Question

show that each statement is false by providing a counterexample.
(a) if the measures of \\( \angle p, \angle q \\), and \\( \angle r \\) sum to \\( 180^{circ} \\), then all of the angles must be acute.
counterexample: \\( m \angle p=\square^{circ}, m \angle q=\square^{circ}, m \angle r=\square^{circ} \\)
(b) if \\( \angle 1 \\) and \\( \angle 2 \\) are complementary angles, then one of them must have a measure greater than \\( 45^{circ} \\).
counterexample: \\( m \angle 1=\square^{circ}, m \angle 2=\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 \\( m \angle a b d=34^{circ} \\), and point \\( c \\) is in the interior of \\( \angle a b d \\), then \\( m \angle a b c=22^{circ} \\) and \\( m \angle c b d=12^{circ} \\).
counterexample: \\( m \angle a b c=\square^{circ}, m \angle c b d=\square^{circ} \\)

Explanation:

(a)

Step1: Recall the definition of acute angle

An acute angle is an angle with measure less than \(90^{\circ}\).

Step2: Find non - acute angles that sum to \(180^{\circ}\)

Let \(m\angle P = 90^{\circ}\), \(m\angle Q= 45^{\circ}\), \(m\angle R = 45^{\circ}\). The sum \(90 + 45+45=180^{\circ}\), but \(\angle P\) is a right angle (not acute).

(b)

Step1: Recall the definition of complementary angles

Complementary angles are two angles whose sum is \(90^{\circ}\).

Step2: Find two non - greater - than - \(45^{\circ}\) angles that are complementary

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

(c)

Step1: Recall the formula for the area of a rectangle

The area of a rectangle \(A = l\times w\), where \(l\) is the length and \(w\) is the width.

Step2: Find non - \(6\times6\) values for length and width with area \(36\)

Let \(l = 9\) and \(w = 4\). Then \(A=9\times4 = 36\).

(d)

Step1: Recall the angle - addition postulate

If \(C\) is in the interior of \(\angle ABD\), then \(m\angle ABD=m\angle ABC + m\angle CBD\).

Step2: Find non - \(22^{\circ}\) and \(12^{\circ}\) angles that sum to \(34^{\circ}\)

Let \(m\angle ABC=30^{\circ}\) and \(m\angle CBD = 4^{\circ}\). Then \(m\angle ABC+m\angle CBD=30 + 4=34^{\circ}\).

Answer:

(a) \(m\angle P = 90^{\circ}\), \(m\angle Q = 45^{\circ}\), \(m\angle R = 45^{\circ}\)
(b) \(m\angle1 = 45^{\circ}\), \(m\angle2 = 45^{\circ}\)
(c) length \(=9\), width \(=4\)
(d) \(m\angle ABC = 30^{\circ}\), \(m\angle CBD = 4^{\circ}\)