Question

sean made the following statement: ∠a and ∠b are supplementary; therefore, one of the angles is acute. determine which statement is a counterexample for the conjecture.
a. ∠a = 32° and ∠b = 108°
b. ∠a = 82° and ∠b = 98°
c. ∠a = 90° and ∠b = 90°
d. ∠a = 65° and ∠b = 115°

Explanation:

Step1: Recall supplementary - angle definition

Two angles are supplementary if their sum is $180^{\circ}$. An acute angle is an angle with a measure between $0^{\circ}$ and $90^{\circ}$.

Step2: Check each option

Option A:

$\angle A = 32^{\circ}$ and $\angle B=108^{\circ}$, $\angle A+\angle B=32^{\circ}+108^{\circ}=140^{\circ}
eq180^{\circ}$, so they are not supplementary.

Option B:

$\angle A = 82^{\circ}$ and $\angle B = 98^{\circ}$, $\angle A+\angle B=82^{\circ}+98^{\circ}=180^{\circ}$, and $82^{\circ}$ is acute. This is not a counter - example.

Option C:

$\angle A = 90^{\circ}$ and $\angle B = 90^{\circ}$, $\angle A+\angle B=90^{\circ}+90^{\circ}=180^{\circ}$, and neither angle is acute. This is a counter - example.

Option D:

$\angle A = 65^{\circ}$ and $\angle B = 115^{\circ}$, $\angle A+\angle B=65^{\circ}+115^{\circ}=180^{\circ}$, and $65^{\circ}$ is acute. This is not a counter - example.

Answer:

C. $\angle A = 90^{\circ}$ and $\angle B = 90^{\circ}$