Finding Angles of Triangle

Video Explanation

Question

In a triangle ABC: \[ \angle A = x^\circ,\quad \angle B = (3x – 2)^\circ,\quad \angle C = y^\circ \] Also, \[ \angle C – \angle B = 9^\circ \] Find the three angles.

Solution

Step 1: Concept

Sum of angles in a triangle = \(180^\circ\)

—

Step 2: Form Equations

Angle sum:

\[ x + (3x – 2) + y = 180 \]

\[ 4x + y = 182 \quad (1) \]

Given condition:

\[ y – (3x – 2) = 9 \]

\[ y – 3x + 2 = 9 \]

\[ y = 3x + 7 \quad (2) \]

—

Step 3: Solve Linear Equations

Substitute (2) into (1):

\[ 4x + (3x + 7) = 182 \]

\[ 7x + 7 = 182 \]

\[ 7x = 175 \Rightarrow x = 25 \]

Then:

\[ y = 3(25) + 7 = 82 \]

—

Step 4: Find Angles

\[ \angle A = 25^\circ \]

\[ \angle B = 3x – 2 = 75 – 2 = 73^\circ \]

\[ \angle C = 82^\circ \]

—

Conclusion

\[ \angle A = 25^\circ,\quad \angle B = 73^\circ,\quad \angle C = 82^\circ \]

Verification

Sum: \(25 + 73 + 82 = 180^\circ\) ✔

Difference: \(82 – 73 = 9^\circ\) ✔