Question

Show that each statement is false by providing a counterexample.

(a) If the length of AC is 46 and point B lies on AC, then A B = 34 and BC = 12.
Counterexample:
AB = BC = 0
(b) If the perimeter of a rectangle is 32, then the length is 8 and the width is 8.
Counterexample:
length width
(c) If _ 1 and 2 are complementary angles, then one of them must have a measure less than 45.
Counterexample:
m1 =
1 m 2 =
(d) If the measures of R, S, and I sum to 180 , then one of the angles must be obtuse.
Counterexample:
m z R = 0°, m 2 S = O, m, m_T =Х

Answer 1

For part (a), a counterexample is when AB = BC = 0, which contradicts the statement. For part (b), a counterexample is when the perimeter is 32 but the length and width are not both 8. For part (c), a counterexample is when angles 1 and 2 have measures less than 45 degrees. For part (d), a counterexample is when angles R, S, and I are all acute.

Step-by-step explanation:

Counterexample for part (a):

If we take AC = 46 and point B lies on AC, but AB = BC = 0, then the statement is false because AB should equal 34 and BC should equal 12.

Counterexample for part (b):

If the perimeter of a rectangle is 32, it does not mean that the length and width are both 8. For example, a rectangle with a length of 10 and a width of 6 also has a perimeter of 32, but the statement is false.

Counterexample for part (c):

If angles 1 and 2 are complementary, it means their sum is 90 degrees. It is possible for both angles to have a measure less than 45 degrees. For example, if angle 1 measures 30 degrees and angle 2 measures 60 degrees, the statement is false.

Counterexample for part (d):

If the measures of angles R, S, and I sum to 180 degrees, it is possible for all three angles to be acute. For example, if angle R measures 60 degrees, angle S measures 60 degrees, and angle I measures 60 degrees, all angles are acute and the statement is false.