to 90 (11c -sb) -tb 0 /10 (s-0 40' f) m z_cof Name: Date:

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1 Measurement of ngles Name: Date: J.1751firitt M.** 11, basic concept in direct measurement is that the unit of measurement must be of the same nature as the thing to be measured. In the measurement of angles, the unit of measurement is an angle of one degree. The number of degrees in an angle is called its measure. In the illustration above, the measure of angle OB is 10. There are 10 in LOB. The symbol for measure of an angle (in degrees), such as angle OB, is m LOB = n where n is a number representing the measure of the angle. The symbol m LOB is read "the measure of angle OB." 1. Use the illustration above to find the measures of the following angles. a) m LOC b) m LOD to 90 c) m LOE d) m LOF e) m z_boc f) m z_cof /10 (s-0 40' (11c -sb) -tb 0 CD Mark Twain Media, Inc., Publishers

2 Name: Date. Measurement n. es n angle may be measured with a protractor. Some protractors are marked in degrees with one scale that is read clockwise and another scale that is read counterclockwise. The sketch below shows a semicircular protractor placed on a set of rays with a common endpoint 0. The base of the protractor is placed evenly on the line segment H. 2. Find the measure in degrees of each angle designated below. B 0 a) m LOB = I 09 h) b) m LDE = c) m LGOH = /0 1 i) d) m Z_FO = UP j) e) m LDO = 6o' k) f) m LCO = 40 g) m LCOD = (0 0 Move a protractor s Q, that the zero ray lies along oe (or 010) 60 m LFOG = Move a protractor so that the zero ray lies along Od (or Or) m LCOB = 3r m LDOG =. m LH0B = ro CD Mark Twain Media, Inc., Publishers 2

3 Name: Date: Measurement of ngles Special Names for ngles moreling to Their Measures n acute angle is an angle whose measure in degrees is less than 90. Verify b measuring each angle below that it is an acute angle. right angle is an angle whose measure in degrees is 90. Verify by measurement that E he following represent right angles. n obtuse angle is an angle whose measure in degrees is greater than 90 but less than 180. Verify by measurement that the following represent obtuse angles. Complementary and Supplementary ngles Two angles are considered complementary angles if the sum of their angles is exactly 90 degrees. Two angles are considered supplementary angles if the sum of their angles is 180 degrees. zle t4 1 >z 3 ez4 3. Which angles in the above diagram are complementary? 40 L2 zzfr Which angles in the above diagram are supplementary? 461 z r 43 rt.* ::D (t). Mark Twain Media. Inc., Publishers 3

4 H/ 112/1 Di"g- it/1 P9 e t'? Meas rement of ngles: Exe Name: Date: T.; 1. Classify the following triangles: a) according to relative lengths of sides. 8 b) according to the measures of the angles Stkt_4_ eri 'demi %,5054e/e4 rikt 2. Several experiments may be used to help students discover that the sum of the measures of the angles of a plane triangle is the same for all triangles. Basic to this discovery is the knowledge of the sum of the measure in degrees of three angles such as those illustrated in the diagram below. a) B is a straight line with P some point on B. ngles 1, 2, and 3 have the same vertex P. Use a protractor to measure the three angles. < )10. The sum of the measures of the three angles is ab 4 b) Draw a triangle and cut it out. Then draw a line and select some point P on the line. Tear off the three vertices of the triangle and arrange them as illustrated with the vertices at P. )11, P sc What does this seem to demonstrate about the sum of the measures of the three angles of a triangle? 714 SUM 6f tit meaksateszi tit ditt. a t s. trig474. is / CD Mar wain Media, e nc., Publishers LL

5 Name: c) Draw a triangle BC and cut it out. Date: Measurement of ngles: Exercises Find the midpoint of C and the midpoint of BC. Draw the _midsegment and fold the triangular region over the midsegment so that C falls on B. Fold again so that angles 1, 2, and 3 are in the position shown below. What does this seem to demonstrate about the sum of the measures of the three angles? it,s1.,* d) Use a protractor to measure each angle of the triangles pictured below. Record the measures in the table and find the sum of the measures for each triangle. ngle Triangle I Triangle II Triangle III Triangle IV L LB gr. fl). 40. tr 1 LC trb Sum (goo /RD lity What does the sum of the measures of the three angles for each triangle appear to be? Now find the average of the four sums. The average is CD Mark Twain Media, Inc., Publishers ita +Ito is/id ttty 4

6 Name - Date: Measurement of ngles: Exer:ises e) BC is a triangle with a line XY through vertex C and parallel to base B. X 3. BC is an equilateral triangle. The three sides of ane iwian le are congruerit Measure each angle and record the information. a) m L ) What do you know about the sum of the measures in degrees of Ll, L2, and L3? t si = ) Using your protractor, find the following measures. m L = 60 m LB = 60 m L1. 6o m L.3 = 3) How do the measures of L and L1 compare? tortyrai., weg- 4) How do the measures of LB and L3 compare? co r44 - ' 5) How does the sum of the measures of L1, L2, and L3 compare with the of the measures of L, LB, and L2? tptz #z j= Z +te74-18 =WO b) m LB c) m LC = 40 B d) What can you conclude about the three angles of an ateral trian CD Mark Twain Media, Inc., Publishers

7 Name: Date: Measurement of ngles: Exercises 4. BC is an isosceles triangle has at least two congruent sides. -6. is congruent to BC. Measure each angle and record the information. a) m L it b) m LB c) m LC 00 d) What can you conclude about L and LB? L=48 ie C h4 a(43 5. BC is a scalene triangle. scalene triangle has no congruent sides. (in inches) and the angles and record the information below. a) longest side 8c- b) next longest side 'VG c) shortest side Measure the sides d) largest angle e) next largest angle 48 f) smallest angle Z.Z g) What conclusions can you draw from this exercise? 16 SitteS* Side- IS WD V, to Smallest intekare tit t sat is WI? st 447es Mark Twain Media, Inc., Publishers

8 Geometry Date. Measurement of ngles: Exercises The sum of the measures in degrees of the three angles of a triangle is 180.This information can be used to determine the sum of the measures of the angles of a polygon with any number of sides (vvi s o easurement). For each polygon that follows, draw all possible diagonals from ssume all polygons to be regular (all sides with the same length and all angles congruent). Complete the following table. a) Number of sides Number of angles Number of diagonals Number of triangles ngle Sum of one triangle ngle Sum of the polygon Measure of each angle Z 3 at $I0 to tro _.?...i, i Co 7 0 s. it 74,-0 foo/ OD lot Or (30 tgo IVO a /to tgoo 11 I/ _ i o _1 12 (1 7 tip>0 1 o I 8.0 alp) t Iv 1 Ti n. d--.3 tt-2- e sv 1 it 01-2 t 4141 State a relationship between the number of sides of any polygon and the number of di4 gonals that can be drawn from one vertex. $ elittloitlf ir 04, 0 Itrit = n b) State a relationsip between the number of sides of a polygon and the number of tr.1 - lied by the sides and by the diagonals trom ore ve tex. * pia fif s fnoted = 1.r " z c) What is the sum of the measures of the ttemm5 of one triangle? i ltd)s3 -": S-4 0 d) What is the sum of the measures of the interior angles of 442.aucaultuicu? CD Mark Twain Media, Inc., Publishers

9 Name: Date: Measurement of ngles: Exercises 7. t8 and CD are a pair EF is called a transverse. EF is a line that intersects each of the parallel lines. Use a protractor to measure each angle in the diagram above. Record the measures in the blanks below. a) ml1= 71)* b) m L.2 = 71). c) m L3 = 7.1. e) m L5 = /fp f) m L6 = g) mz_7= 110 d) m L4 = h) mz18= (100 ID i) re angles with the same measure in degrees congruent? Yes Z./, L 2, L.3, and ter are congruent angles. t-r, 44,, and (16fr are congruent angles. j) Use a color code to show the sets of congruent angles in the picture. k) What is the sum of the measures of angles 2 and 6? (81, a I) What is the sum of the measures of angles 3 and 7? m) What tentative conclusions can you draw about the sum of the measures of two interior angles on the same side of the transversal? 141)* sme.e s # -, Mark Twain Media, Inc., Publishers

10 Name: Date. Measurement of ngles Exe'cisi.s When two parallel lines are cut by a transversal, eight angles are formed. Special names are given to these angles for convenience in discussing them Interior angles: 2, 3, 6, and 7 Exterior angles: 1, 4, 5, and 8 lternate interior angles: 2 and 3, 6 and 7 lternate exterior angles: 5 and 8, 1 and 4 Corresponding angles: 1 and 3, 5 and 7, 2 and 4, 6 and 8 8. In the illustration below, lintkeris parallel to line EF is CD. line a) What is the measure in degrees of each of the eight angles? h) Is line EF also perpendicular to line B 9 Ye) c) If a line is perpendicular to one of two parallel lines, is it perpendicular to both of the parallel lines? Yo d) re two lines parallel if they are both perpendicular to the same line 2 CD (.0 Mark Twain Media, Inc., Publishers Io

11 Date: Measurement of ngle;: Exerci ses 9. Four angles are associated with a pair of intersecting lines. In the illustration below, angles 1 and 2 are referred to as a pair of vertical angles, and angles 3 and 4 are also referred to as a pair of vertical angles. Use a protractor to measure each angle and record the information in the appropriate blanks. a) m L -1 = b) m L2 = c) m L3 = d) m L4 = e) What appears to be true qapair of vertical zalze. 10. In the figure below, angle 1 is referred to as an edenor apaie of the triangle. ngles and B are referred to as the two nonadjacent interior angles with reference to exterior angle 1. How are angles and B related to 1? B Find the measure of each of the following angles. a) m L = 6e (fro p b) m LB = c) m L1 = ID 1) The measure in degrees of an exteriqualeof a triangle is equal to the sum of the rne sures of the two nonadjacent interior angles. CD Mark Twain Media, Inc., Publishers

12 E:iiSC Geometry Date: Measurement of ngles: Exe,:-ciss.s For each of the following fig ures, the number of degrees in one or more of the angles is given. Without measuring, find the number of degrees in each of the other angles. Write the degrees of each angle on the diagrams. a) 147 It I b) c) CD Mark Twain Media, Inc., Publishers /.?