solving for 2 variables

[joedawnl]

Boy do I feel silly. I see "how" to solve this, but never reach the correct answer (yes, I have the answer, but can't work backwards to see how it was done).

There are 2 similar triangles (so we know the corresponding angles are congruent). Trainge ABC is similar to triangle DEF.

Angle A = 50, its corresponding angle D = 2x+5y
Angle B = 102-x
Angle F = 5x+y
Find: the measure of angle F

I can see that we can put 3 angle measurements together since the sum would be 180 degrees and try to solve for one of the variables. Here's how I started:

180 - (2x+5y + 5x+y) = 102-x

I've gone 'round and 'round, getting x = y+13 which doesn't help. I haven't been able to isolate/solve for a single variable. Any thoughts? (btw, I'm told the solution is: x=5, y=8).

Many thanks!! :?

 

[pka]

Here is what I get in triangleABC: \(\displaystyle \L
\angle A = 50,\;\angle B = 102 - x\;\& \;\angle C = 28 + x\)

In triangleDEF: \(\displaystyle \L
\angle D = 2x + 5y,\;\angle E = 180 - 7x - 6y\;\& \;\angle F = 5x + y\).

In proportions: \(\displaystyle \L
\frac{{2x + 5y}}{{50}} = \frac{{180 - 7x - 6y}}{{102 - x}}\quad \& \quad \frac{{2x + 5y}}{{50}} = \frac{{5x + y}}{{x + 28}}\).

Now solve that system and get x=5 & y=8.

 

[Mrspi]

Boy do I feel silly. I see "how" to solve this, but never reach the correct answer (yes, I have the answer, but can't work backwards to see how it was done).

There are 2 similar triangles (so we know the corresponding angles are congruent). Trainge ABC is similar to triangle DEF.

Angle A = 50, its corresponding angle D = 2x+5y
Angle B = 102-x
Angle F = 5x+y
Find: the measure of angle F

I can see that we can put 3 angle measurements together since the sum would be 180 degrees and try to solve for one of the variables. Here's how I started:

180 - (2x+5y + 5x+y) = 102-x

I've gone 'round and 'round, getting x = y+13 which doesn't help. I haven't been able to isolate/solve for a single variable. Any thoughts? (btw, I'm told the solution is: x=5, y=8).

Many thanks!! :?

Here's how I would do it:

Angles A and D are corresponding angles of similar triangles, so their measures must be the same. Thus,
50 = 2x + 5y<------there is one equation in two variables

Angles C and F are corresponding angles as well; since we know that m<F = 5x + y, m<C = 5x + y.

In triangle ABC,
m<A + m<B + m<C = 180
50 + (102 - x) + (5x + y) = 180
152 + 4x + y = 180
4x + y = 28<------there is a second equation in two variables

Now, solve the system.....once you know the values of x and y, you can substitute them into the expression for m<F.

I hope this helps you.