Problem Solving and Solutions?

Question

The sum of the angles of any triangle is 180o. What are the three angles of a triangle if the sum of the first two angles is equal to the third angle and the difference of the first two angles is two-thirds of the third angle?

Answer 1

a + b = c

a - b = 2/3c

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2a = 5/3c

a = 5/6c

-a -b = -c

a - b = 2/3c

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2b = 1/3c

b = 1/6c

a + b + c = 180

5/6c + 1/6c + c = 180

2c = 180

c = 90

a = 5/6c = 5/6(90) = 75

b = 1/6c = 1/6(90) = 15

Answer 2

Let A, B, and C be the three angles. Then you have

A + B + C = 180

A + B = C

A - B = (2/3)C

Now, because A + B = C, you can substitute this into the first equation. Like this:

(A + B) + C = 180

(C) + C = 180

2C = 180

C = 90

Using the second two equations you have

A + B = 90

A - B = 60

Adding these two equations you have

2A = 150

A = 75

Using the first equation (or any other) , you have

A + B + C = 180

75 + B + 90 = 180

B = 180 - 90 - 75 = 15

So

A = 75, B=15, and C = 90

Answer 3

a = the first angle

b = the second angle

c = the third angle

a + b + c = 180° (solve by using substitution)

a + b = c

a - b = 2/3(c)

a - b = 2/3(c)

a - b = 2c/3

3(a - b) = 2c

2c = 3a - 3b

c = (3a - 3b)/2

a + b + c = 180°

a + b + (3a - 3b)/2 = 180°

2[a + b + (3a - 3b)/2] = 2[180°]

2a + 2b + 3a - 3b = 360°

2a + 3a + 2b - 3b = 360°

5a - b = 360°

a + b = c

a + b = (3a - 3b)/2

2[a + b] = 2(3a - 3b)/2

2a + 2b = 3a - 3b

2a - 3a + 2b + 3b = 0

-a + 5b = 0

a = 5b

5a - b = 360°

5(5b) - b = 360°

25b - b = 360°

24b = 360°

b = 360°/24

b = 15°

5a - b = 360°

5a - 15° = 360°

5a = 360° + 15°

5a = 375°

a = 375°/5

a = 75°

a + b + c = 180°

75° + 15° + c = 180°

c = 180° - 90°

c = 90°

∴ a = 75° , b = 15° , c = 90°

Answer 4

a + b + c = 180

a + b = c, 2c = 180 so c = 90

a - b = 60 and we know that a + b = 90

Add these two equations to get a = 75 and b = 15

Answer 5

let's say:

1st angle = x, 2nd angle = y, 3rd angle = z

x + y + z = 180

x + y = z

x - y = 2/3 z

=>

x + y + z = 180

x + y - z = 0

x - y - 2/3 z = 0

add the first 2 equations term by term:

x + y + z + x + y - z = 180 + 0 =>

2(x + y) = 180 => x + y = 90

having this and the first equation:

x + y + z = 180 =? 90 + z = 180 => z = 90

replace this value in the 3rd equation:

x - y = 2/3 * 90 => x - y = 60

take now this result and the one found above:

x + y = 90

x - y = 60

as before, sum the equations:

x + y + x - y = 90 + 60 =>

2x = 150 => x = 75

you have x + y = 90 => 75 + y = 90 => y = 90 - 75 => y = 15

so the 3 angles are:

x = 75

y = 15

z = 90