Question

show that each statement is false by providing a counterexample.
(a) if ∠1 and ∠2 are complementary angles, then one of them must have a measure greater than 45°.
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 complementary - angle definition

Complementary angles add up to 90°. A counter - example for (a) is when both angles are 45° since 45°+45° = 90°.

Step2: Recall rectangle perimeter formula

The perimeter of a rectangle is \(P = 2(l + w)\). Given \(P = 32\), then \(l + w=16\). A counter - example for (b) could be \(l = 10\) and \(w = 6\) since \(2(10 + 6)=32\).

Step3: Consider angle - addition in a larger angle

For (c), if \(m\angle ABD = 56^{\circ}\) and \(C\) is in the interior of \(\angle ABD\), a counter - example could be \(m\angle ABC = 10^{\circ}\) and \(m\angle CBD = 46^{\circ}\) since \(10^{\circ}+46^{\circ}=56^{\circ}\).

Step4: Recall angle - sum and acute - angle definition

Acute angles are less than 90°. For (d), if \(m\angle P=90^{\circ}\), \(m\angle Q = 45^{\circ}\), \(m\angle R = 45^{\circ}\), then \(m\angle P+m\angle Q+m\angle R=180^{\circ}\), but \(\angle P\) is not acute.

Answer:

(a) \(m\angle1 = 45\), \(m\angle2 = 45\)
(b) length \( = 10\), width \( = 6\)
(c) \(m\angle ABC = 10\), \(m\angle CBD = 46\)
(d) \(m\angle P = 90\), \(m\angle Q = 45\), \(m\angle R = 45\)