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 =AB34 and =BC12.=Counterexample:AB=, BC(b)If the perimeter of a rectangle is 12, then the length is 3 and the width is 3.=Counterexample:length=, width(c)If ∠1 and ∠2 are complementary angles, then one of them must have a measure less than 45°.=Counterexample:m∠1=, °m∠2°(d)If the measures of ∠R, ∠S, and ∠T sum to 180°, then one of the angles must be obtuse.=Counterexample:m∠R=, °m∠S=, °m∠T°
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 =AB34 and =BC12.=Counterexample:AB=, BC(b)If the perimeter of a rectangle is 12, then the length is 3 and the width is 3.=Counterexample:length=, width(c)If ∠1 and ∠2 are complementary angles, then one of them must have a measure less than 45°.=Counterexample:m∠1=, °m∠2°(d)If the measures of ∠R, ∠S, and ∠T sum to 180°, then one of the angles must be obtuse.=Counterexample:m∠R=, °m∠S=, °m∠T°
Solution
(a) If the length of AC is 46 and point B lies on AC, then AB=34 and BC=12. Counterexample: Let AB=20, BC=26. Here, AB+BC=AC=46, but AB is not 34 and BC is not 12.
(b) If the perimeter of a rectangle is 12, then the length is 3 and the width is 3. Counterexample: Let length=4, width=2. Here, 2*(length+width)=12, but length is not 3 and width is not 3.
(c) If ∠1 and ∠2 are complementary angles, then one of them must have a measure less than 45°. Counterexample: Let m∠1=50°, m∠2=40°. Here, m∠1+m∠2=90°, but neither of the angles is less than 45°.
(d) If the measures of ∠R, ∠S, and ∠T sum to 180°, then one of the angles must be obtuse. Counterexample: Let m∠R=60°, m∠S=60°, m∠T=60°. Here, m∠R+m∠S+m∠T=180°, but none of the angles is obtuse.
Similar Questions
(b)If ∠1 and ∠2 are complementary angles, then one of them must have a measure greater than 45°.=Counterexample:m∠1=, °m∠2°
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