in the illustrated embodiments , a typical box section formed from wooden boards and with miter joints according to the present invention is shown in fig1 . such hollow box and beam configurations are typical of the uses for the miter joints and mitered boards of this invention , and one or two of the boards may be omitted to form u - or l - sections as needed for diverse purposes . the same boards may be combined as in fig3 a to form flat panel surfaces , and the flat and angular assembly configurations may be combined for more complicated boxing - in purposes . in detail , boards 10 and 12 , wide and narrow respectively , as seen in fig1 and 2a , are typically formed from wood , though other building materials such as plastic , particle board , sheet rock , or otherwise could be used . the miter joints used to join these boards at their edges are characterized by the tongues 14 and grooves 16 , which in simplest form would have plain 90 ° v - shape in cross - section , as shown in fig2 a , for right angle miter joints ( where the miter joint angle a equals 90 °). it is to be understood that the tongue 14 must be slightly smaller than the groove 16 , so that the miter surfaces 18 and 20 on tongued and grooved board edges respectively , may fit fully in contact with each other at assembly . within normal manufacturing practices , this slightly loose fit between tongue and groove provides sufficient alignment capability for assembling the miter joints with neat appearance and sound construction . it is advantageous that the same boards as shown in fig1 and 2a should be capable of assembly into aligned , flat , or paneled configuraton as shown in fig3 a ; therefore , the tongues 14 and the grooves 16 are located centered in the width of their respective mitered surfaces 18 and 20 . also , the tongues 14 and grooves 16 are each in the form of isosceles triangles constructed at the widthwise centers of surfaces 18 and 20 as defined by the dimensions w and w / 2 in fig3 a for example , that is , the tongues and grooves are symmetrical about these centers for interchangeable assembly in angular and flat board relationships . it is frequently desirable for one of the boards of a miter assembly to be assembled or disassembled after the other board or boards have been permanently fixed in place , and it is desirable for this purpose to move the loose board into and out of assembly with a single rectilinear movement perpendicular to the flat sides of the board , as indicated by the board 10 &# 39 ; as shown removed in broken lines in fig2 a . to permit such movement , and at the same time retain the maximum alignment capabilities of the tongue and groove , the tongue and groove included angle b is preferably 90 °, because an angle b less than 90 ° would interfere with the free assembly just mentioned , and an angle greater than 90 ° would provide less than the optimum alignment capability . as a general rule , the tongue and groove included angles b must be equal to or greater than the miter joint angle a , and the miter surface angle c must equal half the miter joint angle a . conventional tongues and grooves frequently have essentially parallel sides , which when assembled together have a locking characteristic rendering them subject to damage ( particularly by splitting or breaking off the tongue or a board edge adjacent the groove ) when any misalignment occurs during assembly of such tongues and grooves . it is notable that the v - tongues and grooves of the present invention have none of this self - locking characteristic and are damage - free from that standpoint . however , in order that the board edge 22 at the acute angle c side of miter surface 20 , be of sufficient thickness for damage - free use , it is desirable that the depth s of the grooves 16 be approximately 20 % of the thickness t of the basic shape of the miter joint boards . the just - mentioned basic shape of the miter joint boards refers to the isosceles trapezoid cross - section of the boards 10 and 12 as shown in fig2 a , the long and shorter , or outer and inner , parallel sides 24 and 26 respectively of the trapezoid being formed by the real or imaginary lines joining the wider spaced and closer spaced ends of the isosceles miter sides or surfaces 18 and 20 . the miter joint boards may have any suitable configuration in modification of their outer or inner sides 24 and 26 , respectively , such as the fancy side 28 , without lessening their utility . as a practical matter , it may be desirable to depart from the plain v - tongue and - groove configurations as shown in fig2 a for ease of manufacturing or other reasons , and various other configurations might well be used to the same effect . a variety of modified , but effectively v - shaped tongue and groove arrangements are shown in fig4 a - 4e . it will be understood that these joints are shown slightly separated for clarity , but in use , the mitered surfaces 18 and 20 would be placed in intimate contact . for all the configurations of fig4 which are intended for use in 90 ° miter joints , the angles b should in each case be effectively 90 ° or more , just as for fig2 a . the &# 34 ; effective &# 34 ; tongue and groove angle is that at which the mating surfaces contact each other , and as shown in fig4 d , the effective angle may be determined by the shape of one member , as at groove 30 in this case , where tongue 32 has no obvious 90 ° included angle b . otherwise , so long as tongue and groove are non - interfering when the miter joint boards are assembled in both angular and aligned configurations , their shapes are immaterial except at the contact surfaces 34 and 36 respectively of the tongues and grooves of fig4 . it is conceivable that multiple tongues and grooves might be formed lengthwise of the miter surface with equal utility , and without departing from the present concept . other miter joint angles such as d in fig2 b or g in fig2 c may be desirable . for any miter joint angle , the tongue and groove angle should be equal or greater , and the miter surface angle such as f and i should equal 1 / 2 of the miter angle such as d and g . thus , in fig2 b , 2f = d ≦ e , and in fig2 c , 2i = g ≦ h . as a practical matter , the various angles called out should nominally be a = 90 °, b = 90 °, c = 45 °; d = 120 °, e = 120 °, f = 60 °; g = 135 °, h = 135 °, i = 67 . 5 °. as illustrated by the broken lines shown in fig2 b , a single side 38 of the hollow hexagonal assembly may be removed without disturbing the other sides , as indicated by 38 &# 39 ;. a similar condition prevails in the hollow octagonal assembly of fig2 c . also , as shown in fig3 tongued and grooved miter joint boards according to the above disclosure may be assembled in aligned relationship , and the miter boards may always be freely removed from the assembly in the direction of the wider flat side thereof , and any miter board prepared for a 90 ° or greater miter joint angle may be removed edgewise from the end position in such an assembly . the various tongue and groove configurations of fig4 or others , may equally well be applied to tongues and grooves of various effective included angles suitable for other miter joint angles . the invention as disclosed above provides means by which prefabricated miter joint boards may be assembled in a variety of configurations , and by which the joints may be readily held in registered alignment for permanent assembly by means of nailing , gluing , screwing , stapling , or otherwise fastening the joints , without problems of the miter surfaces slipping out of register . this is particularly important for lengthy miter joints , where warpage of the boards causes special problems both with alignment and with the tendency of conventional tongue and groove construction to split off tongues or board edges when such tongues and grooves are driven into engagement . the present invention has been described in detail above for purposes of illustration only and is not intended to be limited by this description or otherwise to exclude any variation or equivalent arrangement that would be apparent from , or reasonably suggested by , the foregoing disclosure to the skill of the art .