Question

directions: triangles abc and dfg are given, and bc ≅ fg.
a. without doing any calculations, is there enough information to determine whether △abc ≅ △dfg? explain your reasoning. select an answer
b. it is possible to calculate the measure of ∠fdg? show your work in how to calculate this angle. ∠fdg =
c. now, do you have enough information to determine that △abc is congruent to △dfg? explain your reasoning. ?, △abc ≅ △ select an answer by select an answer
you have 2 attempts to correctly answer this problem.

Explanation:

Step1: Check congruence for part a

For two triangles to be congruent, we need enough corresponding parts (sides and angles) to match. Here, we know \(BC\cong FG\), \(\angle ABC = 90^{\circ}\), \(\angle DFG=90^{\circ}\), but we don't know about other sides or angles. Just one side and no information about other corresponding angles or sides (except the right - angles) is not enough.

Step2: Calculate \(\angle FDG\) for part b

The sum of angles in a triangle is \(180^{\circ}\). In \(\triangle DFG\), we know \(\angle DFG = 90^{\circ}\) and \(\angle DGF=53^{\circ}\).
Using the angle - sum formula for a triangle \(\angle FDG+\angle DFG+\angle DGF = 180^{\circ}\).
Substitute the known values: \(\angle FDG+90^{\circ}+53^{\circ}=180^{\circ}\).
Then \(\angle FDG=180^{\circ}-(90^{\circ} + 53^{\circ})=180^{\circ}-143^{\circ}=37^{\circ}\).

Step3: Check congruence for part c

Now we know \(BC\cong FG\), \(\angle ABC=\angle DFG = 90^{\circ}\), \(\angle BAC=\angle FDG = 37^{\circ}\).
By the Angle - Side - Angle (ASA) congruence criterion. In \(\triangle ABC\) and \(\triangle DFG\), \(\angle BAC=\angle FDG\), \(BC = FG\), \(\angle ABC=\angle DFG\).

Answer:

a. No. We only know one pair of congruent sides (\(BC\cong FG\)) and one pair of congruent angles (\(\angle ABC\cong\angle DFG = 90^{\circ}\)), but not enough to satisfy a congruence criterion (like ASA, SAS, SSS, AAS).
b. \(37\)
c. Yes, \(\triangle ABC\cong\triangle DFG\) by ASA (Angle - Side - Angle).