Question

figure abcd is a parallelogram. what are the measures of angles b and c? ∠b = 15°; ∠c = 165° ∠b = 65°; ∠c = 115° ∠b = 65°; ∠c = 65° ∠b = 15°; ∠c = 15° a b (3n + 20)° d (6n - 25)° c

Explanation:

Step1: Use property of parallelogram

In a parallelogram, adjacent - angles are supplementary, so $\angle B+\angle C = 180^{\circ}$, and opposite - angles are equal, so $\angle B=\angle D=(6n - 25)^{\circ}$ and $\angle A=\angle C=(3n + 20)^{\circ}$. Also, since $\angle B+\angle C = 180^{\circ}$, we have $(6n - 25)+(3n + 20)=180$.

Step2: Solve the equation for n

Combine like - terms: $6n+3n-25 + 20=180$, which simplifies to $9n-5 = 180$. Add 5 to both sides: $9n=180 + 5=185$, then $n=\frac{185}{9}$. But we can also use the adjacent - angle property in another way. Since $\angle B+\angle C = 180^{\circ}$, we know that if we assume $\angle B=(6n - 25)^{\circ}$ and $\angle C=(3n + 20)^{\circ}$, then $6n-25+3n + 20=180$. Combining like terms gives $9n-5 = 180$, so $9n=185$ (this is wrong. Let's use the fact that adjacent angles are supplementary).
In a parallelogram, $\angle B+\angle C = 180^{\circ}$. Let's check the options one by one.
For option A: $15 + 165=180$, but we need to check if they can be formed from the angle expressions.
For option B: $65+115 = 180$.
Let's assume $\angle B = 65^{\circ}$ and $\angle C = 115^{\circ}$.
If $\angle B=(6n - 25)^{\circ}=65^{\circ}$, then $6n-25 = 65$, $6n=65 + 25=90$, $n = 15$.
If $n = 15$, then $\angle C=(3n + 20)^{\circ}=(3\times15+20)^{\circ}=(45 + 20)^{\circ}=65^{\circ}$ (wrong).
Let's assume $\angle B=(3n + 20)^{\circ}$ and $\angle C=(6n - 25)^{\circ}$.
If $\angle B = 65^{\circ}$, then $3n+20 = 65$, $3n=65 - 20=45$, $n = 15$.
If $n = 15$, then $\angle C=(6n - 25)^{\circ}=(6\times15-25)^{\circ}=(90 - 25)^{\circ}=65^{\circ}$ (wrong).
Let's use the property that adjacent angles of a parallelogram are supplementary.
We know that $\angle B+\angle C = 180^{\circ}$.
If $\angle B = 65^{\circ}$ and $\angle C = 115^{\circ}$, it satisfies the adjacent - angle property of a parallelogram.

Answer:

B. $\angle B = 65^{\circ}$; $\angle C = 115^{\circ}$