin and its usethrough history click here:

http://en.wikipedia.org/wiki/Sexagesimal