Question

show that each statement is false by providing a counterexample.
(a) 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} \\)
(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 area of a rectangle is 36, then the length is 6 and the width is 6.
counterexample: length \\( =\square \\) width \\( =\square \\)
(d) if \\( m \angle a b d=50^{circ} \\), and point \\( c \\) is in the interior of \\( \angle a b d \\), then \\( m \angle a b c=30^{circ} \\) and \\( m \angle c b d=20^{circ} \\).
counterexample: \\( m \angle a b c=\square^{\circ}, m \angle c b d=\square^{\circ} \\)

Explanation:

Step1: Analyze part (a)

An acute angle is less than \(90^{\circ}\). Let \(m\angle P = 90^{\circ}\), \(m\angle Q= 45^{\circ}\), \(m\angle R = 45^{\circ}\). The sum \(90 + 45+45=180^{\circ}\), but \(\angle P\) is not acute.

Step2: Analyze part (b)

Supplementary angles sum to \(180^{\circ}\). Let \(m\angle1 = 90^{\circ}\), \(m\angle2=90^{\circ}\). They are supplementary, but neither is obtuse (an obtuse angle is greater than \(90^{\circ}\) and less than \(180^{\circ}\)).

Step3: Analyze part (c)

The area of a rectangle \(A = l\times w\). If \(l = 9\) and \(w = 4\), then \(A=9\times4 = 36\), which is not \(l = 6\) and \(w = 6\).

Step4: Analyze part (d)

If \(m\angle ABD=50^{\circ}\), let \(m\angle ABC = 20^{\circ}\) and \(m\angle CBD=30^{\circ}\). The sum \(20 + 30=50^{\circ}\), but it contradicts the claim \(m\angle ABC = 30^{\circ}\) and \(m\angle CBD = 20^{\circ}\).

Answer:

(a) \(90\), \(45\), \(45\); (b) \(90\), \(90\); (c) \(9\), \(4\); (d) \(20\), \(30\)