Question

1. mathematical connections carolina says that when a figure is dilated using a scale factor of 2, the angle measures in the image are twice the angle measures in the preimage. how could you use the triangle angle sum theorem to explain why this cannot be true?

Explanation:

Step1: Define preimage angles

Let the angles of the preimage triangle be $\angle A$, $\angle B$, $\angle C$. By the Triangle Angle Sum Theorem:
$$\angle A + \angle B + \angle C = 180^\circ$$

Step2: Apply Carolina's claim

If each angle were doubled, the new angles would be $2\angle A$, $2\angle B$, $2\angle C$. Calculate their sum:
$$2\angle A + 2\angle B + 2\angle C = 2(\angle A + \angle B + \angle C)$$

Step3: Substitute preimage sum

Replace $\angle A + \angle B + \angle C$ with $180^\circ$:
$$2(180^\circ) = 360^\circ$$

Step4: Contradict angle sum rule

A triangle's interior angles must sum to $180^\circ$, but $360^\circ
eq 180^\circ$. This is impossible.

Answer:

If the angles were doubled, the sum of the image triangle's angles would be $360^\circ$, which violates the Triangle Angle Sum Theorem (triangle angles must sum to $180^\circ$). Thus, Carolina's claim cannot be true.