Question

abcd is a parallelogram. if $\angle b = 2x - 15^\circ$ and $\angle d = 3x - 30^\circ$. find the measure of $\angle a$ and $\angle c$ in the parallelogram.\
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$\bigcirc$ $\angle a = 90^\circ$ and $\angle c = 90^\circ$\
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$\bigcirc$ $\angle a = 165^\circ$ and $\angle c = 165^\circ$\
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$\bigcirc$ $\angle a = 105^\circ$ and $\angle c = 105^\circ$\
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$\bigcirc$ $\angle a = 45^\circ$ and $\angle c = 15^\circ$

Explanation:

Step1: Recall properties of parallelogram

In a parallelogram, opposite angles are equal. So, $\angle B = \angle D$.
Given $\angle B = 2x - 15^\circ$ and $\angle D = 3x - 30^\circ$, set them equal:
$2x - 15^\circ = 3x - 30^\circ$

Step2: Solve for x

Subtract $2x$ from both sides: $-15^\circ = x - 30^\circ$
Add $30^\circ$ to both sides: $x = 15^\circ$

Step3: Find measure of $\angle B$ and $\angle D$

Substitute $x = 15^\circ$ into $\angle B$: $\angle B = 2(15^\circ) - 15^\circ = 15^\circ$? Wait, no, wait: $2(15) -15 = 30 -15 = 15$? Wait, no, that can't be. Wait, no, wait, opposite angles in parallelogram: $\angle B$ and $\angle D$ are opposite? Wait, no, in parallelogram ABCD, the vertices are in order, so $\angle B$ and $\angle D$: wait, no, in a parallelogram, opposite angles are $\angle A$ and $\angle C$, $\angle B$ and $\angle D$. Wait, so $\angle B = \angle D$. Wait, but when I solved $2x -15 = 3x -30$, I get $x = 15$? Wait, that gives $\angle B = 2*15 -15 = 15$, $\angle D = 3*15 -30 = 15$. Then consecutive angles are supplementary. So $\angle A + \angle B = 180^\circ$. So $\angle A = 180 -15 = 165$? But that's not one of the options. Wait, I must have made a mistake. Wait, maybe $\angle B$ and $\angle D$ are not opposite? Wait, no, in parallelogram ABCD, the order is A, B, C, D, so AB is adjacent to BC, CD, DA. So $\angle A$ and $\angle B$ are adjacent, $\angle B$ and $\angle C$ are adjacent, etc. So opposite angles: $\angle A = \angle C$, $\angle B = \angle D$. Wait, but the options don't have 165. Wait, maybe I messed up the angle labels. Wait, maybe $\angle B$ and $\angle D$ are not opposite? Wait, no, in a parallelogram, opposite angles are equal. Wait, let's check the options again. Wait, maybe I made a mistake in the equation. Wait, let's re-express:

Wait, maybe $\angle B$ and $\angle D$: wait, no, maybe the problem is that $\angle B$ and $\angle D$ are not opposite? Wait, no, in parallelogram ABCD, vertices are A, B, C, D, so sides AB || CD, AD || BC. So angle at B and angle at D: yes, they are opposite. Wait, but when I solved $2x -15 = 3x -30$, I get $x = 15$, then $\angle B = 15$, $\angle D = 15$, then consecutive angles: $\angle A + \angle B = 180$, so $\angle A = 165$, $\angle C = 165$. But one of the options is $\angle A = 165$ and $\angle C = 165$. Wait, the second option: $\angle A = 165^\circ$ and $\angle C = 165^\circ$. Wait, but when I calculated $x = 15$, $\angle B = 2*15 -15 = 15$, $\angle D = 3*15 -30 = 15$. Then $\angle A = 180 -15 = 165$, $\angle C = 165$. So that's the second option. Wait, but let's check again. Wait, maybe I misread the angles. Wait, the problem says $\angle B = 2x -15$ and $\angle D = 3x -30$. So setting them equal: $2x -15 = 3x -30$ → $x = 15$. Then $\angle B = 15$, $\angle D = 15$. Then consecutive angles: $\angle A + \angle B = 180$, so $\angle A = 165$, $\angle C = 165$. So the second option is correct. Wait, but let's check the options again. The options are:

1. $\angle A = 90$ and $\angle C = 90$

1. $\angle A = 165$ and $\angle C = 165$

1. $\angle A = 105$ and $\angle C = 105$

1. $\angle A = 45$ and $\angle C = 15$

So according to the calculation, the second option is correct. Wait, but maybe I made a mistake in the angle labels. Wait, maybe $\angle B$ and $\angle D$ are not opposite? Wait, no, in a parallelogram, opposite angles are equal. So $\angle B = \angle D$. So the calculation seems right. So $\angle A = 180 - \angle B = 180 -15 = 165$, so $\angle A = 165$, $\angle C = 165$ (since $\angle A = \angle C$). So the second option is correct.

Answer:

$\angle A = 165^\circ$ and $\angle C = 165^\circ$ (the second option: $\boldsymbol{\angle A = 165^\circ}$ and $\boldsymbol{\angle C = 165^\circ}$)