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°

Explanation:

Step1: Recall properties of parallelograms

In a parallelogram, opposite angles are equal, and consecutive angles are supplementary (sum to \(180^\circ\)). Also, \(AB \parallel CD\) and \(AD \parallel BC\), so \(\angle B\) and \(\angle D\) are not opposite, wait, actually in parallelogram \(ABCD\), \(\angle A = \angle C\), \(\angle B=\angle D\)? Wait no, wait: in parallelogram \(ABCD\), vertices are in order, so \(AB\) is adjacent to \(BC\), \(BC\) adjacent to \(CD\), etc. So \(\angle B\) and \(\angle D\) are opposite? Wait no, in parallelogram \(ABCD\), \(\angle A\) and \(\angle C\) are opposite, \(\angle B\) and \(\angle D\) are opposite. Wait, but in the diagram, \(\angle B\) is \((3n + 20)^\circ\) and \(\angle D\) is \((6n - 25)^\circ\). Wait, no, actually in a parallelogram, opposite angles are equal. Wait, \(AB \parallel CD\) and \(AD\) is a transversal, so \(\angle A + \angle D = 180^\circ\), but also \(\angle B = \angle D\)? Wait no, let's correct: In parallelogram \(ABCD\), \(AB \parallel CD\) and \(BC \parallel AD\). So consecutive angles are supplementary. So \(\angle B + \angle C = 180^\circ\), \(\angle A + \angle B = 180^\circ\), etc. Also, opposite angles are equal: \(\angle A = \angle C\), \(\angle B = \angle D\). Wait, in the diagram, \(\angle B\) is at vertex \(B\), \(\angle D\) is at vertex \(D\). So \(\angle B\) and \(\angle D\) are opposite angles? Wait, no, vertices are \(A, B, C, D\) in order, so sides are \(AB, BC, CD, DA\). So \(\angle B\) is between \(AB\) and \(BC\), \(\angle D\) is between \(CD\) and \(DA\). So \(AB \parallel CD\) and \(BC \parallel DA\), so \(\angle B\) and \(\angle D\) are equal (opposite angles). Wait, so \(\angle B = \angle D\). So set \(3n + 20 = 6n - 25\).

Step2: Solve for \(n\)

\(3n + 20 = 6n - 25\)

Subtract \(3n\) from both sides: \(20 = 3n - 25\)

Add 25 to both sides: \(45 = 3n\)

Divide by 3: \(n = 15\)

Step3: Find \(\angle B\)

Now, substitute \(n = 15\) into \(\angle B = (3n + 20)^\circ\)

\(\angle B = 3(15) + 20 = 45 + 20 = 65^\circ\)

Step4: Find \(\angle C\)

Since \(\angle B\) and \(\angle C\) are consecutive angles in parallelogram, they are supplementary (sum to \(180^\circ\))

So \(\angle C = 180^\circ - \angle B = 180 - 65 = 115^\circ\)

Answer:

\(\angle B = 65^\circ\); \(\angle C = 115^\circ\) (the second option: \(\boldsymbol{\angle B = 65^\circ; \angle C = 115^\circ}\))