Parallelogram Angle Measures A Step-by-Step Solution

In the world of geometry, parallelograms hold a special place with their unique properties and symmetrical charm. Among these properties, the relationship between their angles stands out as a fundamental concept. Today, we'll dive into a fascinating problem involving a parallelogram-shaped tile, where Jacob is cutting the tile, focusing on the measures of its angles. This exploration will not only enhance our understanding of parallelograms but also sharpen our problem-solving skills. Our task is to determine the two different angle measures of this tile, given that two opposite angles have measures of $(6n-70)^{\circ}$ and $(2n+10)^{\circ}$. Let's embark on this geometric adventure together.

Understanding Parallelograms and Their Angles

Before we delve into the specific problem, let's establish a solid foundation by understanding parallelograms and their angle properties. A parallelogram is a quadrilateral, a four-sided polygon, with two pairs of parallel sides. This seemingly simple definition leads to a cascade of interesting properties, especially when it comes to angles.

Key Angle Properties of Parallelograms:

• Opposite angles are congruent: This means that angles facing each other in a parallelogram have the same measure. This is the cornerstone of our problem-solving approach.
• Consecutive angles are supplementary: Consecutive angles are those that share a side. Supplementary angles add up to $180^{\circ}$. This property will be crucial in finding the second angle measure once we've determined the first.
• The sum of all interior angles is $360^{\circ}$: Like all quadrilaterals, the angles inside a parallelogram add up to $360^{\circ}$. While we won't directly use this in our primary solution, it's a good fact to keep in mind for verifying our answers.

With these properties in mind, we're well-equipped to tackle the problem at hand. We know that Jacob's tile is a parallelogram, and we're given the measures of two opposite angles. Our mission is to find the measures of all the angles, ultimately focusing on the two distinct angle measures present in the parallelogram.

Solving for 'n' and the Angle Measures

Now, let's apply our knowledge of parallelogram properties to the specific problem. We're given that two opposite angles have measures of $(6n-70)^{\circ}$ and $(2n+10)^{\circ}$. Remember, a key property of parallelograms is that opposite angles are congruent. This means the measures of these two angles are equal. This gives us a powerful starting point: we can set up an equation.

Setting up the Equation:

Since the angles are congruent, we can write:

$$(6n - 70) = (2n + 10)$$

This equation is our key to unlocking the value of 'n', which will then allow us to find the angle measures.

Solving for 'n':

Let's solve the equation step-by-step:

1. Subtract $2n$ from both sides:

   $$6n - 2n - 70 = 2n - 2n + 10$$

   $$4n - 70 = 10$$

2. Add 70 to both sides:

   $$4n - 70 + 70 = 10 + 70$$

   $$4n = 80$$

3. Divide both sides by 4:

   $$\frac{4n}{4} = \frac{80}{4}$$

   $$n = 20$$

We've successfully found that $n = 20$. This is a crucial step, as we can now substitute this value back into our original expressions for the angle measures.

Calculating the First Angle Measure:

Let's substitute $n = 20$ into either of the original expressions; we'll use $(2n + 10)^{\circ}$:

$$2(20) + 10 = 40 + 10 = 50$$

So, one of the angle measures is $50^{\circ}$. To double-check, let's substitute $n = 20$ into the other expression, $(6n - 70)^{\circ}$:

$$6(20) - 70 = 120 - 70 = 50$$

As expected, we get the same result. This confirms that our value of 'n' is correct, and one of the angle measures in the parallelogram is indeed $50^{\circ}$.

Finding the Second Angle Measure:

We've found one angle measure, but remember, parallelograms have two distinct angle measures. To find the other, we'll use the property that consecutive angles in a parallelogram are supplementary. This means that the angle we just found ($50^{\circ}$) and its consecutive angle add up to $180^{\circ}$.

Let's call the second angle measure 'x'. We can write the equation:

$$50 + x = 180$$

Subtracting 50 from both sides, we get:

$$x = 180 - 50 = 130$$

Therefore, the second angle measure in the parallelogram is $130^{\circ}$.

The Two Different Angle Measures

After our step-by-step calculations, we've arrived at the answer. The two different angle measures of the parallelogram-shaped tile are $50^{\circ}$ and $130^{\circ}$.

Verification:

Before we conclude, let's quickly verify our answer. In a parallelogram, opposite angles are equal, and consecutive angles are supplementary. We've found angles of $50^{\circ}$ and $130^{\circ}$. If we have a parallelogram with two angles of $50^{\circ}$, there must also be two angles of $130^{\circ}$. Let's check if the sum of all angles is $360^{\circ}$:

$$50 + 50 + 130 + 130 = 360$$

This confirms that our angle measures are correct and consistent with the properties of a parallelogram.

In this exploration, we've successfully navigated the world of parallelograms and their angles. We started with a problem involving Jacob cutting a tile, identified the key properties of parallelograms, set up and solved an equation, and ultimately determined the two different angle measures of the tile. This journey has not only provided us with a solution to a specific problem but also reinforced our understanding of geometric principles. Remember, the beauty of mathematics lies in its ability to connect seemingly disparate concepts and reveal elegant solutions through logical reasoning. Keep exploring, keep questioning, and keep solving!