Question

mathematical practices construct viable arguments and critique the reasoning of others 19. a. construct a viable argument for this conjecture: if two angles are complementary, both angles must be acute. b. what type of reasoning—inductive or deductive—did you use in your argument? explain. c. compare your argument with the argument another student in your group or class has written. do you agree with this student’s argument? why or why not?

Explanation:

Step1: Define complementary angles

Complementary angles sum to 90 degrees, i.e., if $\angle A$ and $\angle B$ are complementary, $\angle A+\angle B = 90^{\circ}$.

Step2: Define acute angles

Acute angles are angles with measure between 0 and 90 degrees.

Step3: Prove the conjecture

Suppose $\angle A$ and $\angle B$ are complementary, so $\angle A+\angle B=90^{\circ}$. Since $\angle A = 90^{\circ}-\angle B$ and $\angle B=90^{\circ}-\angle A$, and both $\angle A>0^{\circ}$ and $\angle B > 0^{\circ}$, also $\angle A<90^{\circ}$ and $\angle B<90^{\circ}$, so both are acute.

Step4: Determine the reasoning type

We used deductive reasoning. We started with known definitions (complementary and acute - angle definitions) and logically derived the conclusion.

Step5: Regarding part c

This part requires comparing with another student's work. Since we don't have another student's argument, we can't provide a specific answer. But in general, if the other student also used correct definitions and logical steps, we may agree. If there are errors in definitions, logic, or steps, we may not agree.

Answer:

a. Complementary angles sum to 90 degrees. Since angles must be positive and less than 90 degrees to sum to 90 degrees, both are acute.
b. Deductive reasoning. We used known definitions to reach the conclusion.
c. Without another student's argument, a specific answer cannot be given. In general, agreement depends on correctness of definitions and logic used in the other - student's argument.