4. The length of overline (CB) is [(A)1(B)5(C)19] ,and mangle BDA is [ 490 (B)67 (C) 23]^circ

To solve this problem, we need to use the information provided in the picture details and the question to find the values of \(\overline{CB}\) and \(m\angle BDA\).Step 1: Determine the value of x.We know that the sum of the angles in a triangle is 180 degrees. Therefore, in triangle ABC, we have:\(\angle CAB + \angle BCA + \angle ABC = 180^\circ\)Given that \(\angle CAB = 9x - 4\), \(\angle BCA = 4x + 1\), and \(\angle ABC = 90^\circ\) (since it's a right triangle at A), we can write:\((9x - 4) + (4x + 1) + 90 = 180\)Combining like terms:\(13x - 3 + 90 = 180\)\(13x + 87 = 180\)Subtract 87 from both sides:\(13x = 93\)Divide by 13:\(x = 93 / 13\)\(x = 7.15\) (approximately)Step 2: Find the length of \(\overline{CB}\).Since \(\angle BCA = 4x + 1\) and we have found that \(x \approx 7.15\), we can calculate \(\angle BCA\):\(\angle BCA = 4(7.15) + 1\)\(\angle BCA = 28.6 + 1\)\(\angle BCA = 29.6^\circ\) (approximately)However, we are given that \(\angle CBD = 23^\circ\). Since \(\angle BCA\) and \(\angle CBD\) are part of the same larger angle \(\angle BCD\), we can find \(\angle BCD\) by adding them together:\(\angle BCD = \angle BCA + \angle CBD\)\(\angle BCD = 29.6^\circ + 23^\circ\)\(\angle BCD = 52.6^\circ\) (approximately)Since \(\angle BCD\) is not a right angle, \(\overline{CB}\) cannot be the hypotenuse of a right triangle, and thus, \(\overline{CB}\) cannot be the longest side of triangle ACD. Therefore, the length of \(\overline{CB}\) cannot be determined solely by the angles given. We need additional information to find the exact length of \(\overline{CB}\). However, the question provides possible lengths as [1, 5, 19], but without additional information, we cannot determine which one is correct.Step 3: Find \(m\angle BDA\).We are given that \(\angle CBD = 23^\circ\). Since \(\angle BDA\) is an extension of \(\angle CBD\), and a straight line measures 180 degrees, we can find \(m\angle BDA\) by subtracting \(\angle CBD\) from 180 degrees:\(m\angle BDA = 180^\circ - \angle CBD\)\(m\angle BDA = 180^\circ - 23^\circ\)\(m\angle BDA = 157^\circ\)However, none of the given options for \(m\angle BDA\) match 157 degrees. It seems there might be a mistake in the question or the options provided. Based on the information given, we cannot match the answer to any of the provided options [90, 67, 23] degrees.Final Answer:The length of \(\overline{CB}\) cannot be determined from the given information. The measure of \(m\angle BDA\) is 157 degrees, which does not match any of the provided options.