Question

show that each statement is false by providing a counterexample. (a) if the area of a rectangle is 100, then the length is 25 and the width is 4. counterexample: length = , width = (b) 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 = (c) if the length of (overline{xz}) is 20 and point y lies on (overline{xz}), then (xy = 10) and (yz = 10). counterexample: (xy = , yz = ) (d) if ∠1 and ∠2 are complementary angles, then one of them must have a measure greater than 45°. counterexample: m∠1 = , m∠2 =

Explanation:

Step1: Recall rectangle area formula

The area of a rectangle is $A = l\times w$. We need to find non - 25 and 4 values for length and width such that $A = 100$.
Let $l = 10$ and $w = 10$, since $10\times10=100$.

Step2: Recall angle - sum property

The sum of angles $\angle P+\angle Q+\angle R = 180^{\circ}$. An acute angle is less than $90^{\circ}$. Let $\angle P=90^{\circ}$, $\angle Q = 45^{\circ}$, $\angle R=45^{\circ}$, since $90 + 45+45=180^{\circ}$ and $90^{\circ}$ is not acute.

Step3: Consider line - segment division

If $XZ = 20$ and $Y$ lies on $XZ$, we can choose non - equal lengths for $XY$ and $YZ$. Let $XY = 5$ and $YZ = 15$, since $5 + 15=20$.

Step4: Recall complementary angles

Complementary angles add up to $90^{\circ}$. Let $\angle1 = 45^{\circ}$ and $\angle2 = 45^{\circ}$, since $45+45 = 90^{\circ}$ and neither is greater than $45^{\circ}$.

Answer:

(a) length = 10, width = 10
(b) $m\angle P=90$, $m\angle Q = 45$, $m\angle R = 45$
(c) $XY = 5$, $YZ = 15$
(d) $m\angle1 = 45$, $m\angle2 = 45$