ure abcd is a parallelogram.
what are the measures of angles b and d?
(2n + 15)°
(3n - 5)°
○ ∠b = 55°; ∠d = 55°
○ ∠b = 55°; ∠d = 125°
○ ∠b = 97°; ∠d = 97°
○ ∠b = 83°; ∠d = 97°

Question

Accepted Answer

# Explanation:
## Step1: Recall parallelogram angle properties
In a parallelogram, consecutive angles are supplementary (sum to $180^\circ$), and opposite angles are equal. So $\angle B$ and $\angle D$ are opposite angles? Wait, no, $\angle B$ and $\angle D$: Wait, in parallelogram $ABCD$, $\angle A$ and $\angle C$ are opposite, $\angle B$ and $\angle D$ are opposite? Wait, no, let's see the labels: $A - B - C - D - A$. So $\angle B$ is at vertex $B$, $\angle D$ is at vertex $D$. Wait, actually, $\angle B$ and $\angle D$ are opposite angles? Wait, no, $\angle B$ and $\angle D$: Wait, in the diagram, $\angle B=(2n + 15)^\circ$ and $\angle D=(3n - 5)^\circ$? Wait, no, wait, in a parallelogram, opposite angles are equal. Wait, but also, consecutive angles are supplementary. Wait, maybe $\angle B$ and $\angle D$ are opposite? Wait, no, let's check the sides. $AB$ is parallel to $CD$, and $AD$ is parallel to $BC$. So $\angle B$ and $\angle D$: Wait, $\angle B$ and $\angle D$ are opposite angles? Wait, no, $\angle A$ and $\angle C$ are opposite, $\angle B$ and $\angle D$ are opposite. Wait, but in the diagram, $\angle D=(3n - 5)^\circ$ and $\angle B=(2n + 15)^\circ$. Wait, maybe I made a mistake. Wait, actually, in a parallelogram, consecutive angles are supplementary. So $\angle B$ and $\angle D$: Wait, no, $\angle B$ and $\angle A$ are consecutive? Wait, no, let's look at the angles given: $\angle B=(2n + 15)^\circ$ and $\angle D=(3n - 5)^\circ$? Wait, no, maybe $\angle B$ and $\angle D$ are opposite? Wait, no, maybe the angles at $B$ and $D$ are opposite? Wait, no, in a parallelogram, opposite angles are equal. Wait, but also, consecutive angles are supplementary. Wait, maybe the angles given are consecutive? Wait, no, the angle at $B$ is $(2n + 15)^\circ$ and the angle at $D$ is $(3n - 5)^\circ$. Wait, maybe I misread. Wait, actually, in parallelogram $ABCD$, $\angle B$ and $\angle D$ are opposite angles? Wait, no, $\angle B$ is adjacent to $\angle A$ and $\angle C$, $\angle D$ is adjacent to $\angle A$ and $\angle C$. Wait, no, let's correct: In parallelogram $ABCD$, $AB \parallel CD$ and $AD \parallel BC$. So $\angle A$ and $\angle B$ are consecutive (supplementary), $\angle B$ and $\angle C$ are consecutive (supplementary), $\angle C$ and $\angle D$ are consecutive (supplementary), $\angle D$ and $\angle A$ are consecutive (supplementary). Also, $\angle A=\angle C$, $\angle B=\angle D$ (opposite angles equal). Wait, but in the diagram, the angle at $D$ is $(3n - 5)^\circ$ and the angle at $B$ is $(2n + 15)^\circ$. Wait, maybe the angles at $B$ and $D$ are opposite, so they should be equal? But that would mean $2n + 15 = 3n - 5$, solving for $n$: $15 + 5 = 3n - 2n$, $n = 20$. Then $\angle B = 2(20)+15 = 55^\circ$, $\angle D = 3(20)-5 = 55^\circ$. But wait, also, consecutive angles should be supplementary. Let's check $\angle A$ and $\angle B$: If $\angle B = 55^\circ$, then $\angle A = 180 - 55 = 125^\circ$, but $\angle D$ is $55^\circ$, which is opposite to $\angle B$, so that works? Wait, but let's check the options. The first option is $\angle B = 55^\circ$; $\angle D = 55^\circ$. But wait, maybe I messed up the opposite angles. Wait, no, in a parallelogram, opposite angles are equal. So $\angle B$ and $\angle D$ are opposite? Wait, maybe the diagram has $\angle B$ and $\angle D$ as opposite. Wait, let's re-express:

Wait, the problem is to find $\angle B$ and $\angle D$. Let's use the property that in a parallelogram, opposite angles are equal. Wait, but also, consecutive angles are supplementary. Wait, maybe the angles given are $\angle B=(2n + 15)^\circ$ and $\angle D=(3n - 5)^\circ$, but actually, $\angle B$ and $\angle D$ are opposite, so they should be equal. So set $2n + 15 = 3n - 5$. Solving:

$2n + 15 = 3n - 5$

Subtract $2n$ from both sides: $15 = n - 5$

Add 5 to both sides: $n = 20$

Then $\angle B = 2(20) + 15 = 55^\circ$, $\angle D = 3(20) - 5 = 55^\circ$. Let's check the options: first option is $\angle B = 55^\circ$; $\angle D = 55^\circ$. But wait, let's check consecutive angles. If $\angle B = 55^\circ$, then the consecutive angle (like $\angle C$) would be $180 - 55 = 125^\circ$, and $\angle D = 55^\circ$ (opposite to $\angle B$), so that's consistent with a parallelogram. Wait, but let's check the other options. The second option has $\angle D = 125^\circ$, which would mean $\angle B$ and $\angle D$ are supplementary, but they are opposite, so they should be equal. Wait, maybe I made a mistake in identifying opposite angles. Wait, maybe $\angle B$ and $\angle D$ are not opposite. Wait, let's look at the labels: $A$ is top-left, $B$ top-right, $C$ bottom-right, $D$ bottom-left. So $\angle B$ is at top-right, $\angle D$ is at bottom-left. So sides $AB$ (top) and $CD$ (bottom) are parallel, sides $AD$ (left) and $BC$ (right) are parallel. So $\angle B$ (top-right) and $\angle D$ (bottom-left) are opposite angles? Wait, no, $\angle A$ (top-left) and $\angle C$ (bottom-right) are opposite, $\angle B$ (top-right) and $\angle D$ (bottom-left) are opposite. So yes, they should be equal. So solving $2n + 15 = 3n - 5$ gives $n = 20$, so $\angle B = 55^\circ$, $\angle D = 55^\circ$, which is the first option. Wait, but let's check the other angles. If $\angle B = 55^\circ$, then $\angle A$ (adjacent to $\angle B$) is $180 - 55 = 125^\circ$, and $\angle D$ is $55^\circ$ (opposite to $\angle B$), so that's correct.

## Step2: Verify with the solution
After solving $n = 20$, $\angle B = 2(20) + 15 = 55^\circ$, $\angle D = 3(20) - 5 = 55^\circ$, which matches the first option.

# Answer:
$\boldsymbol{\angle B = 55^\circ; \angle D = 55^\circ}$ (the first option: $\boldsymbol{\angle B = 55^\circ; \angle D = 55^\circ}$)

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Step1: Recall parallelogram angle properties

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