Question

in triangle abc, the size of angle b is 4 times the size of angle a, and the size of angle c is 16° less than 5 times the size of angle a. find the size of the angles.

Explanation:

Step1: Define variables for angles

Let the measure of angle \( A \) be \( x \) degrees. Then angle \( B = 4x \) degrees and angle \( C=(5x - 16) \) degrees.

Step2: Use triangle angle sum property

The sum of angles in a triangle is \( 180^{\circ} \), so \( x+4x+(5x - 16)=180 \).

Step3: Simplify the equation

Combine like terms: \( 10x-16 = 180 \).

Step4: Solve for \( x \)

Add 16 to both sides: \( 10x=180 + 16=196 \). Then divide by 10: \( x=\frac{196}{10}=19.6^{\circ} \).

Step5: Find other angles

Angle \( B = 4x=4\times19.6 = 78.4^{\circ} \). Angle \( C=5x-16=5\times19.6-16 = 98 - 16=82^{\circ} \).

Answer:

Angle \( A = 19.6^{\circ} \), Angle \( B = 78.4^{\circ} \), Angle \( C = 82^{\circ} \)