Question

Show that each statement is false by providing a counterexample.

(a) If the length of AC is 46 and point B lies on AC, then AB = 34 and BC = 12.
Counterexample: AB = 35, BC = 11

(b) If the perimeter of a rectangle is 32, then the length is 8 and the width is 8.
Counterexample: length = 12, width = 4

(c) If 1 and 2 are complementary angles, then one of them must have a measure less than 45.
Counterexample: m₁ = 50, m₂ = 40

(d) If the measures of R, S, and T sum to 180, then one of the angles must be obtuse.
Counterexample: m_R = 90, m_S = 45, m_T = 45

Answer 1

In part (a), AB = 35 and BC = 11 is a counterexample. In part (b), length = 12 and width = 4 is a counterexample. In part (c), m₁ = 50 and m₂ = 40 is a counterexample. In part (d), m_R = 90, m_S = 45, and m_T = 45 is a counterexample.

Step-by-step explanation:

In part (a), the statement is false. A counterexample is AB = 35 and BC = 11, which satisfies the condition that the length of AC is 46 and point B lies on AC, but AB and BC do not equal 34 and 12, respectively.

In part (b), the statement is also false. A counterexample is length = 12 and width = 4, which satisfies the condition that the perimeter of a rectangle is 32, but the length and width are not equal to 8.

In part (c), the statement is false. A counterexample is m₁ = 50 and m₂ = 40, which satisfies the condition that 1 and 2 are complementary angles, but both angles have a measure greater than 45.

In part (d), the statement is false. A counterexample is m_R = 90, m_S = 45, and m_T = 45, which satisfies the condition that the measures of R, S, and T sum to 180, but none of the angles are obtuse.