In Draw, how can I join two lines of given length with angle of 95.9 degrees?

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As ebot and LeroyG have suggested, i would strongly recommend using Lines. Connectors don't remember their previous angle and aren't constrained by by the Shift key.

My usual method would be to draw one line roughly (constrained or not) then right-click on it and select Position and Size (or F4) . Under the tab Position and Size I would enter the appropriate dimensions and select the appropriate Base point position, OK out.

Copy and paste it in position, right-click Position and Size again and enter the appropriate dimensions for the second line, I would then open the Rotation tab and enter the rotation angle of 95.9 degrees (264.1 deg for clockwise) and carefully select the pivot point such that it will pivot at the point that you want the join, OK out. You should have two lines of length you want at the angle you set with the same origin. Select both and Group

Addition 1: Easier solution: Select both lines, choose menu Shape-Align-Left or Right, then choose menu Shape-Align-Bottom or Top.

You can choose menu Shape-Line instead of Connector.

To rotate the line, select the line and choose menu Format-Position and Size… (or just press F4), and in the Rotation tab give a value to Angle:, OK.

Edit 1: To move both end points: Choose menu Edit-Points (or just press F8), select both points of the line (click and shift+click, or drag a selection frame around both points). Drag one point to join the other line.

Addition 2: No need to select both points, just drag the line to join the other.

To join both lines will be easier if Snap to Object Border or Snap to Object Points are active (to do that choose menu Edit-Snap Guides).

Addition 3: After selecting both lines and choosing menu Shape-Group-Group, you can change individual format of each line. To do that: select the group, press F3 (or choose menu Shape-Group-Enter Group), select one line, make your changes and press Ctrl+F3 (or choose menu Shape-Group-Exit Group).

Addition 4: Selecting both lines and choosing menu Shape-Combine would convert both lines in a polygon. To partialy revert this, choose menu Shape-Split (they will be no more lines, but polygons).

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REM  *****  BASIC  *****

REM There is a conflict: The standard orienatation of the X-Y-axes
REM for a DrawPage in LibreOffice is rightward-downward.
REM The usual standard for geometric drwaings is rightward-upward.
REM Left-turning angles are counted positive in usual geometric drwaings,
REM and should therefore be counted negative in a DrwaPage. 
REM I did NOT accept this consequence, and neededed therefore to apply a ChgSgn
REM to values of the kind Delta-Y. 

Sub drawAngle(Optional pDp as Object, _
              Optional pX0, Optional pY0, _
              Optional pL1, Optional pPhi1, _
              Optional pL2, Optional pAlpha)
If IsMissing(pDp)    Then pDp    = ThisComponent.DrawPages(0)
If IsMissing(pX0)    Then pX0    = 15000
If IsMissing(pY0)    Then pY0    = 25000
If IsMissing(pL1)    Then pL1    = 10000
If IsMissing(pPhi1)  Then pPhi1  = Pi()
If IsMissing(pL2)    Then pL2    = 20000
If IsMissing(pAlpha) Then pAlpha = 95.9 * (Pi() / 180)
REM The default values are arbitrary. For the angle the value 95.6° is taken from the question.
phi2 = pPhi1 - Pi() + pAlpha
polyLsh = ThisComponent.createInstance("com.sun.star.drawing.PolyLineShape")
pDp.add(polyLsh)
Dim angleSequence(2) As New com.sun.star.awt.Point 
angleSequence(0).X = pX0
angleSequence(0).Y = pY0
angleSequence(1).X = angleSequence(0).X + pL1 * COS(pPhi1)
angleSequence(1).Y = angleSequence(0).Y - pL1 * SIN(pPhi1)
angleSequence(2).X = angleSequence(1).X + pL2 * COS(phi2)
angleSequence(2).Y = angleSequence(1).Y - pL2 * SIN(phi2)
polyLsh.Polygon = angleSequence
End Sub

Example: C:\fakepath\ask258072angleByPolyLineShape.odg

Try the CADLO extension. It adds some (simple) CAD functionality to the Draw application. You can draw two crossed lines with the desired angles, and then you can crop the irrelevant part of the lines by the cut feature! image description