Solving Equations with Variables on Both Sides Using Congruent Triangles

🔑 Key Concepts

  • Congruent triangles have equal corresponding side lengths.
  • The congruence statement \(\triangle XYZ\cong\triangle QRS\) identifies the correspondence \(X\leftrightarrow Q\), \(Y\leftrightarrow R\), and \(Z\leftrightarrow S\).
  • Set expressions for corresponding sides equal to create equations.
  • Use inverse operations to solve for each variable.
  • Substitute the variable values into the side-length expressions to find the actual lengths.

✏️ Worked Examples — Finding Side Lengths of Congruent Triangles

🧠 Math Vocabulary

  • Congruent figures: Figures that have the same size and shape.
  • Corresponding sides: Sides that occupy matching positions in congruent figures.
  • Congruence statement: A statement that names congruent figures in corresponding order.
  • Variable expression: A mathematical expression containing a variable, such as \(4a-2\).
  • Inverse operations: Operations that undo one another and help isolate a variable.
  • Substitution: Replacing a variable with its known value to evaluate an expression.

💡 Main Idea

In this lesson, students use congruent triangles to write and solve equations. Because corresponding sides of congruent figures have equal lengths, the expressions representing those sides can be set equal. After solving for each variable, students substitute the values into the side-length expressions to determine the actual lengths of the corresponding sides.

📚 What You Should Already Know

Students should understand how to solve one-step and two-step equations using inverse operations. They should also know that congruent figures have matching side lengths, how to identify corresponding vertices from a congruence statement, and how to evaluate algebraic expressions by substitution.

🚀 What Comes Next

Next, students can use congruent figures to solve more complex equations involving angle measures, perimeter, and multi-step expressions. These skills support later work with triangle congruence theorems, geometric proofs, and algebraic modeling.

🧩 Embed This Video in Your LMS

Teachers can embed this YouTube video directly into Canvas, Google Classroom, Schoology, or another learning platform. Click below to show the embed code.