Question

show that each statement is false by providing a counterexample.
(a) if the length of \\( \overline{xz} \\) is 26 and point \\( y \\) lies on \\( \overline{xz} \\), then \\( xy = 16 \\) and \\( yz = 10 \\).
counterexample: \\( xy = \square \\), \\( yz = \square \\)
(b) if \\( \angle 1 \\) and \\( \angle 2 \\) are supplementary angles, then one of them must be obtuse.
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 48, then the length is 12 and the width is 12.
counterexample: length = \square, width = \square

Explanation:

Step1: Analyze part (a)

A counter - example for the statement "If the length of \(\overline{XZ}\) is 26 and point \(Y\) lies on \(\overline{XZ}\), then \(XY = 16\) and \(YZ=10\)" can be when \(XY = 15\) and \(YZ = 11\) (since \(15+11=26\)).

Step2: Analyze part (b)

A counter - example for the statement "If \(\angle1\) and \(\angle2\) are supplementary angles, then one of them must be obtuse" is when \(m\angle1 = 90^{\circ}\) and \(m\angle2=90^{\circ}\) (supplementary angles sum to \(180^{\circ}\), and right angles are not obtuse).

Step3: Analyze part (c)

A counter - example for the statement "If the measures of \(\angle P\), \(\angle Q\), and \(\angle R\) sum to \(180^{\circ}\), then all of the angles must be acute" is when \(m\angle P = 90^{\circ}\), \(m\angle Q = 60^{\circ}\), \(m\angle R = 30^{\circ}\) (the sum is \(180^{\circ}\), but \(\angle P\) is not acute).

Step4: Analyze part (d)

A counter - example for the statement "If the perimeter of a rectangle is 48, then the length is 12 and the width is 12" is when length \(l = 10\) and width \(w = 14\) (perimeter \(P=2(l + w)=2(10 + 14)=48\)).

Answer:

(a) \(XY = 15\), \(YZ = 11\)
(b) \(m\angle1 = 90^{\circ}\), \(m\angle2 = 90^{\circ}\)
(c) \(m\angle P = 90^{\circ}\), \(m\angle Q = 60^{\circ}\), \(m\angle R = 30^{\circ}\)
(d) length \(= 10\), width \(= 14\)