Question

1. select all the statements that must be true for any scaled copy q of polygon p.

a. the side lengths are all whole numbers.
b. the angle measures are all whole numbers.
c. q has exactly 1 right angle.
d. if the scale factor between p and q is \\( \frac { 1 } { 5 } \\), then each side length of p is multiplied by \\( \frac { 1 } { 5 } \\) to get the corresponding side length of q.
e. if the scale factor is 2, each angle in p is multiplied by 2 to get the corresponding angle in q.
f. q has 2 acute angles and 3 obtuse angles.

Explanation:

Step1: Properties of scaled copies

Scaled copies (similar polygons) have equal corresponding angles and proportional corresponding side - lengths.

Step2: Analyze each option

• Option A:

The side lengths of a scaled copy are proportional. They don't have to be whole numbers. For example, if the original side length is \(3\) and the scale factor is \(\frac{1}{2}\), the new side length is \(1.5\). So, this statement is False.

• Option B:

The angle measures of a scaled copy are equal to the angle measures of the original polygon. They don't have to be whole numbers. For example, if the original angle is \(45.5^{\circ}\), the angle in the scaled copy is also \(45.5^{\circ}\). So, this statement is False.

• Option C:

Looking at polygon \(P\), there is a right - angle (\(90^{\circ}\)). In a scaled copy (similar polygon), corresponding angles are equal. So, if \(P\) has a right - angle, \(Q\) (the scaled copy) has exactly 1 right - angle. This statement is True.

• Option D:

If the scale factor from \(P\) to \(Q\) is \(\frac{1}{5}\), then to get the side length of \(Q\) from \(P\), we multiply the side length of \(P\) by \(\frac{1}{5}\), not \(\frac{1}{3}\). So, this statement is False.

• Option E:

If the scale factor is \(2\), to get the corresponding angle in \(Q\) (scaled copy), the angles of similar polygons are equal. We do not multiply the angles of \(P\) by \(2\). So, this statement is False.

• Option F:

In polygon \(P\), the acute angles are \(35^{\circ}\) and \(80^{\circ}\) (acute angles are angles less than \(90^{\circ}\)), and the obtuse angles are \(125^{\circ}\), \(135^{\circ}\), and \(250 - 180=70\) (wait, no. Let's calculate the sum of interior angles of polygon \(P\). The sum of interior angles of an \(n\) - sided polygon is \((n - 2)\times180^{\circ}\). Let's assume \(P\) is a hexagon (by counting the number of angles). \((6 - 2)\times180^{\circ}=720^{\circ}\). \(90 + 135+80 + 250+125+35=(90 + 135)+(80 + 125)+(250+35)=225+205 + 285=715\) (there may be a mis - count of sides, but using the angle - measure definition: acute (\(<90^{\circ}\)): \(35^{\circ},80^{\circ}\); obtuse (\(>90^{\circ}\)): \(125^{\circ},135^{\circ},250 - 180 = 70\) (no, wrong. Wait, the \(250^{\circ}\) is a reflex angle. But in a scaled copy (similar polygon), angle measures are preserved. So, the number of acute (\(35^{\circ},80^{\circ}\)) and obtuse (\(125^{\circ},135^{\circ}\)) angles is \(2\) acute and \(2\) obtuse (if we consider non - reflex angles. But if we consider the polygon in terms of non - overlapping angles formed by extending sides etc., the number of acute angles is \(2\) (\(35^{\circ}\) and \(80^{\circ}\)) and the number of obtuse angles is \(2\) (\(125^{\circ}\) and \(135^{\circ}\)). So, this statement is False.

Answer:

C. Q has exactly 1 right angle.