Problem 20 - Algebra A Combined Function

Question

The flag of Brazil contains a parallelogram. One angle of the parallelogram is \(15^{\circ}\) less than twice the measure of the angle next to it. Find the measure of each angle of the parallelogram. (Hint: Recall that opposite angles of a parallelogram have the same measure and that the sum of the measures of the angles is \(360^{\circ} .\) )

Step-by-Step Solution

Answer

Each angle of the parallelogram measures \(65^{\circ}\) and \(115^{\circ}\).

Step 1: Define the Variables

Let \( x \) be the measure of one angle of the parallelogram. The angle next to it will then be \( y = 2x - 15^{\circ} \). Since opposite angles in a parallelogram are equal, the two other angles will repeat this pair.

Step 2: Use the Sum of Angles in a Parallelogram

The sum of all angles in a parallelogram is \( 360^{\circ} \). Thus, we have: \[ x + y + x + y = 360^{\circ} \] This equation simplifies to: \[ 2x + 2y = 360^{\circ} \] Which further simplifies to: \[ x + y = 180^{\circ} \]

Step 3: Substitute the Expression for y

Substitute the expression for \( y \) from Step 1 into the equation from Step 2: \[ x + (2x - 15^{\circ}) = 180^{\circ} \] This simplifies to: \[ 3x - 15^{\circ} = 180^{\circ} \]

Step 4: Solve for x

Add \(15^{\circ}\) to both sides of the equation: \[ 3x = 195^{\circ} \] Now, divide by 3 to solve for \( x \): \[ x = 65^{\circ} \]

Step 5: Calculate the Other Angle y

Substitute \( x = 65^{\circ} \) back into the expression for \( y \): \[ y = 2(65^{\circ}) - 15^{\circ} \] Which simplifies to: \[ y = 130^{\circ} - 15^{\circ} = 115^{\circ} \]

Step 6: Verify the Solution

Check that the sum of angles satisfies the condition: \[ x + y = 65^{\circ} + 115^{\circ} = 180^{\circ} \] which is correct. Also, verify that all four angles sum to \(360^{\circ}\), which they do.

Key Concepts

Sum of Angles in a Parallelogram

In geometry, knowing the sum of angles in a shape helps us solve for unknown measurements. A parallelogram is a four-sided shape (quadrilateral) that has opposite sides parallel. One key property of a parallelogram is that its internal angles always add up to a total of 360 degrees.

This means, regardless of the specific angles' measurements, they will always sum to this value. This property is particularly useful in solving problems where some angles are unknown, as it allows us to set up equations using the sum of the angles.

Opposite Angles in a Parallelogram

Understanding the nature of opposite angles in a parallelogram is vital. Parallelograms have a special trait where each pair of opposite angles are equal in measure.

For instance, if one angle is known, its opposite must be the same. This means:

• Angle one equals angle three.
• Angle two equals angle four.

This symmetry can simplify calculations since you only need to work out two different angles before knowing all four.

Algebraic Equations

Algebraic equations provide a systematic method to find unknown values, like angles. In the context of a parallelogram, if an angle's expression involves variables, you can use algebra to solve it.

For example, in this exercise, one angle is described as 15 degrees less than twice another angle. When expressed in an equation, it looks like this:

• Let \( x \) be one angle.
• Then the adjacent angle is \( 2x - 15^{\circ} \).

We can solve this using the property that the sum of certain angles is 180 degrees. This approach leads to finding the precise angle measurements.

Angle Relationships

The relationships between angles tell us how angles interact. In a parallelogram, aside from opposite angles being equal, certain sides and their angles create relationships that form equations.

In our scenario, when two adjacent angles are considered, they sum to 180 degrees. This means when you find an angle, the adjacent one completes the straight line (180 degrees).

• If \( x + y = 180^{\circ} \)
• You can easily determine one angle if you know the other.

Understanding these relationships helps in constructing relevant equations to find unknowns and solve problems efficiently.