What transformations were made to triangle ABC to produce triangle XYZ? Assess whether the two triangles are congruent. How do you know? Show your work.

Answer 1

To determine the transformations made to triangle \( ABC \) to produce triangle \( XYZ \) and assess if the two triangles are congruent, we generally analyze the characteristics of both triangles such as side lengths, angles, and any transformations that might have occurred, such as translations, rotations, reflections, or dilations.

Step 1: Identify the triangles

We need specific information about the triangles \( ABC \) and \( XYZ \). This would include:

• The lengths of the sides \( AB, BC, CA \) for triangle \( ABC \) and \( XY, YZ, ZX \) for triangle \( XYZ \).
• The measurements of the angles \( \angle A, \angle B, \angle C \) for triangle \( ABC \) and \( \angle X, \angle Y, \angle Z \) for triangle \( XYZ \).

Step 2: Assessing congruence

Two triangles are congruent if one can be transformed into the other using rigid transformations (which do not change size or shape), and they must satisfy one of the congruence criteria:

1. Side-Side-Side (SSS): All three corresponding sides are equal.
2. Side-Angle-Side (SAS): Two sides and the included angle are equal.
3. Angle-Side-Angle (ASA): Two angles and the included side are equal.
4. Angle-Angle-Side (AAS): Two angles and a non-included side are equal.
5. Hypotenuse-Leg (HL): In right triangles, the hypotenuse and one leg are equal.

Step 3: Determine transformations

Once the dimensions of the triangles are known, we check for transformations:

• Translation: If the triangles have the same shape and size but are positioned differently.
• Rotation: If the triangles are the same size and shape but oriented differently.
• Reflection: If one triangle is flipped but retains the same size and shape.
• Dilation: If one triangle is a scaled version of the other, then they are not congruent.

Example Analysis

Let's say we have:

• \( AB = 5 \), \( BC = 7 \), \( CA = 9 \) and \( XY = 5 \), \( YZ = 7 \), \( ZX = 9 \)
• Angles \( \angle A = 60^\circ, \angle B = 50^\circ, \angle C = 70^\circ \) and \( \angle X = 60^\circ, \angle Y = 50^\circ, \angle Z = 70^\circ \).

Congruence Test:

1. Check sides:

   • \( AB = XY \)
   • \( BC = YZ \)
   • \( CA = ZX \)

   Therefore, all corresponding sides are equal (SSS criteria).

2. Check angles:

   • \( \angle A = \angle X \)
   • \( \angle B = \angle Y \)
   • \( \angle C = \angle Z \)

   All corresponding angles are equal (AAA criteria).

Conclusion

Since both the sides and angles of triangle \( ABC \) and triangle \( XYZ \) correspond with those of the criteria, we conclude that the triangles are congruent.

Final Answer: Triangle \( ABC \) and triangle \( XYZ \) are congruent through side-side-side (SSS) congruence.