Question

(a) if the measures of ∠p, ∠q, and ∠r sum to 180°, then all of the angles must be acute. counterexample: m∠p = , m∠q = , m∠r = . (b) if the perimeter of a rectangle is 28, then the length is 10 and the width is 4. counterexample: length = , width = . (c) if m∠abd = 60°, and point c is in the interior of ∠abd, then m∠abc = 30° and m∠cbd = 30°. counterexample: m∠abc = , m∠cbd = .

Explanation:

Step1: Analyze part (a)

The statement says if the measures of ∠P, ∠Q, and ∠R sum to 180°, then all of the angles must be acute. A counter - example is when ∠P = 90°, ∠Q = 45°, ∠R = 45°. The sum is 180° but ∠P is a right angle, not acute.

Step2: Analyze part (b)

The statement says if the perimeter of a rectangle is 28, then the length is 10 and the width is 4. The formula for the perimeter of a rectangle is $P = 2(l + w)$. Let $l=8$ and $w = 6$, then $P=2(8 + 6)=28$, so length doesn't have to be 10 and width 4.

Step3: Analyze part (c)

The statement says if $m\angle ABD=60^{\circ}$ and point C is in the interior of $\angle ABD$, then $m\angle ABC = 30^{\circ}$ and $m\angle CBD=30^{\circ}$. Let $m\angle ABC = 20^{\circ}$ and $m\angle CBD = 40^{\circ}$, C is in the interior of $\angle ABD$ and $m\angle ABC+m\angle CBD=m\angle ABD = 60^{\circ}$, but the angles are not both 30°.

Step4: Analyze part (d)

The statement says if ∠1 and ∠2 are complementary angles and measure greater than 45°, but complementary angles add up to 90°. If $m\angle1 = 50^{\circ}$ and $m\angle2=40^{\circ}$, they are complementary and one is not greater than 45°.

Answer:

(a) $m\angle P = 90^{\circ},m\angle Q = 45^{\circ},m\angle R = 45^{\circ}$
(b) length = 8, width = 6
(c) $m\angle ABC = 20^{\circ},m\angle CBD = 40^{\circ}$
(d) $m\angle1 = 50^{\circ},m\angle2 = 40^{\circ}$