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

(2 points)

1 answer

To determine whether triangles ABC and AXY Z are congruent and what transformations were made to triangle AABC to produce triangle AXY Z, we need to analyze the segments and angles of the triangles.

Step 1: Identify the Transformations

  1. Translation: This could involve moving triangle AABC in a specific direction without altering its shape or size.
  2. Rotation: The triangle could be rotated around a point, changing its orientation while maintaining congruency.
  3. Reflection: The triangle could also be flipped over a line, which would yield a mirror image without changing size.
  4. Dilation: If the triangles are not the same size, a scale transformation would lead to a change in dimensions.

Assuming no dilation has occurred:

  • If triangle AXY Z is merely a translation, rotation, or reflection of triangle AABC, then they can be congruent.

Step 2: Assess Congruence

To assess congruence, we can utilize criteria such as:

  • Side-Side-Side (SSS) Congruence: If all three sides of one triangle are equal to the corresponding sides of another triangle, they are congruent.
  • Side-Angle-Side (SAS) Congruence: If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, they are congruent.
  • Angle-Side-Angle (ASA) Congruence: If two angles and the included side of one triangle are equal to the corresponding measures of another triangle, they are congruent.
  • Angle-Angle-Side (AAS) Congruence: If two angles and a non-included side of one triangle are equal to the corresponding measures of another triangle, they are congruent.
  • Hypotenuse-Leg (HL) Congruence: For right triangles, if the hypotenuse and a leg of one triangle are equal to the hypotenuse and a leg of another triangle, they are congruent.

Example Assessment

Assuming we have the lengths of sides and angles, we can compare.

  • Let’s say:
    • Side AB of triangle AABC = 5 units and side AX of AXY Z = 5 units
    • Side AC of triangle AABC = 4 units and side AY of AXY Z = 4 units
    • Angle ∠A in both triangles is 60°

With these values:

  • AB = AX
  • AC = AY
  • ∠A = ∠A

Using the SAS criteria, since we have two sides and the included angle equal in both triangles, we can conclude that:

Conclusion

Triangles AABC and AXY Z are congruent by the SAS postulate. The transformations made were either a translation, rotation, or reflection. Therefore, we establish that AABC ≅ AXY Z.