Question

show that each statement is false by providing a counterexample.
(a) if the measures of ∠r, ∠s, and ∠t sum to 180°, then one of the angles must be obtuse.
counterexample: ( mangle r = square^circ ), ( mangle s = square^circ ), ( mangle t = square^circ ).
(b) if ( mangle wxz = 60^circ ), and point y is in the interior of ( angle wxz ), then ( mangle wxy = 40^circ ) and ( mangle yxz = 20^circ ).
counterexample: ( mangle wxy = square^circ ), ( mangle yxz = square^circ ).
(c) if the area of a rectangle is 36, then the length is 6 and the width is 6.
counterexample: length = ( square ), width = ( square )
(d) if ( angle 1 ) and ( angle 2 ) are complementary angles, then one of them must have a measure less than 45°.
counterexample: ( mangle 1 = square^circ ), ( mangle 2 = square^circ ).

Explanation:

(a)

Step1: Recall the definition of obtuse angle

An obtuse angle is an angle greater than \(90^{\circ}\) but less than \(180^{\circ}\). A counter - example 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" is when all angles are non - obtuse.

Step2: Choose non - obtuse angles

Let \(m\angle R = 60^{\circ}\), \(m\angle S=60^{\circ}\), \(m\angle T = 60^{\circ}\). Then \(m\angle R+m\angle S + m\angle T=60 + 60+60=180^{\circ}\), and none of the angles is obtuse.

(b)

Step1: Use the angle - addition formula

If \(\angle WXY=\angle WXZ+\angle YXZ\) (angle - addition postulate). The statement is "If \(m\angle WXZ = 60^{\circ}\), and point \(Y\) is in the interior of \(\angle WXZ\), then \(m\angle WXY = 40^{\circ}\) and \(m\angle YXZ=20^{\circ}\)"

Step2: Provide a counter - example

Let \(m\angle WXY = 30^{\circ}\) and \(m\angle YXZ = 30^{\circ}\). Then \(m\angle WXZ=m\angle WXY + m\angle YXZ=30 + 30=60^{\circ}\), but \(m\angle WXY
eq40^{\circ}\)

(c)

Step1: Use the area formula for a rectangle

The area formula for a rectangle is \(A = l\times w\), where \(A\) is the area, \(l\) is the length and \(w\) is the width. The statement is "If the area of a rectangle is \(36\), then the length is \(6\) and the width is \(6\)"

Step2: Provide a counter - example

Let \(l = 9\) and \(w = 4\). Then \(A=l\times w=9\times4 = 36\), but \(l
eq6\) and \(w
eq6\)

(d)

Step1: Recall the definition of complementary angles

Two angles \(\angle1\) and \(\angle2\) are complementary if \(m\angle1 + m\angle2=90^{\circ}\). The statement is "If \(\angle1\) and \(\angle2\) are complementary angles, then one of them must have a measure less than \(45^{\circ}\)"

Step2: Provide a counter - example

Let \(m\angle1 = 45^{\circ}\) and \(m\angle2=45^{\circ}\). Then \(m\angle1 + m\angle2=45 + 45=90^{\circ}\), and neither angle is less than \(45^{\circ}\)

Answer:

(a) \(m\angle R = 60^{\circ}\), \(m\angle S = 60^{\circ}\), \(m\angle T=60^{\circ}\)
(b) \(m\angle WXY = 30^{\circ}\), \(m\angle YXZ = 30^{\circ}\)
(c) length \(l = 9\), width \(w = 4\)
(d) \(m\angle1 = 45^{\circ}\), \(m\angle2 = 45^{\circ}\)