# November 10, 2004 : Fax:

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1 Honors Geometry Issue Super Mathter November 0, 004 : Fax: For class info, visit irect your questions and comments to Name: Peter Lin Peter Lin SUPPLEMENTRY N MPLEMENTRY NGLES... VERTIL NGLES... 3 NGLE ISETRS... 6 PRLLEL LINES N NGLES...0 STNR TEST...

2 Supplementary and omplementary ngles When the two sides of an angle form a straight line, the angle is called a straight angle or flat angle. Specifically, the angle is measured to be o Honors Geometry Issue [Linear Pair Postulate] Two angles of a linear pair are supplementary to each other.. What should be the value of x in the figure? x 0 o Straight angle is 80. Right angle is 90. right angle notation, it stands for 90 o right angle is special, it also has a special notation. Instead of writing 90 at the angle, we draw a small square near the angle vertex. oth and are right angles in the figure. Each angle of a rectangle is a right angle. 90 o 90 o 90 o If the sum of two angles is 90, we called them complementary angles. Namely, + = 90 and are complementary. If the sum of two angles is 80, we called them supplementary angles. Namely, + = 80 and are supplementary.

3 Vertical ngles efinition: vertical angle In the figure, and are called vertical angles since their sides form two crossing lines and they stay on the opposite sides of the vertex. 3 4 Honors Geometry Issue Question set [4-7] vertical angles NT vertical angles Two lines L and M intersect. Given that = L 4. Find the measure of and state the reason. M Then =, namely, vertical angles are congruent.. For, name the vertical angle and adjacent angle.. Find the measure of 3 and state the reason and 3 are called _, so are and 4. Each pair of angles are _. 3. List all pairs of adjacent and vertical angles Find the measure of 4 and state the reason. 3

4 Honors Geometry Issue Question set [8-7]. n angle is equal to its complementary onceptual and computational problems. angle, what is the measure of this angle? 8. If two angles are complementary to the same angle, then they must be congruent. Is this statement always true? 3. What is the complementary angle for 3? What is the supplementary for 3? 9. If an angle has 60 as its complementary angle, what is its supplementary angle? 0. n angle is twice of its complementary angle, what is the measure of this angle? 4. n angle is two times the complementary angle of 40. What is the measure of this angle?. n angle is twice its supplementary angle, what is the measure of this angle?. Given: line EF bisects as shown in the figure, namely, 3 = 4. Prove: =. E 3 4 F 4

5 Honors Geometry Issue 6. oth and are right angles. = 40. Find the measure of. 40 o 7. s the figure shows, bisects, which is a right angle. Find the measures of. E 8. What is the measure of? (60 - x) o o (3x + ) o

6 ngle isectors efinition: angle bisector ngle bisector divides an angle equally. The ray divides the angle evenly into two congruent angles: and, so is called the angle bisector of. Honors Geometry Issue. What should be the value for x in the figure? 60 o x bisects an angle Question set [9 - ] Find the value of x in each of the following. 9. What is the measure of x in the figure?. Find the value of x. x- 3x+ x 40 o 0. is a straight angle (80 ), and divide the entire angle into three congruent angles, what should be the value for x? Question set [3-8] onceptual and computational problems. 3. and are called. x x x 6

7 4. Two angles are if they add up to be 90. Honors Geometry Issue 8. Is it true that a straight angle is twice a right angle?. Two angles are if they add up to be 80. Question set [9-30] and are linear pair. bisects and E bisects. E 6. What angles are supplementary to? 9. Given that = 40, what is the measure of? Prove that + = 90 regardless of the measure of. 7. If bisects, what is the measure of? o o Question set [3-3] omputational problems. 7

8 3. bisects. If = 30, find the value for x. Honors Geometry Issue 34. E is bisected by and is bisected by. Find the value for x. x o 30 o E 0 o x 3. is bisected by in the figure. If E = 00, find the value for x. x o 00 o E 3. P is a point on E. P bisects P. P bisects PE. Find the measure of P. E o P o x x 33. s in the following figure, is it true that x = y? Name of an angle: y o x o 40 o 40 o n angle is formed by one vertex and two sides connected by the vertex. s in the figure, the angle can be expressed as or in short. efinition: congruent angle ongruent angles are equal in measure. 8

9 Honors Geometry Issue is congruent to is not congruent to 3 4 When two angles and are measured to be the same, we called them congruent angles, or we say is congruent to. 9

10 Honors Geometry Issue Parallel Lines and ngles There are two such pairs: ( 3, 6) and ( 4, ). efinition: corresponding angles lternate exterior angles: Two lines L and L (not necessarily parallel) are cut by a transversal. Get familiar with the following terms. L 3 4 L 7 8 L L There are two such pairs: (, 7) and (, 8). There are four such pairs: (, ), ( 3, 7), (, 6), ( 4, 8). THEREM [orresponding ngles Postulate] If L and L are parallel and cut by a transversal then corresponding angles are congruent. onsecutive exterior angles: 7 8 There are two such pairs: (, 7) and (, 8). L L onsecutive interior angles: 3 4 L onsecutive and alternate angles The term consecutive pair refers to both angles falling on the same side of the transversal. 6 There are two such pairs: ( 3, ), ( 4, 6). lternate interior angles: L consecutive consecutive The term alternate pair refers to either of the angle falling at the opposite side of the transversal. L L L L alternate alternate L L 0

11 Honors Geometry Issue Interior and exterior angles The term interior pair refers to both angles falling in the interior strip formed by L and L. Interior pairs L L The term exterior pair refers to both angles falling in the exterior strip formed by L and L. exterior pairs L L 36. In each of the following problems use the information to name the segments that must be parallel. If there is no such segment, write none. 4 3 F 3 4 E G Given Parallel Reason segments a) = 8 //EG corr. angles b) + = 7+ 8 c) = 80 d) 8 = e) 3 = 4 f) =80 g) + = 80 h) =

12 Standard Test 37. square is a special case of all of the following geometric figures EXEPT a () parallelogram () rectangle () rhombus () trapezoid Honors Geometry Issue 40. triangle has two congruent sides, and the measure of one angle is 40. Which of the following types of triangles is it? () isosceles () equilateral () right () scalene 38. polygon is a plane figure composed of connected lines. How many connected lines must there be to make a polygon? () 3 or more () 4 or more () or more () 6 or more 4. triangle has one 30 angle and one 60 angle. Which of the following types of triangles is it? () isosceles () equilateral () right () scalene 39. Which of the following statements is true? () Parallel lines intersect at right angles. () Parallel lines never intersect. () Perpendicular lines never intersect. () Intersecting lines have two points in common. 4. triangle has angles of 7 and 6. Which of the following best describes the triangle? () acute scalene () obtuse scalene () acute isosceles () obtuse isosceles

13 43. Which of the following does NT have parallel two pairs of line segments? () a rhombus () a square () a trapezoid () a rectangle Honors Geometry Issue 46. If pentagon E is similar to pentagon FGHIJ, and = 0, =, and FG = 30, what is IH? () 3. () () () In a triangle, angle is 70 and angle is 30. What is the measure of angle? () 90 () 70 () 80 () What is the greatest area possible enclosed by a quadrilateral with a perimeter of 4 feet? () 6 square feet () 4 square feet () 36 square feet () 48 square feet 4. What is a quadrilateral with two parallel sides and an angle of 4? () triangle () rectangle () square () parallelogram 48. What is the difference in area between a square with a base of 4 feet and a circle with a diameter of 4 feet? () 6 - π square feet () 6-4π square feet () 8π -6 square feet () 6π - 6 square feet 3

14 49. What is the difference in perimeter between a square with a base of 4 feet and a circle with a diameter of 4 feet? () 8 - π feet () 6 - π feet () 6-4π feet () 6-8π feet Honors Geometry Issue. What is the perimeter of the regular hexagon shown below? () 0 () 4 () 8 () rectangle s topmost side is 3 times that of the leftmost side. If the leftmost side is inches long, what is the area of the rectangle? () 3 () 6 () 3 () 6 3. What is the perimeter of the following figure? a 4 o () a () a + a () ( + )a () 4a. What is the perimeter of the triangle shown below? 8 0 () 4 () 8 () 6 () 4 4

15 Honors Geometry Issue 4. The perimeter of a rectangle is 48 feet. Its two longest sides add up to 86 feet. What is the length of each of its two shortest sides? () 3 feet () 4 feet () 6 feet () 7 feet. How many feet of ribbon will a theatrical company need to tie off a performance area that is 34 feet long and 0 feet wide? () 4 () 68 () 88 () What is the outer perimeter of the doorway shown below? 0 4 () () 4 () 0 + 4π () 4 + π

16 Honors Geometry Issue nswer Key Supplementary and omplementary ngles. x +0 = 80 x = 30. Vertical ngles. vertical: 4 adjacent: 3. Vertical angle pairs: (, 4), (, ), ( 3, 6). djacent pairs (, 6), ( 6, ), (, 4), ( 4, 3), ( 3, ), (, ) = 7 since and are linear pair with respect to line M = 0 since and 3 are linear pair with respect to line L. 6. vertical angles congruent = = 7 for vertical angles. 8. Yes, it is always true The angle is 30 since 90-60=30. Therefore, its supplementary angle is = The complementary angle for 60 is ivide 80 into + parts. Each part gets 60. The angle in the question should get two parts, thus it has The complementary angle for 4 is still The complementary angle of 3 is. The supplementary angle of 3 is The complementary angle of 40 is 0. The angle is two times 0, so it must be 00.. = 4, (vertical angle) 4 = 3 (given) 3 = (vertical angle) = (transitivity) = 90 (Given) = 0 = + (angle addition property) = = = = 3 since the bisector divides the right angle to two angles with x = + 3x x = x 0 = 4x x = = (60 - x) = 6

17 ngle isectors 9. x = 0 x+40 =90 x = 0 0. x = 60 since 80 3 = 60.. x = 0. x - + 3x + = 90 x = 90 x = 8 3. Linear pair 4. complementary. supplementary 6. and 4 are supplementary to Yes. Straight angle has 80 and a right angle is = = 40 = = (40 ) = = 80 (linear pair) = (bisector) = (bisector) + = ( + ) = x = 80 - (30 ) = 0 3. = 80, therefore, x = (80) = Yes, since x = y = E = 40. =40. Thus, x = (40) = Honors Geometry Issue Parallel Lines and ngles 36. The answer is listed in the following table. Given Parallel segments Reason a) = 8 //EG corr. angles b) + = F/ corr. angles. c) = 80 E//G consec. int. d) 8 = None e) 3 = 4 E//G corr. angles f) =80 F// consec. int. g) + = 80 None h) = //EG alt. int. angles Standard Test =

18 4.. Honors Geometry Issue 6. 8

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