1Cademy - Solving a Right Triangle Angle Problem Using a System of Equations by Substitution

Learn Before

Problem-Solving Strategy for Systems of Linear Equations

Example

Solving a Right Triangle Angle Problem Using a System of Equations by Substitution

Apply the seven-step problem-solving strategy for systems of linear equations to a geometry word problem involving the angles of a right triangle, using substitution to solve the resulting system.

Problem: The measure of one of the small angles of a right triangle is ten more than three times the measure of the other small angle. Find the measures of both angles.

Read the problem and draw a right triangle with the two unknown acute angles labeled $a$ and $b$ .
Identify: The measures of the two small (acute) angles.
Name: Let $a$ = the measure of the first angle and $b$ = the measure of the second angle.
Translate into a system of equations. The relationship between the angles gives the first equation, and the triangle angle sum property (with one angle equal to $90°$ ) gives the second:

$\left\{\begin{array}{l} a = 3b + 10 \\ a + b + 90 = 180 \end{array}\right.$

Solve using substitution. Because the first equation is already solved for $a$ , substitute $3b + 10$ for $a$ in the second equation:

$(3b + 10) + b + 90 = 180$

Combine like terms: $4b + 100 = 180$ . Subtract $100$ : $4b = 80$ . Divide by $4$ : $b = 20$ .

Substitute $b = 20$ into the first equation: $a = 3(20) + 10 = 60 + 10 = 70$ .

Check: Do angles measuring $70°$ , $20°$ , and $90°$ sum to $180°$ ? $70 + 20 + 90 = 180$ ✓. Is $70$ ten more than three times $20$ ? $3(20) + 10 = 70$ ✓.
Answer: The measures of the small angles are $20°$ and $70°$ .

This example extends the systems approach from abstract number problems to triangle geometry. The two equations arise from two different facts: the verbal relationship between the angles ( $a = 3b + 10$ ) and the triangle angle sum property applied to a right triangle ( $a + b + 90 = 180$ ). Because the first equation already expresses $a$ in terms of $b$ , substitution can begin immediately. Compare this with the single-variable approach to similar right-triangle problems, where both unknown angles must be expressed through one variable before writing a single equation — the systems method avoids that extra translation step.

Updated 2026-04-21

Contributors are:

Who are from:

References

Learn Before

Related

Learn After