If you draw a line across the big triangle like the one you drew, but parallel to the base, then we will have $y=w$, so $x+y=x+w$. Since $x+y$ is an integer greater than $110$, the smallest possible value of $x+y$, and hence of $x+w$, is $111$. If the line meets the base somewhere to the right of the base, that is, has negative slope if we think of the base as horizontal, then $w>y$, so $w+x$ will be greater than $111$. Show If the line across the triangle has positive slope, then $w$ can be substantially less than $y$, so $w+x$ can be substantially less than $x+y$. Remark: If we change the problem to asking about $w+z$, then indeed the smallest possible value of $w+z$ is $111$. For the angles complementary to $x$ and $y$ are $180^\circ -x$ and $180^\circ -y$. Thus the sum of the angles of the quadrilateral "below" our line is $(180^\circ-x)+(180^\circ -y)+w+z.$ But the sum of the angles of a convex quadrilateral is $360^\circ$. It follows that $$(180^\circ-x)+(180^\circ -y)+w+z=360^\circ,$$ from which we see that $w+z=x+y$. Since the smallest allowed $x+y$ is $111^\circ$, this is also the smallest allowed value of $w+z$.
The side lengths are roughly the same, so we expect all angles to be somewhat near $\frac \pi3$. We know that the sines are proportional to the sidelengths (sine theorem, $\frac{\sin\alpha}a=\frac{\sin\beta}b=\frac{\sin\gamma}c$), hence the fact that the shortest side is $12.5\%$ smaller (and the longest is $12.5\%$ longer) than the middle side, suggests that the sine of the smallest angle is about $12.5\%$ smaller than the sine of $\frac \pi3$ (similarly, the sine of the largest is about as much larger, and themiddle angle is still about sixty degrees). But at that angle, the slope of sine is $\frac 12$, so we raise to $25\%$ and thus expect an angle about $\frac\pi3-\frac\pi{12}=\frac\pi4=45^\circ$. Honestly, we probably deducted too much, so by simply applying these rule-of-thumb calculations (no calculator, no slide-rule, no trigonometric table), we pick D from the suggested options. Remark: Using a calculator to numerically determine the three angles, we find that they are $$ 48^\circ11'22.9''\qquad 58^\circ24'42.7''\qquad 73^\circ23'54.4''$$ whichjustifies our approximations, but shows an error in the problem statement. Note however that the smallest angles is approximately $48.19^\circ$, which is not the same as the stated $48^\circ19'$.
"SSS" means "Side, Side, Side"
We use the "angle" version of the Law of Cosines: cos(C) = a2 + b2 − c2 2ab cos(A) = b2 + c2 − a2 2bc cos(B) = c2 + a2 − b2 2ca (they are all the same formula, just different labels)
 In this triangle we know the three sides: Use The Law of Cosines first to find one of the angles. It doesn't matter which one. Let's find angle A first: cos A = (b2 + c2 − a2) / 2bc cos A = (62 + 72 − 82) / (2×6×7) cos A = (36 + 49 − 64) / 84 cos A = 0.25 A = cos−1(0.25) A = 75.5224...° A = 75.5° to one decimal place. Next we will find another side. We use The Law of Cosines again, this time for angle B: cos B = (c2 + a2 − b2)/2ca cos B = (72 + 82 − 62)/(2×7×8) cos B = (49 + 64 − 36) / 112 cos B = 0.6875 B = cos−1(0.6875) B = 46.5674...° B = 46.6° to one decimal place Finally, we can find angle C by using 'angles of a triangle add to 180°': C = 180° − 75.5224...° − 46.5674...° C = 57.9° to one decimal place Now we have completely solved the triangle i.e. we have found all its angles. The triangle can have letters other than ABC:
This is also an SSS triangle. In this triangle we know the three sides x = 5.1, y = 7.9 and z = 3.5. Use The Law of Cosines to find angle X first: cos X = (y2 + z2 − x2)/2yz cos X = ((7.9)2 + (3.5)2 − (5.1)2)/(2×7.9×3.5) cos X = (62.41 + 12.25 − 26.01)/55.3 cos X = 48.65/55.3 = 0.8797... X = cos−1(0.8797...) X = 28.3881...° X = 28.4° to one decimal place Next we will use The Law of Cosines again to find angle Y: cos Y = (z2 + x2 − y2)/2zx cos Y = −24.15/35.7 = −0.6764... cos Y = (12.25 + 26.01 − 62.41)/35.7 cos Y = −24.15/35.7 = −0.6764... Y = cos−1(−0.6764...) Y = 132.5684...° Y = 132.6° to one decimal place. Finally, we can find angle Z by using 'angles of a triangle add to 180°': Z = 180° − 28.3881...° − 132.5684...° Z = 19.0° to one decimal place Another MethodLargest Angle?Why do we try to find the largest angle first? That way the other two angles must be acute (less than 90°) and the Law of Sines will give correct answers.
The Law of Sines is difficult to use with angles above 90°. There can be two answers either side of 90° (example: 95° and 85°), but a calculator will only give you the smaller one. So by calculating the largest angle first using the Law of Cosines, the other angles are less than 90° and the Law of Sines can be used on either of them without difficulty.
B is the largest angle, so find B first using the Law of Cosines: cos B = (a2 + c2 − b2) / 2ac cos B = (11.62 + 7.42 − 15.22) / (2×11.6×7.4) cos B = (134.56 + 54.76 − 231.04) / 171.68 cos B = −41.72 / 171.68 cos B = −0.2430... B = 104.1° to one decimal place Use the Law of Sines, sinC/c = sinB/b, to find angle A: sin C / 7.4 = sin 104.1° / 15.2 sin C = 7.4 × sin 104.1° / 15.2 sin C = 0.4722... C = 28.2° to one decimal place Find angle A using "angles of a triangle add to 180": A = 180° − (104.1° + 28.2°) A = 180° − 132.3° A = 47.7° to one decimal place So A = 47.7°, B = 104.1°, and C = 28.2° Copyright © 2017 MathsIsFun.com
Just like regular numbers, angles can be added to obtain a sum, perhaps for the purpose of determining the measure of an unknown angle. Sometimes we can determine a missing angle because we know that the sum must be a certain value. Remember -- the sum of the degree measures of angles in any triangle equals 180 degrees. Below is a picture of triangle ABC, where angle A = 60 degrees, angle B = 50 degrees and angle C = 70 degrees. If we add all three angles in any triangle we get 180 degrees. So, the measure of angle A + angle B + angle C = 180 degrees. This is true for any triangle in the world of geometry. We can use this idea to find the measure of angle(s) where the degree measure is missing or not given. Finding a Missing AngleIn triangle ABC below, angle A = 40 degrees and angle B = 60 degrees. What is the measure of angle C? We know that the sum of the measures of any triangle is 180 degrees. Using the fact that angle A + angle B + angle C = 180 degrees, we can find the measure of angle C.
angle A = 40 angle B = 60 angle C = we don't know. To find angle C, we simply plug into the formula above and solve for C.
A + B + C = 180 C = 180 - A - B C = 180 - 40 - 60 C = 80 To check if 80 degrees is correct, let's add all three angle measures. If we get 180 degrees, then our answer for angle C is right. Here we go:
40 + 60 + 80 = 180 You don't always have to plug in those values to the equation and solve. Once you're comfortable with this sort of problem you'll be able to say "okay, 40 + 60 =100, so the other angle has to be 80!" and it's much quicker. Equilateral TrianglesIf a triangle is equilateral, what is the degree measure of each of its angles? Remember, all sides of an equilateral triangle have equal measure. They also, as you'll learn, have equal angles! Let x = the degree measure of each angle. Triangles have three angles and so we will add x THREE times. We have this:
x + x + x = 180 3x = 180 x = 60 Makes sense, right? If all the angles are equal, and they add up to 180, then it has to be 60 degrees! A Ratio of Angles The degree measures of the angles of a triangle are in the ratio 4 : 5 : 9. Notice that the smallest angle is represented by the smallest number in the ratio given. The smallest number given is 4, right? Since this is a ratio, we have to multiply all those values (4,5,9) by some common factor to get the actual angles. (For example, 60 and 80 are in a 3:4 ratio with a factor of 20) Let 4x = the measure of the smallest angle of the triangle. We can now say that 5x and 9x = the degree measures of the remaining angles of the triangle. We simply add 4x + 5x + 9x, equated the sum to 180 degrees and solve for x. After finding x, we plug the value of x into 4x, and simplify to find the measure of the smallest angle of the triangle at hand.
4x + 5x + 9x = 180 9x + 9x = 180 18x = 180 x = 180/18 x = 10 We found the value of x but it does NOT mean we are done. The answer is 40 degrees. Remember, the sum of the angles of a triangle is 180 degrees. Just take what you are given in a problem and try to determine what will make the final angle add up to 180 degrees. A lesson provided by Mr. Feliz Try the "Triangle Calculator" below: |
How to calculate the smallest angle of a triangle
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