# N triangle​ ABC, the size of angle B is 4 times the size of angle​ A, and the size of angle C is 7° less than 5 times the size of angle A?

Jan 16, 2018

$A = 18.7$, $B = 74.8$, and $C = 86.5$

#### Explanation:

Where A, B and C represent their respective angles:

We are given that:
$B = 4 A$
$C = 5 A - 7$

And since ABC make up a triangle, we know that:
$A + B + C = 180$

Since we have three variables and three equations, we can solve for A, B, and C. To solve for A:

Take the equations $B = 4 A$ and $C = 5 A - 7$, and plug B and C into the last equation:

$A + B + C = 180$
$A + 4 A + 5 A - 7 = 180$
$10 A = 187$
$A = \frac{187}{10} = 18.7$

Then we can plug this value into the first two equations to solve for B and C:

$B = 4 A$
$B = 4 \left(18.7\right) = 74.8$

$C = 5 A - 7$
$C = 5 \left(18.7\right) - 7 = 93.5 - 7 = 86.5$

So $A = 18.7$, $B = 74.8$, and $C = 86.5$

Jan 16, 2018

$A$ is $18.7$ degrees, $B$ is 74.8 degrees and $C$ is 86.5 degrees.

#### Explanation:

We'll use $A , B , C$ for the angles in degrees.

$B = 4 A$
$C = 5 A - 7$
$A + B + C = 180$

(Because a triangle's angles, in degrees, sum up to 180)

Now we can substitute the values for $B , C$ in terms of $A$ in the third equation to solve for $A$:

$A + 4 A + \left(5 A - 7\right) = 180$
$10 A - 7 = 180$
$10 A = 187$
$A = 18.7$

And now use this value for $A$ to solve the first two equations.

$B = 4 \cdot 18.7 = 74.8$
$C = 5 \cdot 18.7 - 7 = 86.5$

This problem teaches us two things: "Translating" words and expressions into equations (possibly a system of equations) and utilizing certain known facts that can help with the problem (here we had to notice that there was no way to find angle B with just the first two equations, but because of the third one, we now know, if not already, that we only need to ever find the measure of two angles in a triangle to know all of them)