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 = (square), m∠2 = (square). (b) if the perimeter of a rectangle is 28, then the length is 10 and the width is 4. counterexample: length = (square), width = (square). (c) if m∠abd = 60°, and point c is in the interior of ∠abd, then m∠abc = 30° and m∠cbd = 30°. counterexample: m∠abc = (square), m∠cbd = (square). (a) if the measures of ∠p, ∠q, and ∠r sum to 180°, then all of the angles must be acute. counterexample: m∠p = (square), m∠q = (square), m∠r = (square)

Explanation:

Step1: Analyze (a)

Complementary angles sum to 90°. If ∠1 = 46° and ∠2=44°, they are complementary and neither is greater than 45°.

Step2: Analyze (b)

Perimeter of a rectangle is given by \(P = 2(l + w)\). If \(l = 10\) and \(w = 4\), \(P=2(10 + 4)=28\), which is a valid rectangle with given perimeter, length and width.

Step3: Analyze (c)

If point C is in the interior of ∠ABD, and \(m\angle ABD=60^{\circ}\), \(m\angle ABC\) and \(m\angle CBD\) can have various values. For example, if \(m\angle ABC = 40^{\circ}\) and \(m\angle CBD=20^{\circ}\), it contradicts the statement.

Step4: Analyze (a) (angles summing to 180°)

If \(\angle P = 90^{\circ}\), \(\angle Q= 45^{\circ}\), \(\angle R = 45^{\circ}\), they sum to 180° but \(\angle P\) is not acute.

Answer:

(a) \(m\angle1 = 46^{\circ}\), \(m\angle2 = 44^{\circ}\)
(b) length \(= 10\), width \(= 4\)
(c) \(m\angle ABC = 40^{\circ}\), \(m\angle CBD = 20^{\circ}\)
(a) (angles summing to 180°) \(m\angle P=90^{\circ}\), \(m\angle Q = 45^{\circ}\), \(m\angle R=45^{\circ}\)