Questions: Problem Set C 18 Given: triangle ABC ~ triangle DEF, m angle A=50, m angle D=2x+5y m angle F=5x+y, m angle B=102-x Find: m angle F

Problem Set C 18 Given: triangle ABC ~ triangle DEF, m angle A=50, m angle D=2x+5y m angle F=5x+y, m angle B=102-x Find: m angle F
Transcript text: Problem Set C 18 Given: $\triangle \mathrm{ABC} \sim \triangle \mathrm{DEF}$, \[ \begin{array}{l} \mathrm{m} \angle \mathrm{~A}=50, \mathrm{~m} \angle \mathrm{D}=2 \mathrm{x}+5 \mathrm{y} \\ \mathrm{~m} \angle \mathrm{~F}=5 \mathrm{x}+\mathrm{y}, \mathrm{~m} \angle \mathrm{~B}=102-\mathrm{x} \end{array} \] Find: $m \angle F$
failed

Solution

failed
failed

Solution Steps

Step 1: Establish relationships between angles

Since ΔABC ~ ΔDEF, corresponding angles are congruent. Thus, m∠A = m∠D, m∠B = m∠E, and m∠C = m∠F.

Step 2: Set up equations

We are given m∠A = 50, m∠D = 2x + 5y, m∠F = 5x + y, and m∠B = 102 - x. Since m∠A = m∠D, we have 50 = 2x + 5y. The sum of angles in a triangle is 180 degrees. In ΔABC, m∠A + m∠B + m∠C = 180, so 50 + (102 - x) + m∠C = 180, which simplifies to m∠C = 28 + x. Similarly, in ΔDEF, m∠D + m∠E + m∠F = 180. Since m∠B = m∠E, we have (2x + 5y) + (102 - x) + (5x + y) = 180, which simplifies to 6x + 6y + 102 = 180, or x + y = 13.

Step 3: Solve for x and y and find m∠F

We have the system of equations: 2x + 5y = 50 x + y = 13 Multiplying the second equation by 2 gives 2x + 2y = 26. Subtracting this from the first equation gives 3y = 24, so y = 8. Substituting y = 8 into x + y = 13 gives x + 8 = 13, so x = 5. Now we can find m∠F = 5x + y = 5(5) + 8 = 25 + 8 = 33.

Final Answer

m∠F = 33°

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
Suppose you make a deposit of P into a savings account that earns interest at a rate of 100 r % per year. a. Show that if interest is compounded once per year, then the balance after t years is B(t)=P(1+r)^(1/2). b. If interest is compounded m times per year, then the balance after t years is B(T)=P(1+r/m)^(mt) In the limit m -> infinity, the compounding is said to be continuous. Show that with continuous compounding, the balance after t years is B(t)=Pe. a. Begin by stating the compound interest formula. B(t)=P(1+r/n)^(nt), where P is the principal, r is the annual rate of interest as a decimal, t is the number of years the amount is deposited for, and n is the number of times the interest is compounded per year.
failedAnswered
Employees often have very high job expectations when they start a job, which are sometimes not met. These unmet expectations can lead to turnover. One way to lower these expectations is to: Make sure everyone is treated fairly Use a realistic job preview Tailor the job to employee's interests Pay employees more
failedAnswered
If total liabilities increased by 9000, then - assets and owner's equity each increased by 4500. - assets must have decreased by 9000. - owner's equity must have increased by 9000. - assets must have increased by 9000, or owner's equity must have decreased by 9000.
failedAnswered
A manufacturer of chocolate chips would like to know if its bag filling machine works correctly at the 438 gram setting. It is believed the machine is under filling the bags, thus cheating customers. Thirty seven bags were sampled and found to have a mean of 435 grams. Assume the population standard deviation is 14. Using a 0.05 significance level, determine if the bag filling machine is under filling the bags (that is, less than 438 grams).
failedAnswered
At time t=0, a tank contains 25 oz of salt dissolved in 50 gallons of water. Then brine containing 40 z of salt per gallon of brine is allowed to enter the tank at a rate of 5 gal / min and the mixed solution is drained from the tank at the same rate. (a) How much salt is in the tank at an arbitrary time? oz. (b) How much salt is in the tank at time 15 min ? oz.