The angle measures can be ordered from least to greatest as follows: m∠BDC > m∠B > m∠C.
The side lengths can be ordered from least to greatest as follows: DF < EF < DE.
Part (a): Ordering the angle measures
To order the angle measures, we can use the fact that the sum of the angles in a triangle is 180 degrees. Therefore, we have:
m∠B + m∠C + m∠BDC = 180°
From the diagram, we can see that ∠BDC is an exterior angle of triangle ABC. An exterior angle of a triangle is equal to the sum of the two remote interior angles. In this case, the remote interior angles are ∠B and ∠C. Therefore, we have: m∠BDC = m∠B + m∠C
Substituting this into the first equation, we get:
m∠B + m∠C + (m∠B + m∠C) = 180°
2(m∠B + m∠C) = 180°
m∠B + m∠C = 90°
We are given that m∠F = 57°. Therefore, we can order the angle measures as follows:
m∠BDC > m∠B > m∠C
Part (b): Ordering the side lengths
To order the side lengths, we can use the Law of Sines. The Law of Sines states that the ratio of two sides of a triangle is equal to the ratio of the sine of their opposite angles. Therefore, we can write the following equations: DE/sin(m∠F) = EF/sin(m∠D) DE/sin(57°) = EF/sin(62°)
Since sin(57°) > sin(62°), we have DE > EF. We can also use the Law of Sines to compare DE and DF. We have:
DE/sin(m∠BDC) = DF/sin(m∠C)
From part (a), we know that m∠BDC > m∠C. Therefore, sin(m∠BDC) > sin(m∠C). Therefore, DE > DF.
Ordering from least to greatest, we get:
DF < EF < DE