Question

find the measure of \\( \angle a b c \\).

Explanation:

Step1: Find the measure of ∠ACB

Since ∠ACB and ∠ACD are supplementary angles (they form a linear - pair), we use the formula \(∠ACB + ∠ACD=180^{\circ}\).
Given \(∠ACD = 145^{\circ}\), then \(∠ACB=180^{\circ}-145^{\circ}=35^{\circ}\).

Step2: Use the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). For \(\triangle ABC\), we have \(∠A + ∠B+∠ACB = 180^{\circ}\).
We know \(∠A = 95^{\circ}\), \(∠B = 2x^{\circ}\), and \(∠ACB = 35^{\circ}\). Substitute these values into the formula: \(95^{\circ}+2x^{\circ}+35^{\circ}=180^{\circ}\).
First, simplify the left - hand side: \(130^{\circ}+2x^{\circ}=180^{\circ}\).
Then, solve for \(x\): \(2x^{\circ}=180^{\circ}-130^{\circ}\), so \(2x^{\circ}=50^{\circ}\), and \(x = 25\).

Step3: Calculate the measure of ∠ABC

Since \(∠ABC=2x^{\circ}\) and \(x = 25\), then \(∠ABC=2\times25^{\circ}=50^{\circ}\).

Answer:

\(50\)