Problem 21 - Algebra A Combined Function

Question

The sum of the measures of the angles of a parallelogram is \(360^{\circ} .\) In the parallelogram below, angles \(A\) and \(D\) have the same measure as well as angles \(C\) and \(B\). If the measure of angle \(C\) is twice the measure of angle \(A,\) find the measure of each angle.

Step-by-Step Solution

Answer

Angles A and D each measure 60°, while angles B and C each measure 120°.

Step 1: Understand the Problem

We have a parallelogram with four angles: \( A, B, C, \) and \( D. \) The sum of all angles is \( 360^{\circ}. \) Angles \( A \) and \( D \) are equal, angles \( C \) and \( B \) are equal, and angle \( C \) is twice angle \( A. \) We need to find the measure of each angle.

Step 2: Set Up Equations

Let \( A = D = x^{\circ} \) and \( B = C = y^{\circ}. \) From the problem, \( y = 2x. \) Also, the sum of the angles in the parallelogram is \( 360^{\circ}, \) so \( A + B + C + D = 360. \) This gives the equation \( 2x + 2y = 360. \)

Step 3: Substitute Known Relations

Substitute \( y = 2x \) into the equation \( 2x + 2y = 360 \). This gives us the equation: \[ 2x + 2(2x) = 360 \] simplifying to \[ 2x + 4x = 360. \]

Step 4: Solve for x

Combine like terms: \( 6x = 360 \). Divide both sides by 6 to find \( x: \) \[ x = \frac{360}{6} = 60. \] So, \( A = D = 60^{\circ}. \)

Step 5: Solve for y using x

Since \( y = 2x, \) substitute \( x = 60 \) into \( y = 2x: \) \[ y = 2 \times 60 = 120. \] So, \( B = C = 120^{\circ}. \)

Key Concepts

Sum of Angles

In any parallelogram, the sum of all internal angles is always \(360^{\circ}\). This is because a parallelogram can be divided into two triangles, and since each triangle has angle measures that add up to \(180^{\circ}\), the entire shape's angles sum to \(360^{\circ}\). Knowing this property helps us solve many problems related to finding unknown angle measures in parallelograms. To solve these types of problems, remember:

• Every parallelogram has four angles.
• The sum of these angles always equals \(360^{\circ}\).
• This property is used often in algebraic equations to solve for unknown angle measures.

Breaking down the angles into manageable equations using this property simplifies finding their measures.

Angle Relationships

Understanding the relationship between angles in a parallelogram is crucial in solving angle-related problems. In this specific exercise, you are told that angles \(A\) and \(D\) are equal, and so are angles \(C\) and \(B\). This symmetry is a natural trait of parallelograms. Additionally, angle \(C\) is twice angle \(A\). Here are some key points to consider:

• Opposite angles in a parallelogram are equal. So, \(A = D\) and \(B = C\).
• This equality creates a convenient setup to use algebraic equations to find actual measurements.
• Considering that angle \(C\) is twice \(A\), you immediately set up a proportion (e.g., \(y = 2x\)).

With these relationships, you can effectively use your known angles to figure out the unknown ones by setting up algebraic equations.

Algebraic Equations

Solving problems involving angles in a parallelogram can be simplified using algebraic equations. The strategy often involves setting up an equation based on known angle relationships. In this instance:

• We let \(A = D = x^{\circ}\).
• Since \(C = B\) and \(C = 2A\), we also let \(C = B = y^{\circ}\), giving us the equation \(y = 2x\).
• Knowing the sum of the angles is \(360^{\circ}\), we frame the equation: \(2x + 2y = 360\).

Substituting \(y = 2x\) into the equation gives us \(2x + 4x = 360\), simplifying to \(6x = 360\). Solving for \(x\), you find that \(x = 60^{\circ}\). Using this, calculate \(y = 120^{\circ}\). Consequently, the angles are \(A = 60^{\circ}, D = 60^{\circ}, B = 120^{\circ}, C = 120^{\circ}\). This step-by-step method using algebraic equations is key to unlocking the measures of unknown angles in many geometric problems.