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Are the triangles congruent? Why or why not?

An image displaying a triangle DEF of lengths of sides DE equals 3 units, EF equals 5 units and DF equals 4 units. Angle D equals 90 degree, angle E equals 53 degree and angle F equals 37 degree. An image displaying a triangle ABC of lengths of sides AB equals 4 units, BC equals 5 units and AC equals 3 units. Angle A equals 90 degree, angle B equals 53 degree and angle C equals 37 degree.
A.
Yes. The corresponding sides and angles have the same measures.
B.
No. The triangles are not oriented the same way.
C.
No. The angles have the same measures but the sides do not.
D.
Yes. Both are right triangles.

1 answer

To determine if the triangles DEF and ABC are congruent, we can analyze both triangles based on their sides and angles.

Given:

  • Triangle DEF has sides DE = 3, EF = 5, DF = 4 units, and angles D = 90°, E = 53°, and F = 37°.
  • Triangle ABC has sides AB = 4, BC = 5, AC = 3 units, and angles A = 90°, B = 53°, and C = 37°.

To check for congruence, we can use the Side-Angle-Side (SAS) or Angle-Angle-Side (AAS) criteria, as both triangles are right triangles:

  1. Right Angles: Both triangles have a right angle (90°).

  2. Corresponding Angles: Both triangles have a 53° angle and a 37° angle. Since all angles in a triangle add up to 180°, if two angles are the same in both triangles, the third angle must also be the same, which it is (90°).

  3. Corresponding Sides: When comparing the sides:

    • In triangle DEF, sides are 3, 4, and 5 units.
    • In triangle ABC, sides are also 3, 4, and 5 units.

However, upon close examination, the lengths of the corresponding sides do not match:

  • In triangle DEF, the longest side (hypotenuse) is 5 units (EF).
  • In triangle ABC, the longest side (hypotenuse) is also 5 units (BC).

But the important point raised is the order of the sides when compared directly:

  • DE = 3 in DEF corresponds to AC = 3 in ABC,
  • DF = 4 in DEF corresponds to AB = 4 in ABC,
  • EF = 5 in DEF (hypotenuse) corresponds to BC = 5 in ABC.

This correspondence indicates that the two triangles have congruent sides and angles.

Thus, the correct answer is:

A. Yes. The corresponding sides and angles have the same measures.