Question

In \(\triangle A B C, \overline{A B} \cong \overline{A C}\). If \(m \angle B=5 x-7\) and \(m \angle C=4 x+2\), find \(m \angle B\) and \(m \angle C\).

Short answer

\( m \angle B = 38\) and \( m \angle C = 38\).

Step-by-step solution

Understand the Properties of an Isosceles Triangle

An isosceles triangle has two sides of equal length, and the angles opposite those sides are equal. In triangle \( \triangle ABC \), \( \overline{AB} \cong \overline{AC} \) implies \( m \angle B = m \angle C \).

Set Up the Equation

Since \( m \angle B = m \angle C \), you can set the measures equal to each other: \[ 5x - 7 = 4x + 2 \]

Solve for x

Rearrange the equation to isolate \( x \). Start by subtracting \( 4x \) from both sides:\[ 5x - 4x - 7 = 4x - 4x + 2 \]Which simplifies to:\[ x - 7 = 2 \]Then, add 7 to both sides:\[ x = 9 \]

Find \( m \angle B \) and \( m \angle C \)

Substitute \( x = 9 \) back into the equations for \( m \angle B \) and \( m \angle C \):\[ m \angle B = 5(9) - 7 = 45 - 7 = 38 \]\[ m \angle C = 4(9) + 2 = 36 + 2 = 38 \]

Verify the Solution

Check that both angles are equal, confirming the properties of an isosceles triangle. Since \( m \angle B = 38 \) and \( m \angle C = 38 \), the calculation is correct.

Key concepts

Triangle Properties

Triangles are fascinating shapes with unique characteristics. A triangle with two sides of equal length is known as an isosceles triangle. In such triangles, the angles opposite the equal sides are also equal. This means if you know two sides of a triangle are equal, you can immediately conclude that the angles opposite those sides are equal.
In our exercise, triangle \( \triangle ABC \) is an isosceles triangle because sides \( \overline{AB} \) and \( \overline{AC} \) are congruent. Therefore, angle B is equal to angle C, or \( m \angle B = m \angle C \). Recognizing these properties helps simplify many problems involving triangles, guiding us towards setting up important equations for solving various triangle-based exercises.

Angle Calculation

Calculating angles in an isosceles triangle involves using the properties we've identified. We set up an equation reflecting that the angles in question are equal. In the given example, \( m \angle B \) is expressed as \( 5x - 7 \), and \( m \angle C \) as \( 4x + 2 \). Since these angles are equal, we can equate them:
\[ 5x - 7 = 4x + 2 \]This critical step ties the abstract algebraic expressions back to the geometric properties of the triangle. Once the equation is established, it sets the stage for solving it, revealing the actual values of the angles. These values offer the concrete numerical insights needed to fully understand the triangle's characteristics.

Equation Solving

Solving equations is a fundamental skill in mathematics, crucial for finding the unknown variable \( x \) in our problem. We start with the equation \( 5x - 7 = 4x + 2 \). The goal is to isolate \( x \), transforming the equation into a simpler form.
First, subtract \( 4x \) from both sides for clarity:\[ 5x - 4x - 7 = 4x - 4x + 2 \]This simplifies to \( x - 7 = 2 \).
Next, add 7 to each side to solve for \( x \):\[ x = 9 \]Once we find \( x \), we can substitute it back into the expressions for angles B and C. This calculation gives us \( m \angle B = 38 \) and \( m \angle C = 38 \), confirming our understanding of both algebraic manipulation and geometric principles. Validating the solution ensures that our logic aligns with the properties of isosceles triangles, providing a complete picture of the triangle's internal angles.