Final answer:
To determine which statement about the angles of triangle RST is true, we can use the Triangle Sum Theorem and the Law of Cosines to calculate the measures of the angles. Based on the calculations, none of the angles in triangle RST are congruent.
Step-by-step explanation:
To determine which statement about the angles of triangle RST must be true, we can use the Triangle Sum Theorem, which states that the sum of the measures of the angles in a triangle is always 180 degrees.
Let's calculate the measure of angle R. Since RS = 20 and TR = 19, the longest side is RS. Therefore, angle R is the largest angle in the triangle. Using the Law of Cosines, we can find the measure of angle R:
RS^2 = ST^2 + TR^2 - 2(ST)(TR) cos(R)
20^2 = 3^2 + 19^2 - 2(3)(19) cos(R)
Solving for cos(R), we find that cos(R) = -2/57. Taking the inverse cosine of -2/57, we find that angle R is approximately 97.25 degrees. Therefore, statement a) ∠R is acute is false.
Similarly, we can calculate the measures of angles S and T. Angle S can be found using the Law of Cosines:
ST^2 = TR^2 + RS^2 - 2(TR)(RS) cos(S)
3^2 = 19^2 + 20^2 - 2(19)(20) cos(S)
Solving for cos(S), we find that cos(S) = 163/760. Taking the inverse cosine of 163/760, we find that angle S is approximately 66.25 degrees. Therefore, statement b) ∠S is obtuse is false.
Lastly, we can calculate the measure of angle T by subtracting the measures of angles R and S from 180 degrees:
Angle T = 180 - Angle R - Angle S
Substituting the values we found, we have:
Angle T = 180 - 97.25 - 66.25
Angle T is approximately equal to 16.5 degrees. Therefore, statement c) ∠T is a right angle is false.
Since none of the statements a), b), or c) are true, the correct answer is d) None of the angles is congruent.