Question

show that each statement is false by providing a counterexample.
(a) if the length of \\( \overline{xz} \\) is 52 and point y lies on \\( \overline{xz} \\), then \\( xy = 40 \\) and \\( yz = 12 \\).
counterexample: \\( xy=\square, yz = \square \\)
(b) if \\( \angle 1 \\) and \\( \angle 2 \\) are complementary angles, then one of them must have a measure greater than \\( 45^{circ} \\).
counterexample: \\( m \angle 1=\square^{\circ}, m \angle 2=\square^{\circ} \\)
(c) if the measures of \\( \angle p, \angle q \\), and \\( \angle r \\) sum to \\( 180^{circ} \\), then all of the angles must be acute.
counterexample: \\( m \angle p=\square^{\circ}, m \angle q=\square^{\circ}, m \angle r=\square^{\circ} \\)
(d) if the perimeter of a rectangle is 12, then the length is 3 and the width is 3.
counterexample: length \\( =\square \\) width \\( =\square \\)

Explanation:

(a)

Step1: Recall the property of a line segment

If \(XZ = 52\) and \(Y\) lies on \(XZ\), then \(XY+YZ = XZ\). We need to find non - \(40\) and non - \(12\) values that sum to \(52\). Let \(XY = 30\) and \(YZ=22\) (since \(30 + 22=52\)).

(b)

Step1: Recall the definition of complementary angles

Complementary angles are two angles whose sum is \(90^{\circ}\). If \(m\angle1 = 45^{\circ}\) and \(m\angle2 = 45^{\circ}\), then \(\angle1+\angle2=45 + 45=90^{\circ}\), and neither angle is greater than \(45^{\circ}\).

(c)

Step1: Recall the definition of acute and non - acute angles

Acute angles have a measure less than \(90^{\circ}\). Let \(m\angle P=90^{\circ}\), \(m\angle Q = 45^{\circ}\), \(m\angle R = 45^{\circ}\). Then \(m\angle P+m\angle Q+m\angle R=90 + 45+45 = 180^{\circ}\), but \(\angle P\) is a right angle (not acute).

(d)

Step1: Recall the formula for the perimeter of a rectangle

The perimeter of a rectangle is \(P = 2(l + w)\). Given \(P = 12\), then \(l + w=\frac{12}{2}=6\). Let \(l = 4\) and \(w = 2\) (since \(2\times(4 + 2)=2\times6 = 12\)), and \(l
eq3\) and \(w
eq3\).

Answer:

(a) \(XY = 30\), \(YZ = 22\)
(b) \(m\angle1 = 45\), \(m\angle2 = 45\)
(c) \(m\angle P=90\), \(m\angle Q = 45\), \(m\angle R = 45\)
(d) length \(=4\), width \(=2\)