find the value of ( x ), ( y ), and ( z ) in the parallelogram below.\n\nanswer attempt 1 out of 2\n( x =…

Explanation:

Step1: Recall properties of parallelograms

In a parallelogram, opposite angles are equal, and consecutive angles are supplementary (sum to (180^\circ)).

Step2: Solve for (x)

One angle is (74^\circ) and its opposite angle is ((10x + 6)^\circ)? Wait, no, wait. Wait, in a parallelogram, opposite angles are equal. Wait, the angle (74^\circ) and the angle ((-y - 6)^\circ) – no, wait, let's look at the diagram. Wait, the angle (74^\circ) and the angle ((-y - 6)^\circ) – no, wait, actually, in a parallelogram, opposite angles are equal. So the angle (74^\circ) should be equal to the angle ((-y - 6)^\circ)? Wait, no, maybe I got the angles wrong. Wait, the angle ((10x + 6)^\circ) and the angle ((-z + 8)^\circ) – no, wait, let's re-examine.

Wait, in a parallelogram, consecutive angles are supplementary. So the angle (74^\circ) and the angle ((10x + 6)^\circ) are consecutive? Wait, no, the angle (74^\circ) and the angle ((10x + 6)^\circ) – let's see, the angle (74^\circ) and the angle ((-y - 6)^\circ) – no, maybe the angle (74^\circ) and the angle ((10x + 6)^\circ) are supplementary? Wait, no, in a parallelogram, opposite angles are equal, and consecutive angles are supplementary. So let's identify the angles:

Looking at the parallelogram, the angles are: (74^\circ), ((10x + 6)^\circ), ((-y - 6)^\circ), and ((-z + 8)^\circ).

So opposite angles: (74^\circ) should be equal to ((-y - 6)^\circ), and ((10x + 6)^\circ) should be equal to ((-z + 8)^\circ). Also, consecutive angles: (74^\circ + (10x + 6)^\circ = 180^\circ) (since consecutive angles in a parallelogram are supplementary).

Ah, that makes sense. So let's start with (x).

So (74 + (10x + 6) = 180)

Step3: Solve the equation for (x)

Combine like terms: (74 + 6 + 10x = 180)

(80 + 10x = 180)

Subtract 80 from both sides: (10x = 180 - 80 = 100)

Divide both sides by 10: (x = \frac{100}{10} = 10)? Wait, no, (180 - 80 = 100), so (10x = 100), so (x = 10)? Wait, but let's check: (10x + 6 = 10*10 + 6 = 106). Then (74 + 106 = 180), which is correct for consecutive angles. Good.

Now, for (y): The angle (74^\circ) and the angle ((-y - 6)^\circ) are opposite angles? Wait, no, opposite angles are equal. Wait, the angle ((-y - 6)^\circ) should be equal to (74^\circ)? Wait, no, maybe the angle ((-y - 6)^\circ) is equal to (74^\circ)? Wait, let's see:

If ((-y - 6) = 74), then (-y = 74 + 6 = 80), so (y = -80). Wait, that seems odd, but let's check.

Wait, maybe the angle ((-y - 6)^\circ) is equal to (74^\circ) because they are opposite angles. So:

(-y - 6 = 74)

(-y = 74 + 6 = 80)

(y = -80)

Now for (z): The angle ((10x + 6)^\circ) and the angle ((-z + 8)^\circ) are opposite angles, so they should be equal. We found (x = 10), so (10x + 6 = 106). Therefore:

(-z + 8 = 106)

(-z = 106 - 8 = 98)

(z = -98)

Wait, but let's verify all angles:

Angles: (74^\circ), (106^\circ), (74^\circ), (106^\circ). Let's check consecutive angles: (74 + 106 = 180), which is correct for a parallelogram. Opposite angles: (74 = 74) and (106 = 106), which is correct. Also, the angle ((-y - 6)^\circ = -(-80) - 6 = 80 - 6 = 74^\circ), which matches. The angle ((-z + 8)^\circ = -(-98) + 8 = 98 + 8 = 106^\circ), which matches (10x + 6 = 106^\circ). So that works.

Wait, but let's re-examine the steps:

1. For (x): Consecutive angles are supplementary. So (74 + (10x + 6) = 180)

(10x + 80 = 180)

(10x = 100)

(x = 10)

2. For (y): Opposite angles are equal. So (-y - 6 = 74)

(-y = 80)

(y = -80)

3. For (z): Opposite angles are equal. So (-z + 8 = 10x + 6 = 106)

(-z = 98)

(z = -98)

Yes, that seems correct.

Answer:

(x = 10), (y = -80), (z = -98)