Question

show that each statement is false by providing a counterexample.
(a) if the length of (xz) is 46 and point (y) lies on (xz), then (xy = 14) and (yz = 12).
counterexample: (xy=square), (yz=square)
(b) if (angle1) and (angle2) are complementary angles, then one of them must have a measure greater than (45^{circ}).
counterexample: (mangle1=square^{circ}), (mangle2=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: (mangle p=square^{circ}), (mangle q=square^{circ}), (mangle r=square^{circ})
(d) if the perimeter of a rectangle is 28, then the length is 7 and the width is 7.
counterexample: length=square, width=square

Explanation:

Step1: Analyze part (a)

If \(XZ = 46\), we can choose non - given values for \(XY\) and \(YZ\) such that \(XY+YZ = 46\). Let \(XY = 20\) and \(YZ=26\).

Step2: Analyze part (b)

Complementary angles add up to \(90^{\circ}\). We can have two angles both equal to \(45^{\circ}\), so \(m\angle1 = 45^{\circ}\) and \(m\angle2 = 45^{\circ}\).

Step3: Analyze part (c)

Acute angles are less than \(90^{\circ}\). If \(m\angle P=90^{\circ}\), \(m\angle Q = 40^{\circ}\), \(m\angle R = 50^{\circ}\), their sum is \(180^{\circ}\) but \(\angle P\) is not acute.

Step4: Analyze part (d)

The perimeter of a rectangle \(P = 2(l + w)\). If \(P = 28\), then \(l + w=14\). We can choose \(l = 10\) and \(w = 4\).

Answer:

(a) \(XY = 20\), \(YZ = 26\)
(b) \(m\angle1 = 45\), \(m\angle2 = 45\)
(c) \(m\angle P = 90\), \(m\angle Q = 40\), \(m\angle R = 50\)
(d) length \(=10\), width \(=4\)