Question

show that each statement is false by providing a counterexample.
(a) 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 } \\)
(b) if the perimeter of a rectangle is 44, then the length is 17 and the width is 5.
counterexample: length \\( = \square \\) width \\( = \square \\)
(c) 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 = 17 ^ { \circ } \\) and \\( m \angle c b d = 17 ^ { \circ } \\).
counterexample: \\( m \angle a b c = \square ^ { \circ } , m \angle c b d = \square ^ { \circ } \\)
(d) 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 } \\)

Explanation:

Step1: Understand complementary angles

Complementary angles sum to \(90^{\circ}\).

Step2: Find counter - example for (a)

Let \(m\angle1 = 30^{\circ}\) and \(m\angle2=60^{\circ}\). Both \(30^{\circ}<45^{\circ}\) and \(60^{\circ}>45^{\circ}\), but they are complementary.

Step3: Understand rectangle perimeter formula

Perimeter of a rectangle \(P = 2(l + w)\).

Step4: Find counter - example for (b)

If \(l = 10\) and \(w = 12\), then \(P=2(10 + 12)=44\), but \(l
eq17\) and \(w
eq5\).

Step5: Understand angle addition

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

Step6: Find counter - example for (c)

Let \(m\angle ABD = 34^{\circ}\) and \(m\angle CBD = 20^{\circ}\), then \(m\angle ABC=34^{\circ}+20^{\circ} = 54^{\circ}
eq17^{\circ}\).

Step7: Understand triangle angle sum

Sum of angles in a triangle is \(180^{\circ}\), but they don't have to be equal.

Step8: Find counter - example for (d)

Let \(m\angle P=30^{\circ}\), \(m\angle Q = 60^{\circ}\), \(m\angle R = 90^{\circ}\), sum is \(180^{\circ}\) but angles are not equal.

Answer:

(a) \(m\angle1 = 30^{\circ}\), \(m\angle2 = 60^{\circ}\); (b) length \(= 10\), width \(= 12\); (c) \(m\angle ABD = 34^{\circ}\), \(m\angle CBD = 20^{\circ}\); (d) \(m\angle P=30^{\circ}\), \(m\angle Q = 60^{\circ}\), \(m\angle R = 90^{\circ}\)