Question

show that each statement is false by providing a counterexample.
(a) if the length of \\( \overline { x z } \\) is 40 and point \\( y \\) lies on \\( \overline { x z } \\), then \\( x y = 26 \\) and \\( y z = 14 \\).
counterexample: \\( x y = \square, y z = \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:

Step1: Analyze part (a)

If \(XY = 26\) and \(YZ=14\), then \(XY + YZ=26 + 14=40\). But if \(Y\) is not between \(X\) and \(Z\) (for example, if the order is \(X,Z,Y\)), the length of \(\overline{XZ}\) is not \(XY + YZ\). Let \(XY = 10\), \(YZ = 10\) and \(XZ=10\) (where \(Y\) is not between \(X\) and \(Z\))

Step2: Analyze part (b)

Complementary angles sum to \(90^{\circ}\). Let \(m\angle1 = 30^{\circ}\) and \(m\angle2=60^{\circ}\), both are less than \(45^{\circ}\)

Step3: Analyze part (c)

Let \(\angle P = 90^{\circ}\), \(\angle Q= 60^{\circ}\), \(\angle R = 30^{\circ}\). \(\angle P+\angle Q+\angle R=90 + 60+30=180^{\circ}\), but \(\angle P\) is not acute

Step4: Analyze part (d)

The perimeter of a rectangle is \(P = 2(l + w)\). If \(l = 4\) and \(w = 2\), then \(P=2(4 + 2)=12\) (not \(l = 3\) and \(w = 3\) as \(P=2(3 + 3)=12\) is a square, but a square is a special rectangle)

Answer:

(a) \(XY = 10\), \(YZ = 10\), \(XZ = 10\) (b) \(m\angle1=30^{\circ}\), \(m\angle2 = 60^{\circ}\) (c) \(m\angle P=90^{\circ}\), \(m\angle Q = 60^{\circ}\), \(m\angle R=30^{\circ}\) (d) length \(=4\), width \(=2\)