Question

what are the measures of ∠abc and ∠bca? use the number pad to enter your answers in the boxes. m∠abc = m∠bca =

Explanation:

Step1: Use exterior - angle property

The exterior - angle of a triangle is equal to the sum of the two non - adjacent interior angles. So, \(106=(3x + 3)+(4x-2)\).

Step2: Simplify the equation

Combine like terms: \(106 = 3x+3 + 4x-2\), which simplifies to \(106=7x + 1\).

Step3: Solve for \(x\)

Subtract 1 from both sides: \(106 - 1=7x\), so \(105 = 7x\). Then divide both sides by 7: \(x=\frac{105}{7}=15\).

Step4: Find \(m\angle ABC\)

Substitute \(x = 15\) into the expression for \(\angle ABC\): \(m\angle ABC=3x + 3=3\times15+3=45 + 3=48\).

Step5: Find \(m\angle BCA\)

Substitute \(x = 15\) into the expression for \(\angle BCA\): \(m\angle BCA=4x-2=4\times15-2=60 - 2=58\).

Answer:

\(m\angle ABC = 48\)
\(m\angle BCA = 58\)