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 = 45°, m∠2 = 45°

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 with area 100.

Step2: Find counter - example for rectangle

Let length $l = 10$ and width $w = 10$, since $10\times10=100$.

Step3: Recall angle sum and acute angle definition

Acute angles are less than $90^{\circ}$. We need to find angles that sum to $180^{\circ}$ with non - acute angles.

Step4: Find counter - example for angles

Let $m\angle P=90^{\circ}$, $m\angle Q = 45^{\circ}$, $m\angle R=45^{\circ}$, since $90 + 45+45 = 180^{\circ}$ and $90^{\circ}$ is not acute.

Step5: Recall line - segment length property

If $Y$ lies on $\overline{XZ}$ with length 20, $XY$ and $YZ$ don't have to be equal.

Step6: Find counter - example for line - segment

Let $XY = 5$ and $YZ = 15$, since $5 + 15=20$.

Answer:

(a) length = 10, width = 10
(b) $m\angle P = 90^{\circ}$, $m\angle Q = 45^{\circ}$, $m\angle R = 45^{\circ}$
(c) $XY = 5$, $YZ = 15$
(d) $m\angle 1 = 45^{\circ}$, $m\angle 2 = 45^{\circ}$