Question

show that each statement is false by providing a counter - example.
(a) if the area of a rectangle is 16, then the length is 8 and the width is 2.
counterexample: length = □, width = □
(b) if ∠1 and ∠2 are complementary angles, then one of them must have a measure less than 45°.
counterexample: m∠1 = □°, m∠2 = □°
(c) if the length of \\( \overline{ac} \\) is 32 and point b lies on \\( \overline{ac} \\), then ab = 16 and bc = 16.
counterexample: ab = □, bc = □
(d) if the measures of ∠r, ∠s, and ∠t sum to 180°, then one of the angles must be obtuse.
counterexample: m∠r = □°, m∠s = □°, m∠t = □°

Explanation:

(a)

Step1: Recall the formula for the area of a rectangle

The area of a rectangle is \(A = length\times width\). We need to find non - \(8\) and non - \(2\) positive real numbers whose product is \(16\).

Step2: Choose values

Let \(length = 4\) and \(width = 4\). Then \(A=4\times4 = 16\)

(b)

Step1: Recall the definition of complementary angles

Complementary angles are two angles whose sum is \(90^{\circ}\). We need to find two angles \(\angle1\) and \(\angle2\) such that \(\angle1+\angle2 = 90^{\circ}\) and neither is less than \(45^{\circ}\)

Step2: Choose angle measures

Let \(m\angle1=45^{\circ}\) and \(m\angle2 = 45^{\circ}\). Then \(m\angle1 + m\angle2=45^{\circ}+45^{\circ}=90^{\circ}\)

(c)

Step1: Recall the segment addition postulate

If \(B\) lies on \(\overline{AC}\), then \(AC=AB + BC\). We know \(AC = 32\). We need to find \(AB\) and \(BC\) such that \(AB+BC = 32\) and \(AB
eq16\) and \(BC
eq16\)

Step2: Choose segment lengths

Let \(AB = 10\) and \(BC=22\). Then \(AB + BC=10 + 22=32\)

(d)

Step1: Recall the definition of acute, right, and obtuse angles

An acute angle has a measure between \(0^{\circ}\) and \(90^{\circ}\), a right angle has a measure of \(90^{\circ}\), and an obtuse angle has a measure between \(90^{\circ}\) and \(180^{\circ}\). We need to find three angles \(\angle R\), \(\angle S\), \(\angle T\) such that \(\angle R+\angle S+\angle T = 180^{\circ}\) and none of them is acute

Step2: Choose angle measures

Let \(m\angle R = 60^{\circ}\), \(m\angle S=60^{\circ}\), \(m\angle T = 60^{\circ}\). Then \(m\angle R+m\angle S+m\angle T=60^{\circ}+60^{\circ}+60^{\circ}=180^{\circ}\)

Answer:

(a) length \( = 4\), width \( = 4\)
(b) \(m\angle1 = 45^{\circ}\), \(m\angle2=45^{\circ}\)
(c) \(AB = 10\), \(BC = 22\)
(d) \(m\angle R=60^{\circ}\), \(m\angle S = 60^{\circ}\), \(m\angle T=60^{\circ}\)