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Four students were asked to label angles in pairs of triangles. Their drawings are shown below.

Tatyanna
Triangle A B C. Angle B is 54.3 degrees and angle C is 61.2 degrees. Triangle X Y Z. Angle Z is 64.5 degrees and angle Y is 54.3 degrees.[Not drawn to scale]

Simon
Triangle A B C. Angle A is 63.2 degrees and angle B is 58.4 degrees. Triangle X Y Z. Angle Z is 63.2 degrees and angle Y is 58.4 degrees.[Not drawn to scale]

Anders
Triangle A B C. Angle A is 59.8 degrees and angle B is 60.6 degrees. Triangle X Y Z. Angle X is 59.8 degrees and angle Z is 60.6 degrees.[Not drawn to scale]

Kai
Triangle A B C. Angle B is 60.9 degrees and angle C is 57.7 degrees. Triangle X Y Z. Angle X is 61.4 degrees and angle Z is 57.7 degrees.
[Not drawn to scale]

Which student labeled the triangles such that Triangle A B C is similar to X Y Z?
Tatyanna
Simon
Anders
Kai

1 answer

To determine which student's triangles are similar, we need to check if the angles of Triangle ABC and Triangle XYZ are equal.

Recall that triangles are similar if their corresponding angles are equal.

  1. Tatyanna

    • Triangle ABC: Angle B = 54.3°, Angle C = 61.2°
      • Angle A can be found: \[ \text{Angle A} = 180° - 54.3° - 61.2° = 180° - 115.5° = 64.5° \]
    • Triangle XYZ: Angle Y = 54.3°, Angle Z = 64.5°
      • Angle X can be found: \[ \text{Angle X} = 180° - 54.3° - 64.5° = 180° - 118.8° = 61.2° \]
    • Angles are:
      • Triangle ABC: 54.3°, 61.2°, 64.5°
      • Triangle XYZ: 54.3°, 61.2°, 64.5°
    • They are equal. Tatyanna's triangles are similar.

  2. Simon

    • Triangle ABC: Angle A = 63.2°, Angle B = 58.4°
      • Angle C can be found: \[ \text{Angle C} = 180° - 63.2° - 58.4° = 180° - 121.6° = 58.4° \]
    • Triangle XYZ: Angle Z = 63.2°, Angle Y = 58.4°
      • Angle X can be found: \[ \text{Angle X} = 180° - 63.2° - 58.4° = 58.4° \]
    • Angles are:
      • Triangle ABC: 63.2°, 58.4°, 58.4°
      • Triangle XYZ: 63.2°, 58.4°, 58.4°
    • They are equal. Simon's triangles are similar.

  3. Anders

    • Triangle ABC: Angle A = 59.8°, Angle B = 60.6°
      • Angle C can be found: \[ \text{Angle C} = 180° - 59.8° - 60.6° = 180° - 120.4° = 59.6° \]
    • Triangle XYZ: Angle X = 59.8°, Angle Z = 60.6°
      • Angle Y can be found: \[ \text{Angle Y} = 180° - 59.8° - 60.6° = 180° - 120.4° = 59.6° \]
    • Angles are:
      • Triangle ABC: 59.8°, 60.6°, 59.6°
      • Triangle XYZ: 59.8°, 60.6°, 59.6°
    • They are equal. Anders's triangles are similar.

  4. Kai

    • Triangle ABC: Angle B = 60.9°, Angle C = 57.7°
      • Angle A can be found: \[ \text{Angle A} = 180° - 60.9° - 57.7° = 180° - 118.6° = 61.4° \]
    • Triangle XYZ: Angle X = 61.4°, Angle Z = 57.7°
      • Angle Y can be found: \[ \text{Angle Y} = 180° - 61.4° - 57.7° = 180° - 119.1° = 60.9° \]
    • Angles are:
      • Triangle ABC: 60.9°, 57.7°, 61.4°
      • Triangle XYZ: 61.4°, 57.7°, 60.9°
    • They are not equal. Kai's triangles are not similar.

In conclusion, the students whose triangles are similar are Tatyanna and Simon and Anders. However, since the question is asking for which single student demonstrated similarity, the example that fits possibly best in terms of angle matching and order is Tatyanna.