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.11.1 Fundamental vocabulary terms Measuring angles The magnitude of an angle does not depend on the length of its sides; it only depends on the relative direction of the two sides. E.g. the adjacent edges of a postcard are at an angle of 90– with respect to each other. But so is the Empire State Building in New York C... |
(180–). An angle twice the measure of a straight line is 360–. An angle measuring 360– looks identical to an angle of 0–, except for the labelling. All angles after 360– also look like we have seen them before. Angles that measure more than 360– are largely for mathematical convenience to maintain continuity in our en... |
angles. E.g., (a;b) and (c;d) are adjacent angles. Linear pairs Deflnition: The adjacent angles formed by two intersecting straight lines are said to form a linear pair. E.g. (a;b), (b;c), (c;d) and (d;a) all form linear pairs. Since the non common sides of a linear pair are part of the same straight line, the total an... |
middle of the intersection is the common point between the streets. It is possible that two lines that lie on the same plane never intersect even when extended to inflnity in either direction. Such lines are termed parallel lines. E.g. the tracks of a straight railway line are parallel lines. We wouldn’t want the track... |
angles. E.g. (1;5), (4;8) and (3;7), etc., are pairs of corresponding angles. In order to prove relationships between the angles deflned above, we will assume the following postulate regarding parallel lines. Euclid’s Parallel Line Postulate: Postulate: If a straight line falling on two straight lines makes the two int... |
, 2 = 4. Theorem 3: If two parallel lines are intersected by a transversal, the corresponding angles are equal. Proof: Again using Theorem 1, in the flgure above, (1 + 4) = 180– Also, since EF is a straight line, (4 + 5) = 180– So 1 = 5, etc. Theorem 4: The sum of the three angles in a triangle is 180. Proof: Consider t... |
of alternate interior angles are equal, the lines are parallel. Proof: Left as an exercise. Theorem 7: If two lines are intersected by a transversal such that a pair of alternate corresponding angles are equal, the lines are parallel. Proof: Left as an exercise. Theorem 8: Prove that if a line AB is parallel to CD, an... |
, r, we flnd that the radian angle subtended by the complete circumference, (or in other words the number of radians in a full circle) is 2…r r = 2…. This means that 2… radians is the same as 360–. With this in mind we can easily work out how to convert between degrees and radians. Deflnition: (rad) = (–) 2… 360 £ or (–)... |
cosine and tangent. These functions, known as trigonometric functions, relate the lengths of the sides of a triangle to its interior angles. How to remember the deflnitions Difierent people have difierent ways of remembering these ratios. One way involves deflning opposite to be side of the triangle opposite to the angle,... |
us look back at our values for sin { 0– 0 30– 1 2 45– 1 p2 60– p3 2 90– 1 180– 0 sin As you can see, the function sin has a value of 0 at = 0–. Its value then smoothly increases until = 90– when its value is 1. We then know that it later decreases to 0 when = 180–. Putting all this together we can start to picture the... |
followed by a reection in the y axis due to a negative sign in front of the. However, because cosine is an even function (i.e. symmetric about the y axis) this reection doesn’t really matter! 3The dotted lines in the tangent graph are known as asymptotes and the graph is said to display asymptotic behaviour. This mean... |
angle on this circle. Now let us add 360– to our angle so that our line sweeps all the way around the circle and ends up back where it started as indicated in the diagram. There is no way of knowing whether we have swept around the circle in this way as everything ends up exactly where it started. In other words, if w... |
� 180 360 As you can see, if we shift the whole graph by 360– left or right it will end up back on top of itself. The sine of an angle is therefore completely unchanged. Identities of this form can be very useful. We shall consider a few such identities here using the ideas of chapter (NOTE: Add in correct ref for tran... |
We can however still consider the black graph as a translation of 180– to the left or right. As before this gives us an identity{ cos = cos( 180–) § ¡ One flnal and very important class of translation identity are those that convert sine into cosine and vice versa. We have already seen one of these when we looked at th... |
any question. They do not need to be used in a triangle they will work for any angle in any situation. Deflnition: Pythagorean Identities{ cos2 + sin2 = 1 1 + cot2 = csc2 tan2 + 1 = sec2 7.4.3 Sine Rule So far we have only dealt with the trigonometry of right angled triangles where we are able to use our deflnitions of ... |
tell us that we have not proved the sine rule fully. Consider the following triangle{ b p a A B c This type of triangle which has all of its angles smaller than 90– is called an acute triangle. We have already proved the sine rule for acute triangles. If we ‘fold’ this acute triangle along the perpendicular we obtain ... |
of the triangle we flnd{ ¡ sin A = p b p = b sin A and and cos A = x b x = b cos A ) Now we will use Pythagorases theorem on the right hand side of the triangle. a2 = p2 + (c = p2 + c2 x)2 2cx + x2 ¡ ¡ Substituting in p = b sin A and x = b cos A a2 = b2 sin2 A + c2 = b2(sin2 A + cos2 A) + c2 ¡ 2cb cos A + b2 cos2 A 2bc... |
20) (7.21) (7.22) (7.23) Putting these expressions togeher we flnd that the base of the triangle, x + y, has a length{ x + y = sin + cos sin ` cos ` 113 + ` 1 90– sin + cos sin ` cos ` h ` ¡ We can now use the sine rule on the two angles that we know{ sin( + `) sin + cos sin ` cos ` = sin(90– 1 `) ¡ As we learnt in sect... |
cos( `) + sin sin( `) ¡ = cos cos ` ¡ sin sin ` ¡ (7.32) (7.33) 114 Similar equations can be found for the tangemt function. Proof of these is left to the reader6. Deflnition: Addition and Subtraction Formulae{ ¡ ¡ sin ( + `) = sin cos ` + cos sin ` cos sin ` sin ( `) = sin cos ` sin sin ` cos ( + `) = cos cos ` ¡ cos ... |
in the deflnition you need to divide everything through by a factor. The fact there is a 1 in the denominator should give you a clue as to what the factor is! Once you have the addition forumula you need to remember that tan(¡) = ¡ tan to flnd the subtraction formula. 115 Now that we have the double angle formulae it is... |
cos 2 ¡ 1+cos 2 = §r 1 cos 1 + cos ¡ As with all identities the half angle formulae can be expressed in a number of ways. Some of these will be proven in the worked example and more 116 given in the summary of identities at the end of the chapter. Deflnition: Half Angle Formulae cos sin tan 2 2 2 = = = §r §r §r 1 1 + c... |
`0 ¡ 2 ¶ 7.4.10 Solving Trigonometric Identities A standard type of question in an exam is of the form \show that sin(2) tan = 2 cos2 ". As well as being important in examinations being able to prove 117 38:7– 100m Figure 7.9: Determining the height of a building using trigonometry. identities is a key mathematical sk... |
80m £ tan 38:7– (7.65) (7.66) (7.67) (7.68) 7.5.2 Maps and Plans Maps and plans are usually scale drawings. This means that they are an enlagement (usually with a negative scale factor so that they are smaller than the original) so all angles are unchanged. We can use this to make use of maps and plans by adding infor... |
line represents sin = 0:7. ¡ can see from flgure 7.12, there are four possible angles with a sine of 0:7 between 360– and 360–. If we were to extend the range of the sine graph to inflnity we would in fact see that there are an inflnite number of solutions to this equation! This di–culty (which is caused by the periodici... |
¡ 121 1 0 1 ¡ 1st 2nd 3rd 4th 90 180 270 360 – 180 +VE +VE -VE -VE – 90 2nd +VE 1st +VE 3rd -VE 4th -VE – 270 – 0 /360 – Figure 7.13: The graph and unit circle showing the sign of the sine function. 1 90 180 270 360 360 ¡ 270 ¡ 180 ¡ 90 ¡ 1 ¡ This method can be time consuming and inexact. We shall now look at how to s... |
diagram as the letter, taken anticlock- 180– 90– S T A C 270– 0–/360– S T A C Figure 7.15: The two forms of the CAST diagram. wise from the bottom right, read C-A-S-T. The letter in each quadrant tells us which trigonometric functions are positive in that quadrant. The ‘A’ in the 1st quadrant stands for all (meaning s... |
.4. Using the same logic the 3rd quadrant solution can be seen to be (180– + ) and the 4th quadrant solution (360– ). It is now left to the reader to show, using similar graphs for cosine and tangent, that these relationships are true for all three of the trigonometric functions. These rules can be expressed in a simpl... |
• halving, our flnal answer gave us solutions in the range 0– 180–. There are two ways of dealing with this. We could redo the problem looking the the range 0– 720–. This will work but there is a simpler method. We know that all the trigonometric functions are periodic with a period of 360–. This means we can add (or s... |
That should look rather more familiar so that you can immediately write down the factorised form and the solutions: (y + 1)(y + 2) = 0 ¡ Next one just substitutes back for the temporary variable: ) y = 1 OR y = 2 ¡ tan (2x + 1) = 1 ¡ or tan (2x + 1) = 2 ¡ And then we are left with two linear trigonometric equations. B... |
(90– sec(90– ¡ ¡ ¡ ¡ ¡ ¡ ) = cos ) = sin ) = cot ) = tan ) = sec ) = csc Double Angle Identities Addition/Subtraction Identities Half Angle Identities sin(2) = 2 sin cos sin ( + `) = sin cos ` + cos sin ` sin ( ¡ `) = sin cos ` cos sin ` ¡ cos (2) = cos2 ¡ cos (2) = 2 cos2 cos (2) = 1 sin2 1 ¡ 2 sin2 ¡ tan (2) = 2 tan ... |
, or what he scored! Obviously you are trying everything to get him to tell you, and he decides to tease you and makes you work it out for yourself. He says the following: \I have 2 marks more than you and the sum of both our marks is equal to 14. How much did we get?" Now if the numbers are simple like in the example,... |
(8.6) (8.7) (8.8) (8.9) (8.10) (8.11) (8.12) If x is multiplied by a fraction we need to divide both sides of the equal sign with that fraction to get x alone. We do that by ipping the fraction around and then multiplying both sides with it Example 1: )x( 2 3 14 3 x = ) = 7( 2 3 ) ) ) (8.13) (8.14) (8.15) These are th... |
+ 34x = 2x ¡ 6x + 34x = 2x 6x + 34x ¡ 2x = 24 64 ¡ 24 ¡ 64 ¡ 24 ¡ 64 ¡ ¡ ¡ 12 12 ) ¡ ) ¡ ¡ 100 ) ) ) 26x = x = x = ¡ ¡ ¡ 100 26 50 13 (8.27) (8.28) (8.29) (8.30) (8.31) (8.32) (8.33) (8.34) (8.35) (8.36) (8.37) (8.38) (8.39) (8.40) (8.41) (8.42) We simplifled the answer - but this is not necessarily a required step TIP... |
26x = 87 x = 87 26 (8.55) (8.56) (8.57) (8.58) And that is it for our examples. This covers all the types of linear equations you can be expected to solve. It’s the best to always keep your priority of steps in mind and then just simply do them one by one. If you are unsure about your answer you can just substitute it... |
{ write the problem in the form ax2 + bx + c = 0 (with a positive) { write down two brackets, with an x in each, leaving room for a num- ber on each side ( x )( x ) (8.72) 136 { write out your options (in a table at the side) for multiplying two numbers together to give a. these numbers should go in front of the xs in... |
change anything. Therefore y = ax2 + bx + b2 4a ¡ b2 4a + c Taking out a factor of a then gives y = a(x2 + b a + ( b 2a )2) + c b2 4a ¡ (8.73) (8.74) 137 25 20 15 10 Figure 8.1: Graph of y = x2 x 6 ¡ ¡ The expression in brackets is then a perfect square so that y = a(x + ( b 2a ))2 + c ¡ = a(x ( ¡ ¡ b 2a ))2 + (c b2 4... |
quadratic in the form f (x) = a(x f (x x2 + 6x (x2 (x2 (x2 (x2 ¡ ¡ ¡ ¡ 6x + 5) 6x + 9 6x + 9) + 4 3)2 + 4 ¡ 9 + 5) p)2 + q. ¡ (8.78) (8.79) (8.80) (8.81) (8.82) ¡ Therefore the turning point is (3,4) and the axis of symmetry is x = 3 (in other words p = 3 and q = 4). ¡ Now to plot the graph we need to know the interce... |
factorising the quadratic function. 3) = 0 means that either x = 6 = (x + 2)(x ¡ ¡ ¡ ¡ x Knowing how to factorise a quadratic takes some practice, but here some general ideas which are useful. { First divide the entire equation by any common factor of the coe–cients, so as to obtain an equation of the form ax2 + bx + ... |
� 2. This gives A: We must flnd the solutions to the equation f (x) = ¡ First we divide both sides of the equation be a factor of the equation ¡ Now, let us assume that x2 ¡ 2x + 1 = 0 (8.89) x2 ¡ 2x + 1 = (x + s)(x + v) = x2 + (s + v)x + sv (8.90) Then sv = 1 and therefore either s = v = 1 or s = v = s + v = 2, it foll... |
possible to factorize a quadratic function. We shall now derive a general formula, which gives the solutions to any quadratic equation. Consider a general quadratic equation ax2 + bx + c = 0. Adding and subtracting b2 equation. Thus 4a from the left-hand side does not change the ax2 + bx + b2 4a ¡ b2 4a + c = 0 (8.94)... |
102) (8.103) (8.104) Therefore the two roots of the quadratic function are x = ¡ 3+2p14 4 and 3 2p14 ¡ 4 ¡. Example 2: Q: Solve for the solutions to the quadratic equation x2 5x + 8. ¡ A: Again it is not possible to factorise this equation. The general formula shows that b x = ¡ § 4ac pb2 2a ¡ 5)2 ( ¡ 2(1) p ( = ¡ ¡ 5(... |
, these are given by x = b 2a ¡ (8.110) Unequal Roots: There will be 2 unequal roots if ¢ > 0. The roots of f (x) are rational if ¢ is a perfect square (a number which is the square of a rational number), since, in this case, p¢ is rational. Otherwise, if ¢ is not a perfect square, then the roots are irrational. Imagin... |
8.113) ¡ 0 because this is a perfect square. Therefore we know that ¡ ¡ Now (b ¢ ¡ 4 > 0. 4)2 ‚ ‚ We can thus say that f (x) has real unequal roots. We do not know whether ¢ is a perfect square, since we do not know that value of the constant b, and therefore we cannot say whether the roots are rational or irrational. ... |
feasible region. Now sometimes x and y must satisfy more than one inequality. In this case, we consider each inequality separately and then the feasible region is where the feasible regions of each inequality overlap. 146 3 2 1 3 2 1 2 3 4 Figure 8.4: Graph of y = 2x + 3. Points satisfying y ¡ 2x + 3 are shaded ‚ ¡ Wo... |
3. We can see from flgure 8.6 that f (x) is above/on the x-axis when x or x 2. 3 ‚ Therefore the solution to the quadratic inequality is or in interval notation ( [3; ;2] ). x : x f ‚ 3 or x 2 g • ¡1 [ 1 Note: The x-intercepts are included in this solution, since the f (x) inequality includes the solution f (x) = 0. 0 ... |
. • ¡ 149!! 6 4 2 x1 4 ¡ 3 ¡ 2 ¡ 1 ¡ x2 1 Figure 8.7: Graph of f (x) = x2 ¡ ¡ 3x + 5 1:5 1:0 0:5 0:5 1:0 Figure 8.8: Graph of f (x) = 4x2 4x + 1 ¡ A: Let f (x) = 4x2 f (x) = (2x ¡ 4x + 1. Factorising this quadratic function gives 1)2, which shows that f (x) = 0 only when x = 1 2. The function f (x) lies below/on the x-... |
AandB) ¡ { identify dependent and independent events and calculate the prob of 2 independent events occurring by applying the product rule for independent events P (AandB) = P (A):P (B) { identify mutually exclusive events. calculate prob of the events occuring by applying additive rule for mutually exclusive events P ... |
to the best options (pyramid and micro lenders schemes) 9.6 Worked Examples TODO 9.7 Exercises TODO 154 Part II Old Maths 155 Chapter 10 Worked Examples 10.1 Exponential Numbers (NOTE: All of these worked examples need to be updated to use the FHSST internal environments and also to use the new rules (i changed the or... |
following paragraph Aside: A fraction to the power of a negative exponent is the same as the inverse fraction to the power of the corresponding positive exponent. Therefore ( a b )¡ n = 1 a b = a an bn = ( b a )n (10.18) 157 Example 1: Q: Simplify the expression ( 1 A: 2 )¡ 3. ( 1 2 )¡ 3 = 1 ( 1 2 )3 = 1 1 23 = 23 = 8... |
40) Worked Example: Q: Which of the numbers 3p100 and p20 is bigger? (You may not use a calculator to answer this question.) A: The two numbers must flrst be converted into like surds. Since we have a cube root and a square root, we must flrst flnd the lowest common multiple of 2 and 3 which is 6. We then convert each of ... |
x2 + 6x + 9 = 6x + 13 x2 = 4 x = 2 or x = 2 ¡ ) ) ) (10.51) (10.52) (10.53) (10.54) When we square both sides of the equation, it is possible that we introduce extra solutions, which may not actually satisfy the original equation. This is the reason one should always check the answers by substituting these into the or... |
10.66) (10.67) (10.68) = 299790000 m s Example 2: The approximate radius of the hydrogen atom (called the Bohr radius a0) is a0 = 5:3 = 5:3 = 5:3 £ £ £ 11 m 10¡ 1 1011 m 1 10000000000 0:00000000001 m m = 5:3 = 0:000000000053 m £ (10.69) (10.70) (10.71) (10.72) (10.73) Note to self: check the above number... I think a0 ... |
plus four. We can check whether the formula is valid by going through the flrst few terms, and seeing whether the terms in the sequence correspond to the If we substitute n = 1 into the terms given by the formula. formula, we should get the flrst term of the sequence, and this is indeed the case: 2 = 2 + 4(1 1). If we s... |
for the nth term, as was previously the case. So, substituting n = 1, we get that a1+1 = a1 + 2(1) = 3 + 2 = 5: To get the next term, we must substitute n = 2: a2+1 = a2 + 2(2) = 5 + 4 = 9. Lastly, we calculate that a4 = 9 + 2(3) = 15. So the flrst four terms of the sequence are 3;5;9;15;:::. Worked Example 13 : Checki... |
for the nth term of the sequence 6; 17; 28; 39;:::. Which term in the sequence equals 688? 164 Answer:First we must check that the given sequence is in fact an arithmetical one, otherwise we can’t use the formula. This is easily seen, since the difierence between successive terms is 11. Now we must use (??). We know th... |
2(3n). If we work out the ratio between successive terms, we get that an+1 an = 2(3 n+1 = 3, which is the same answer that we got using the previous method. 2(3n¡1) = 3n ¡ n ) Worked Example 16 : Calculating the Formula for the nth term of a Sequence 165 Question:Find a formula for the nth term of the sequence 2;4;8;1... |
= n 1)] Answer:Let us start by writing out the flrst few terms of Sn. S1 equals the flrst term of the series, so S1 = 2. S2 is the sum of the flrst two terms, so S2 = 2 + 4 = 6. S3 is the sum of the flrst three terms, namely 12. Continuing in this fashion, we can see that the flrst few terms of Sn are 2;6;12;20;28;:::.What... |
. Answer:We wish to flnd S16 (since 46 is the 16th term of the series). Of course we can do this with a calculator, but there is a much quicker way. S16 = 1 + 4 + ::: + 43 + 46 S16 = 46 + 43 + ::: + 4 + 1 2S16 = 47 + 47 + ::: + 47 + 47 2S16 = 16 47 = 752 £ S16 = 752 2 = 376 Worked Example 22 : Using the Formula for Sn Q... |
a1 +a1 +:::+a1 = na1. (10.74) Sn = ¡ 1 1) Worked Example 24 : Using the Formula for Sn Question:What is 2 + 4 + 8 + 16 + ::: + 32768? Answer:We are dealing with a geometric series, so we have to use equation (10.74). In order to use it we need to know which values to put in for a1, r, and n. Since the series starts at... |
We don’t really have any systematic way of working this out yet, but we can easily guess by using our calculators. { Working out the flrst few terms of the sum, we get S1 = 1, S2 = 1 + 2 = 3, S3 = 1 + 2 + 3 = 6, S4 = 1 + 2 + 3 + 4 = 10, S5 = 1 + 2 + 3 + 4 + 5 = 15. Clearly this is getting larger and larger, and it would... |
We are asked to determine the sum of the inflnite series, which is 1 and 1, so we can use the formula. The formula between ¡ = a gives S 1 ¡ P 8 + :::. The common ratio is clearly 1 r = 1 = 2. 1 ¡ 1k=1( 1 2 )k 1 2 1 1 ¡ 10.3 functions Q: State the domain and range of the function y = x2 notation and interval notation. ¡... |
ax2 + b) b ¡ (10.80) (10.81) (10.82) 171 $ $ Therefore the slope describes the change in y (sometimes called ¢y = x1) between any two y2 ¡ difierent points on the line, i.e. y1) divided by the change in x (¢x = x2 ¡ a = ¢y y2 ¡ y1 x2 ¡ x1 ¢x = 0) because one cannot x1 6 We have used the fact that x1 6 = x2 (i.e. x2 ¡ = ... |
, for a general parabola of the form f (x) = ax2 + c where a is positive, the term ax2 is always positive so the function is at a minimum when x = 0. Therefore the arms of the parabola point upwards. Otherwise, if a is negative, then ax2 is always negative and thus the function is maximum at x = 0. This means that the ... |
A: A table of x and y values is as follows: 1. x : y = 1 x : -4 1 4 -2 1 2 -4 1 2 -2 1 -1 2 - 1 2 4 - 1 4 This gives the graph shown below Figure 10.8: Graph of the hyperbola xy = 1 ¡ We can see that a hyperbola has no x or y-intercepts. However, there are two general forms for hyperbolae, depending on whether a is po... |
the line y = x are (-3,-3) and (3,3). It is also clear that the points (-1,-9), (-9,-1), (1,9) and (9,1) are part of the hyperbola. Thus the graph is 177 Figure 10.10: Graph of the hyperbola xy = 9 1:5 1:0 0:5 0:5 1:0 1:5 1:5 ¡ 1:0 ¡ 0:5 ¡ 0:5 ¡ 1:0 ¡ 1:5 ¡ 1 2 ; 1 2 ) and Figure 10.11: Graph of a hyperbolic function ... |
the semi-circle y = p9 the domain is [ x2, 3;3] and the range is [0;3] and, for the semi-circle y = ¡ p9 ¡ ¡ ¡ x2, the domain is [ ¡ 3;3] and the range is [-3,0]. 179 3 3 ¡ 3 3 ¡ 0 3 3 ¡ Figure 10.13: Semi-circles of radius 3, centered at the origin. The equation for the left semi-circle is y = p9 x2 whereas the right... |
p case", we need to use 1 p ¡ ¡ ¡ ref a lot more) n For the flrst semi-circle, the domain is [0;1] and the range is [ the case of the second semi-circle, the domain is [ [ ¡ 1;1]. ¡ 1;1] and in ¡ 1;0] and the range is Worked Example: Q: Plot a graph of the function f (x) = x + 2 j j ¡ 5. A: Let us flrst work out a table... |
2( 2x x + 1) ¡ 2 = 0 ) ¡ ¡ 2x = 10.111) (10.112) (10.113) (10.114) (10.115) Therefore the graph of the absolute values function is as follows Figure 10.16: Graph of the absolute values function y = 2 j 4 1 j ¡ ¡ Example 2: Q: Plot a graph of the function f (x) = turning point and axis of symmetry. x+3 1 showing the in... |
) (10.118) (10.119) (10.120) But this is just the equation for a circle centered at the origin with radius 1 (x2 + y2 = 1), where x and y have been replaced by x 1 and y + 2. So we can see that this is a circle which has been moved 1 to the right and 2 downwards. Therefore this relation describes a circle centered at t... |
the equation for the function which can be obtained by stretching f (x) vertically by a factor of a, then shifting this function to the right by p and upwards by q? A: We start with the function y = x2 and must be very careful to apply these changes in the right order. First we stretch f (x) vertically by a factor of ... |
left) and the flnal function y = x + 3 (right) ¡ Finally we must look for two transformations which give this same result. We suspect that there must be a reection involved, because the flnal funcx and the initial function contains x (also the above graph tion involves shows that the straight lines are perpendicular). Th... |
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of ABC, mABD 15, and mDBC 90. a. Find mABC. b. Name an acute angle. A D c. Name a right angle. d. Name an obtuse angle. B C 14365C01.pgs 7/9/07 4:37 PM Page 18 18 Essentials of Geometry Solution a. mABC mABD mDBC 15 90 105 b. ABD is an acute angle. c. DBC is a right angle. d. ABC is an obtuse angle. A D B C Answers a.... |
, ABC and DEF have the same measure. This is written as mABC mDEF. Since congruent angles are angles that have the same measure, we may also say that ABC and DEF are congruent angles, symbolized as ABC DEF. We may use either notation. ABC DEF mABC mDEF The angles are congruent. The measures of the angles are equal. Not... |
called the foot of the perpendicular. The distance from P to R length of PR ). Adding and Subtracting Angles T P S R DEFINITION If point P is a point in the interior of RST and RST is not a straight angle, or if P is any point not on straight angle RST, then RST is the sum of two angles, RSP and PST. 14365C01.pgs 7/9/... |
Write a conclusion that states the congruence of two angles. c. Write a conclusion that states the equality of the measures of two angles. h CD h AC 14. 15. bisects ACB. is the bisector of DAB. 16. If a straight angle is bisected, what types of angles are formed? 17. If a right angle is bisected, what types of angles ... |
and crease the paper. Label two points, D and E, on the crease with B between D and E. 3. Measure each angle formed by the crease and one of the opposite rays that form the straight angle. What is true about these angles? 4. What is true about DE and AC? 1-7 TRIANGLES DEFINITION A polygon is a closed figure in a plane... |
there is one vertex of the triangle that is not an endpoint of that side. For example, in ABC, C is not an endpoint of. We say that. AB AB is opposite A Similarly, and A is opposite is the side opposite C and that C is the angle opposite side AC is opposite B and B is opposite ; also AB AC BC BC. The length of a side ... |
two sides of the triangle, are called the that form the right angle, GH legs of the right triangle. The third side of the triangle,, the side opposite the right angle, is called the hypotenuse. and HL GL G g e l hyp oten use H leg L EXAMPLE 1 E In DEF, mE 90 and EF ED. D F a. Classify the triangle according The triang... |
angle. (4) QTR is isosceles. No (5) QRW is adjacent to WRU. Yes Yes No P V Q W S RT U (6) Q is between V and T. (7) PQR and VQT intersect. (8) R is the midpoint of. TU Yes Yes No (9) QT PS No (10) mQRW mVQP No (11) PQTS is a trapezoid. No Exercises Writing About Mathematics 1. Is the statement “A triangle consists of ... |
of the triangle? 17. The lengths of the sides of an isosceles triangle are represented by 2x 5, 2x 6, and 3x 3. What are the lengths of the sides of the triangle? (Hint: the length of a line segment is a positive quantity.) 18. The measures of the sides of an isosceles triangle are represented by x 5, 3x 13, and 4x 11... |
• An angle is a set of points that is the union of two rays having the same endpoint. • A straight angle is an angle that is the union of opposite rays and whose degree measure is 180. • An acute angle is an angle whose degree measure is greater than 0 and less than 90. • A right angle is an angle whose degree measure... |
a 5 1 Distributive Property a(b c) ab ac Identity Property Inverse Property Multiplication Property ab 0 if and only if a 0 or b 0 of Zero VOCABULARY 1-1 Undefined term • Set • Point • Line • Straight line • Plane 1-2 Number line • Coordinate • Graph • Numerical operation • Closure property of addition • Closure prope... |
7. Two rays of the same line with a common endpoint and no other points in common. 8. Angles that have the same measure. 9. A triangle that has two congruent sides. 10. A set of points consisting of two points on a line and all points on the line between these two points. 11. In right triangle LMN, mM 90. Which side o... |
sphere. How does this point compare to a point in Euclidean geometry? 2. The shortest distance between two points is called a geodesic. In Euclidean geometry, a line is a geodesic. Draw a second point on your sphere. Connect the points with a geodesic and extend it as far as possible. How does this geodesic compare to... |
07 4:40 PM Page 35 2-1 SENTENCES, STATEMENTS, AND TRUTH VALUES Sentences, Statements, and Truth Values 35 Logic is the science of reasoning. The principles of logic allow us to determine if a statement is true, false, or uncertain on the basis of the truth of related statements. We solve problems and draw conclusions b... |
open sentences, that is, sentences that contain a variable. The truth value of the open sentence depended on the value of the variable. For example, the open sentence x 2 5 is true when x 3 and false for all other values of x. In some sentences, a pronoun, such as he, she, or it, acts like a variable and the name that... |
a true sentence, a false sentence, an open sentence, or not a mathematical sentence at all. Answers a. Football is a water sport. False sentence b. Football is a team sport. True sentence c. He is a football player. Open sentence: the variable is he. d. Do you like football? Not a mathematical sentence: this is a ques... |
. It is not true that a carpenter works with wood. (True) (False) (False) Both negations express the same false statement. Logic Symbols The basic element of logic is a simple declarative sentence. We represent this basic element by a single, lowercase letter. Although any letter can be used to represent a sentence, th... |
. m: Massachusetts is a city. (True) (False) For each sentence given in symbolic form: a. Write a complete sentence in words to show what the symbols represent. b. Tell whether the statement is true or false. (1) k (2) m Answers a. Oatmeal is not a cereal. a. Massachusetts is not a city. b. False b. True EXAMPLE 3 Exer... |
27. Its capital is Albany. 28. It does not border on or touch an ocean. 29. It is on the east coast of the United States. 30. It is one of the states of the United States of America. 31. It is one of the last two states admitted to the United States of America. 14365C02.pgs 7/9/07 4:40 PM Page 41 Sentences, Statements... |
likes spring. For each sentence given in symbolic form: a. Write a complete sentence in words to show what the symbols represent. b. Tell whether the sentence is true, false, or open. 57. p 61. (p) 58. q 62. (q) 59. r 63. (r) 60. s 64. (s) 14365C02.pgs 7/9/07 4:40 PM Page 42 42 Logic 2-2 CONJUNCTIONS We have identifie... |
q is true. q is false p is false and q is false. These four possible combinations of the truth values of p and q can be displayed in a chart called a truth table. The truth table can be used to show the possible truth values of a compound statement that is made up of two simple statements. For instance, write a truth ... |
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