Question

show that each statement is false by providing a counter - example.
(a) if ∠1 and ∠2 are supplementary angles, then one of them must be obtuse.
counterexample: m∠1 = , m∠2 =
(b) if the perimeter of a rectangle is 32, then the length is 13 and the width is 3.
counterexample: length = , width =
(c) if m∠abd = 56°, and point c is in the interior of ∠abd, then m∠abc = 28° and m∠cbd = 28°.
counterexample: m∠abc = , m∠cbd =
(d) if the measures of ∠p, ∠q, and ∠r sum to 180°, then all of the angles must be acute.
counterexample: m∠p = , m∠q = , m∠r =

Explanation:

Step1: Recall supplementary - angle definition

Supplementary angles add up to 180°. A right - angle is 90°. Two right - angles are supplementary and neither is obtuse.
So for (a), $m\angle1 = 90^{\circ}$, $m\angle2=90^{\circ}$.

Step2: Recall rectangle perimeter formula

The perimeter of a rectangle $P = 2(l + w)$. Given $P = 32$, then $l + w=16$. We can choose $l = 10$ and $w = 6$.

Step3: Consider angle - interior cases

Just because point $C$ is in the interior of $\angle ABD$ doesn't mean the angles are bisected. We can choose $m\angle ABC = 10^{\circ}$ and $m\angle CBD = 46^{\circ}$ since $10 + 46=56$.

Step4: Recall angle - sum property of a triangle

The sum of the measures of the angles in a triangle is 180°. We can have an obtuse - angled triangle. For example, $m\angle P = 100^{\circ}$, $m\angle Q = 30^{\circ}$, $m\angle R = 50^{\circ}$.

Answer:

(a) $m\angle1 = 90^{\circ}$, $m\angle2 = 90^{\circ}$
(b) length = 10, width = 6
(c) $m\angle ABC = 10^{\circ}$, $m\angle CBD = 46^{\circ}$
(d) $m\angle P = 100^{\circ}$, $m\angle Q = 30^{\circ}$, $m\angle R = 50^{\circ}$