Triangle ABC is a right triangle. If the measure of angle PAB is x degrees and the measure of angle ACB is expressed in the form Mx+N with M=1, what is the value of M+N?
+2440 There is a theorem that would be very helpful to know here. It is called the triangle exterior angle theorem. It states that the exterior angle is equal to the measure of the two remote (angle opposite the exterior) angles. This definition may be hard to understand, but a diagram would probably be useful. It may not be the neatest drawing, but you should understand the concept.
Now, let's use this to our advantage. Because of the triangle exterior angle theorem, we know that \(m\angle PAB=m\angle B+m\angle C\). Now that we have an equation that relates all the angles together, we can do the process of solving!
| \(m\angle PAB=m\angle B+m\angle C\) | We have already derived this equation. Now, substitute the values given for each angle. |
| \(x=90+mx+n\) | Now, substitute 1 for m, as given in the original description of the equation. |
| \(x=90+x+n\) | Subtract x from both sides. |
| \(0=90+n\) | Subtract 90 from both sides. |
| \(n=-90\) | |
Now, of course, we know that m=1 and n=-90. If you add those 2 values together, you get that \(m+n=1+(-90)=-89\)
+2440 There is a theorem that would be very helpful to know here. It is called the triangle exterior angle theorem. It states that the exterior angle is equal to the measure of the two remote (angle opposite the exterior) angles. This definition may be hard to understand, but a diagram would probably be useful. It may not be the neatest drawing, but you should understand the concept.
Now, let's use this to our advantage. Because of the triangle exterior angle theorem, we know that \(m\angle PAB=m\angle B+m\angle C\). Now that we have an equation that relates all the angles together, we can do the process of solving!
| \(m\angle PAB=m\angle B+m\angle C\) | We have already derived this equation. Now, substitute the values given for each angle. |
| \(x=90+mx+n\) | Now, substitute 1 for m, as given in the original description of the equation. |
| \(x=90+x+n\) | Subtract x from both sides. |
| \(0=90+n\) | Subtract 90 from both sides. |
| \(n=-90\) | |
Now, of course, we know that m=1 and n=-90. If you add those 2 values together, you get that \(m+n=1+(-90)=-89\)
