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 less than 45°. counterexample: m∠1 = , m∠2 = (b) if the measures of ∠r, ∠s, and ∠t sum to 180°, then one of the angles must be obtuse. counterexample: m∠r = , m∠s = , m∠t = (c) if m∠wxz = 42°, and point y is in the interior of ∠wxz, then m∠wxy = 21° and m∠yxz = 21°. counterexample: m∠wxy = , m∠yxz = (d) if the perimeter of a rectangle is 48, then the length is 20 and the width is 4. counterexample: length = , width =

Explanation:

Step1: Recall complementary - angle definition

Complementary angles sum to 90°. A counter - example is when both angles are equal, so \(m\angle1 = 45^{\circ}\), \(m\angle2=45^{\circ}\).

Step2: Recall angle - sum property of a triangle

Angles in a triangle sum to 180°. An equilateral triangle has all angles equal to 60°, so \(m\angle R = 60^{\circ}\), \(m\angle S = 60^{\circ}\), \(m\angle T = 60^{\circ}\).

Step3: Recall angle - addition postulate

If \(Y\) is in the interior of \(\angle WXZ\), \(m\angle WXZ=m\angle WXY + m\angle YXZ\). A counter - example could be \(m\angle WXY = 20^{\circ}\), \(m\angle YXZ = 22^{\circ}\) (since \(20 + 22=42\)).

Step4: Recall rectangle perimeter formula

The perimeter of a rectangle \(P = 2(l + w)\). Given \(P = 48\), then \(l + w=24\). A counter - example is \(l = 15\), \(w = 9\) (since \(2(15 + 9)=2\times24 = 48\)).

Answer:

(a) \(m\angle1 = 45^{\circ}\), \(m\angle2 = 45^{\circ}\)
(b) \(m\angle R = 60^{\circ}\), \(m\angle S = 60^{\circ}\), \(m\angle T = 60^{\circ}\)
(c) \(m\angle WXY = 20^{\circ}\), \(m\angle YXZ = 22^{\circ}\)
(d) length \(=15\), width \(=9\)