Question

1. which statement is true? (preserves – does not change; it stays the same)

a. rotations will preserve the figure’s side lengths but will not preserve the angle measurements
b. rotations will not preserve the figure’s side lengths and will not preserve the angle measurements
c. rotations will preserve the figure’s side lengths and will preserve the angle measurements
d. rotations will not preserve the figure’s side lengths but will preserve the angle measurements

1. which of the following must be true?

a. $m + t + x = 180$
b. $t + x + h = 180$
c. $m + a + t + x = 180$
d. $x + t + a = 180$
(image of a triangle with angles labeled x, t, a and points m, h)

Explanation:

Question 2

Rotation is a rigid transformation. Rigid transformations preserve both side lengths (congruence) and angle measurements (similarity in terms of angle preservation, and congruence for rigid motions). So rotations keep side lengths and angle measures the same. Option A says angles aren't preserved (wrong), B says both aren't preserved (wrong), D says side lengths aren't preserved (wrong). Only C is correct.

The figure is a triangle, so by the triangle angle - sum theorem, the sum of the interior angles of a triangle is \(180^{\circ}\). The interior angles of the triangle are \(X\), \(T\), and \(A\). So \(X + T+ A=180\). Option A includes \(M\) (an exterior - related angle), B includes \(H\) (an exterior angle), C includes \(M\) and has 4 angles, which is incorrect. Only D follows the triangle angle - sum theorem.

Answer:

C. Rotations will preserve the figure’s side lengths and will preserve the angle measurements

Question 3