Show that each statement is false by providing a counterexample. (a) If the length of \( \overline{A C} \) is 26 and point \( B \) lies on \( \overline{A C} \), then \( A B=16 \) and \( B C=10 \). Counterexample: \( A B=\square, B C=\square^{\circ} \) (b) If the perimeter of a rectangle is 32 , then the length is 8 and the width is 8. Counterexample: length \( =\square \), width \( =\square \) (c) If \( \angle 1 \) and \( \angle 2 \) are supplementary angles, then one of them must be acute. (d) If the measures of \( \angle R, \angle S \), and \( \angle T \) sum to \( 180^{\circ} \), then one of the angles must be obtuse. Counterexample: \( m \angle R=\square^{\circ}, m \angle S=\square^{\circ}, m \angle T=\square^{\circ} \)

Let's provide counterexamples for each statement to show that they are false. (a) **Statement**: If the length of \( \overline{A C} \) is 26 and point \( B \) lies on \( \overline{A C} \), then \( A B=16 \) and \( B C=10 \). **Counterexample**: Let \( A B = 20 \) and \( B C = 6 \). Then, \( A C = A B + B C = 20 + 6 = 26 \). This shows that \( A B \) and \( B C \) can have different values while still satisfying the condition that \( A C = 26 \). (b) **Statement**: If the perimeter of a rectangle is 32, then the length is 8 and the width is 8. **Counterexample**: Let the length be \( 10 \) and the width be \( 6 \). The perimeter is calculated as \( P = 2(\text{length} + \text{width}) = 2(10 + 6) = 2 \times 16 = 32 \). This shows that the rectangle can have different dimensions while still having a perimeter of 32. (c) **Statement**: If \( \angle 1 \) and \( \angle 2 \) are supplementary angles, then one of them must be acute. **Counterexample**: Let \( \angle 1 = 120^\circ \) and \( \angle 2 = 60^\circ \). Both angles are not acute, as \( \angle 1 \) is obtuse. This shows that it is possible for both angles to be non-acute while still being supplementary (their sum is \( 180^\circ \)). (d) **Statement**: If the measures of \( \angle R, \angle S \), and \( \angle T \) sum to \( 180^{\circ} \), then one of the angles must be obtuse. **Counterexample**: Let \( m \angle R = 60^\circ \), \( m \angle S = 60^\circ \), and \( m \angle T = 60^\circ \). The sum is \( 60 + 60 + 60 = 180^\circ \), and all angles are acute. This shows that it is possible for all angles to be acute while still summing to \( 180^\circ \).