MathPlotAI Logo MathPlotAI

SAS Congruence by Salvador Martinez

SAS Congruence by Salvador Martinez
Loading the math board and drawing, please wait…
2017-05-24 14:24
TriangleVectorGeometric Construction
This example illustrates the SAS (Side-Angle-Side) Congruence Theorem. It shows how to construct congruent triangles given two sides and the included angle. Commands used include Polygon, Angle, Mirror, PolyLine, Segment, Vector, and Translate. Suitable for middle school and above students to learn about geometric construction and the concept of congruent triangles.

GeoGebra Commands & Parameters

  1. Polygon(A, B, C) → poly1, c, a, b
  2. Angle(B, A, C) → α
  3. Mirror(A, D) → A'
  4. Mirror(B, D) → B'
  5. Mirror(C, D) → C'
  6. Polygon(A', B', C') → poly1', c', a', b'
  7. Angle(C', A', B') → β
  8. PolyLine((-0.44744730521427306, 6.120370737084371), (NaN, NaN), (0.1267545700145157, 7.161111635936551), (0.23441742161991358, 7.125224018734753), (0.27030503882171286, 7.0175611671293545), (0.3061926560235122, 6.909898315523956), (0.3420802732253115, 6.802235463918558), (0.37796789042711076, 6.69457261231316), (0.4497431248307094, 6.586909760707762), (0.48563074203250867, 6.479246909102364), (0.5574059764361072, 6.371584057496966), (0.5932935936379066, 6.2639212058915685), (-0.44744730521427306, 6.120370737084371), (0.6291812108397058, 6.15625835428617), (0.6650688280415051, 6.048595502680772), (0.7009564452433045, 5.940932651075374), (NaN, NaN), (-0.19623398480167797, 6.479246909102364), (NaN, NaN), (-0.19623398480167797, 6.479246909102364), (-0.016795898792681486, 6.479246909102364), (0.1267545700145157, 6.479246909102364), (0.27030503882171286, 6.479246909102364), (-0.33978445360887516, 6.228033588689769), (0.41385550762891005, 6.479246909102364), (0.5574059764361072, 6.479246909102364), (NaN, NaN), (-2.851917657734826, -0.0881870388269147), (NaN, NaN), (-2.851917657734826, -0.0881870388269147), (-2.851917657734826, -0.23173750763411205), (-2.8160300405330267, -0.3752879764413094), (-2.780142423331227, -0.5547260624503061), (-2.744254806129428, -0.6623889140557041), (-0.3038968364070759, 6.335696440295167), (-2.7083671889276286, -0.8059393828629015), (-2.6724795717258294, -0.9136022344682995), (-2.6724795717258294, -1.0571527032754968), (NaN, NaN), (-2.887805274936625, -0.2676251248359114), (NaN, NaN), (-2.887805274936625, -0.2676251248359114), (-2.851917657734826, -0.0881870388269147), (-2.744254806129428, -0.01641180442331602), (-2.63659195452403, 0.019475812778483318), (-0.23212160200347726, 6.479246909102364), (-2.528929102918632, 0.09125104718208199), (-2.385378634111435, 0.09125104718208199), (-2.385378634111435, -0.052299421625115354), (-2.385378634111435, -0.23173750763411205), (-2.385378634111435, -0.3752879764413094), (-2.385378634111435, -0.5188384452485068), (-2.4212662513132344, -0.6265012968539048), (-2.493041485716833, -0.7341641484593028), (-2.63659195452403, -0.7341641484593028), (-2.528929102918632, -0.6982765312575034), (-0.16034636759987866, 6.622797377909562), (-2.4212662513132344, -0.6623889140557041), (-2.277715782506037, -0.6623889140557041), (-2.1700529309006393, -0.6982765312575034), (-2.062390079295241, -0.8059393828629015), (-2.062390079295241, -0.9494898516700988), (-2.062390079295241, -1.093040320477296), (-2.13416531369884, -1.2007031720826942), (-2.241828165304238, -1.2365907892844934), (-2.349491016909636, -1.308366023688092), (-2.493041485716833, -1.308366023688092), (-0.08857113319628009, 6.802235463918558), (-2.63659195452403, -1.308366023688092), (-2.6724795717258294, -1.2007031720826942), (-2.744254806129428, -1.093040320477296), (-2.744254806129428, -0.9494898516700988), true, (0.01909171840911781, 6.945785932725755), (0.054979335610917106, 7.053448784331153)) → stroke1
  9. PolyLine((4.935695275055621, 0.019475812778483318), (NaN, NaN), (4.2179429310196355, -0.2676251248359114), (4.182055313817836, -0.3752879764413094), (4.182055313817836, -0.5188384452485068), (4.2179429310196355, -0.6265012968539048), (4.289718165423234, -0.7341641484593028), (4.397381017028632, -0.7700517656611021), (4.50504386863403, -0.8059393828629015), (4.648594337441226, -0.8059393828629015), (4.792144806248424, -0.8059393828629015), (4.899807657853821, -0.7700517656611021), (4.935695275055621, 0.019475812778483318), (5.007470509459219, -0.7341641484593028), true, (4.828032423450223, 0.05536342998028266), (4.7203695718448255, 0.09125104718208199), (4.576819103037629, 0.12713866438388133), (4.433268634230431, 0.12713866438388133), (4.325605782625034, 0.09125104718208199), (4.289718165423234, -0.01641180442331602), (4.2179429310196355, -0.12407465602871404)) → stroke2
  10. PolyLine((13.692273872294649, -0.7341641484593028), (NaN, NaN), (13.835824341101846, -1.918455516118681), (13.871711958303646, -2.026118367724079), (13.871711958303646, -2.1696688365312764), (NaN, NaN), (13.62049863789105, -0.7341641484593028), (NaN, NaN), (13.62049863789105, -0.7341641484593028), (13.728161489496449, -0.6982765312575034), (13.907599575505445, -0.6982765312575034), (14.087037661514442, -0.6982765312575034), (13.692273872294649, -0.7341641484593028), (14.230588130321639, -0.6982765312575034), (14.338250981927036, -0.7700517656611021), (14.374138599128836, -0.8777146172665001), (14.445913833532435, -0.9853774688718981), (14.517689067936033, -1.093040320477296), (14.517689067936033, -1.2365907892844934), (14.517689067936033, -1.3801412580916907), (14.517689067936033, -1.5595793441006875), (14.517689067936033, -1.703129812907885), (14.445913833532435, -1.8466802817150823), (13.692273872294649, -0.8777146172665001), (14.374138599128836, -1.9543431333204802), (14.266475747523438, -2.0620059849258783), (14.19470051311984, -2.1696688365312764), (14.051150044312642, -2.3132193053384738), (13.979374809909043, -2.4208821569438714), (13.871711958303646, -2.4926573913474703), (13.728161489496449, -2.4926573913474703), (NaN, NaN), (16.958047037658385, 5.689719330662779), (NaN, NaN), (13.728161489496449, -0.9853774688718981), (16.958047037658385, 5.689719330662779), (16.958047037658385, 5.546168861855581), (16.958047037658385, 5.402618393048384), (16.958047037658385, 5.259067924241187), (16.958047037658385, 5.11551745543399), (16.958047037658385, 4.971966986626792), (16.993934654860183, 4.864304135021394), (17.10159750646558, 4.828416517819595), (17.24514797527278, 4.828416517819595), (17.388698444079978, 4.828416517819595), (13.728161489496449, -1.1289279376790955), (17.532248912887173, 4.828416517819595), (17.67579938169437, 4.828416517819595), (NaN, NaN), (17.065709889263783, 5.187292689837588), (NaN, NaN), (17.065709889263783, 5.187292689837588), (17.17337274086918, 5.259067924241187), (17.31692320967638, 5.259067924241187), (17.460473678483574, 5.259067924241187), (NaN, NaN), (13.728161489496449, -1.3442536408898915), (16.993934654860183, 5.689719330662779), (NaN, NaN), (16.993934654860183, 5.689719330662779), (17.137485123667382, 5.689719330662779), (17.281035592474577, 5.689719330662779), (17.424586061281776, 5.689719330662779), (17.568136530088974, 5.689719330662779), true, (13.728161489496449, -1.523691726898888), (13.764049106698248, -1.703129812907885), (13.799936723900046, -1.8107926645132828)) → stroke3
  11. PolyLine((9.134546487666139, 5.04374222103039), (NaN, NaN), (9.134546487666139, 6.048595502680772), (9.278096956473336, 6.048595502680772), (9.385759808078733, 6.084483119882571), (NaN, NaN), (9.06277125326254, 5.582056479057381), (NaN, NaN), (9.06277125326254, 5.582056479057381), (9.206321722069736, 5.582056479057381), (9.349872190876935, 5.582056479057381), (9.457535042482332, 5.546168861855581), (9.134546487666139, 5.04374222103039), true, (9.09865887046434, 5.1514050726357885), (9.09865887046434, 5.294955541442986), (9.06277125326254, 5.438506010250183), (9.02688363606074, 5.546168861855581), (9.02688363606074, 5.689719330662779), (9.02688363606074, 5.833269799469976), (9.02688363606074, 5.976820268277174)) → stroke4
  12. Polygon(E, F, G) → poly2, g, e, f
  13. Segment(H_1, I_1) → h
  14. Mirror(E, h) → H
  15. Mirror(F, h) → I
  16. Mirror(G, h) → J
  17. Polygon(H, I, J) → poly2', g', e', f'
  18. PolyLine((-3.237544781750333, -18.97841390281207), (NaN, NaN), (-1.8130158448660982, -20.235351200062865), (-1.8130158448660982, -20.570534479329744), true, (-3.237544781750333, -18.97841390281207), (-2.9023615024834544, -18.97841390281207), (-2.5671782232165756, -18.97841390281207), (-2.231994943949697, -18.97841390281207), (-1.896811664682818, -18.97841390281207), (-1.8130158448660982, -19.22980136226223), (-1.8130158448660982, -19.564984641529108), (-1.8130158448660982, -19.900167920795987)) → stroke5
  19. PolyLine((21.314630424548543, -22.16265505584742), (NaN, NaN), (22.655363541616058, -20.8219219387799), (22.990546820882937, -20.8219219387799), true, (21.314630424548543, -22.16265505584742), (21.314630424548543, -21.82747177658054), (21.314630424548543, -21.492288497313663), (21.314630424548543, -21.15710521804678), (21.314630424548543, -20.8219219387799), (21.649813703815422, -20.8219219387799), (21.9849969830823, -20.8219219387799), (22.32018026234918, -20.8219219387799)) → stroke6
  20. Polygon(K, L, M) → poly3, m, k, l
  21. Angle(K, L, M) → γ
  22. Vector(N_1, O_1) → u
  23. Translate(K, u) → N
  24. Translate(L, u) → O
  25. Translate(M, u) → P
  26. Polygon(N, O, P) → poly3', m', k', l'
  27. Angle(N, O, P) → δ