Question

show that each statement is false by providing a counterexample.
(a) if ∠1 and ∠2 are supplementary angles, then one of them must be acute.
counterexample: ( mangle 1 = square^circ ), ( mangle 2 = square^circ )
(b) if the measures of ∠r, ∠s, and ∠t sum to ( 180^circ ), then one of the angles must be obtuse.
counterexample: ( mangle r = square^circ ), ( mangle s = square^circ ), ( mangle t = square^circ )
(c) if ( mangle wxz = 50^circ ), and point y is in the interior of ∠wxz, then ( mangle wxy = 25^circ ) and ( mangle yxz = 25^circ ).
counterexample: ( mangle wxy = square^circ ), ( mangle yxz = square^circ )
(d) if the perimeter of a rectangle is 28, then the length is 10 and the width is 4.
counterexample: length ( = square ), width ( = square )

Explanation:

Step1: Analyze part (a)

Supplementary angles sum to \(180^{\circ}\). A counter - example for the statement "If \(\angle1\) and \(\angle2\) are supplementary angles, then one of them must be acute" is when \(\angle1 = 90^{\circ}\) and \(\angle2=90^{\circ}\) (since \(90 + 90=180\) and neither is acute).

Step2: Analyze part (b)

For the statement "If the measures of \(\angle R,\angle S,\) and \(\angle T\) sum to \(180^{\circ}\), then one of the angles must be obtuse", a counter - example is \(\angle R = 60^{\circ},\angle S = 60^{\circ},\angle T = 60^{\circ}\) (since \(60+60 + 60=180\) and none are obtuse).

Step3: Analyze part (c)

For the statement "If \(m\angle WXZ = 50^{\circ}\), and point \(Y\) is in the interior of \(\angle WXZ\), then \(m\angle WXY = 25^{\circ}\) and \(m\angle YXZ=25^{\circ}\)", a counter - example is when \(m\angle WXY = 10^{\circ}\) and \(m\angle YXZ = 40^{\circ}\) (since \(10 + 40=50\)).

Step4: Analyze part (d)

For the statement "If the perimeter of a rectangle is \(28\), then the length is \(10\) and the width is \(4\)", using the formula for the perimeter of a rectangle \(P = 2(l + w)\). Let \(l = 9\) and \(w=5\), then \(P=2(9 + 5)=2\times14 = 28\).

Answer:

(a) \(m\angle1 = 90^{\circ}\), \(m\angle2 = 90^{\circ}\)
(b) \(m\angle R=60^{\circ}\), \(m\angle S = 60^{\circ}\), \(m\angle T=60^{\circ}\)
(c) \(m\angle WXY = 10^{\circ}\), \(m\angle YXZ = 40^{\circ}\)
(d) length \(=9\), width \(=5\)