Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Euler angles wikipedia , lookup
Noether's theorem wikipedia , lookup
Rational trigonometry wikipedia , lookup
Reuleaux triangle wikipedia , lookup
Trigonometric functions wikipedia , lookup
Brouwer fixed-point theorem wikipedia , lookup
History of trigonometry wikipedia , lookup
Four color theorem wikipedia , lookup
Euclidean geometry wikipedia , lookup
Menu Teoirim 1 Tá uillinneacha rinnurchormhaireacha ar coimhéid Select the proof required then click mouse key to view proof. Theorem 2 The measure of the three angles of a triangle sum to 1800 . Theorem 3 An exterior angle of a triangle equals the sum of the two interior opposite angles in measure. Theorem 4 If two sides of a triangle are equal in measure, then the angles opposite these sides are equal in measure. Theorem 5 The opposite sides and opposite sides of a parallelogram are respectively equal in measure. Theorem 6 A diagonal bisects the area of a parallelogram Teoirim 7 Tá tomhas na hullinne ag lár ciorcail cothrom le dhá oiread thomhas na uillinne ag an imlíne ag seasamh ar an stual céanna Theorem 8 A line through the centre of a circle perpendicular to a chord bisects the chord. Theorem 9 If two triangles are equiangular, the lengths of the corresponding sides are in proportion. Theorem 10 In a right-angled triangle, the square of the length of the side opposite to the right angle is equal to the sum of the squares of the other two sides. Constructions Sketches Quit Teoirim 1: Tá uillinneacha rinnurchomhaireacha ar comhéid T&T 2 ltch 205: Dein cliceáil chun na céimeanna A fheiscint diaidh ar ndiaidh 1 4 2 3 Le Cruthu 1 = 3 and 2 = 4 1 + 2 = 1800 Cruthú: ………….. 2 + 3 = 1800 ………….. Líne Díreach Líne Díreach 1 + 2 = 2 + 3 1 = 3 Ar an modh céanna 2 = 4 Q.E.D. Constructions Sketches Menu Quit Teoirim 2: The measure of the three angles of a triangle sum to 1800 . Use mouse clicks to see proof Given: Triangle To Prove: 1 + 2 + 3 = 1800 Construction: Draw line through 3 parallel to the base 4 3 5 3 + 4 + 5 = 1800 Proof: Straight line 1 = 4 and 2 = 5 Alternate angles 1 2 3 + 1 + 2 = 1800 1 + 2 + 3 = 1800 Q.E.D. Constructions Sketches Menu Quit Theorem 3: An exterior angle of a triangle equals the sum of the two interior opposite angles in measure. Use mouse clicks to see proof 3 4 To Prove: Proof: 1 2 1 = 3 + 4 1 + 2 = 1800 ………….. 2 + 3 + 4 = 1800 Straight line ………….. Theorem 2. 1 + 2 = 2 + 3 + 4 1 = 3 + 4 Q.E.D. Constructions Sketches Menu Quit Theorem 4: If two sides of a triangle are equal in measure, then the angles opposite these sides are equal in measure. a Use mouse clicks to see proof 3 4 Given: Triangle abc with |ab| = |ac| To Prove: 1 = 2 Construction: Construct ad the bisector of bac Proof: b d In the triangle abd and the triangle adc 3 = 4 ………….. |ab| = |ac| |ad| = |ad| ………….. ………….. Construction Given. Common Side. The triangle abd is congruent to the triangle adc 1 = 2 Constructions 2 1 ……….. SAS = SAS. Q.E.D. Sketches Menu Quit c Theorem 5: The opposite sides and opposite sides of a parallelogram are respectively equal in measure. Use mouse clicks to see proof b c 3 Given: Parallelogram abcd To Prove: |ab| = |cd| and |ad| = |bc| 4 and Construction: 1 a 2 d Proof: abc = adc Draw the diagonal |ac| In the triangle abc and the triangle adc 1 = 4 …….. Alternate angles 2 = 3 ……… Alternate angles |ac| = |ac| …… Common The triangle abc is congruent to the triangle adc ……… ASA = ASA. |ab| = |cd| and |ad| = |bc| and abc = adc Q.E.D Constructions Sketches Menu Quit Theorem 6: A diagonal bisects the area of a parallelogram b a c Use mouse clicks to see proof d x Given: Parallelogram abcd To Prove: Area of the triangle abc = Area of the triangle adc Construction: Proof: Draw perpendicular from b to ad Area of triangle adc = ½ |ad| x |bx| Area of triangle abc = ½ |bc| x |bx| As |ad| = |bc| …… Theorem 5 Area of triangle adc = Area of triangle abc The diagonal ac bisects the area of the parallelogram Constructions Sketches Menu Q.E.D Quit Teoirim 7: (T&T 2 ltch 265) . Tá tomhas na hullinne ag lár ciorcail cothrom le dhá oiread thomhas na uillinne ag an imlíne ag seasamh ar an stual céanna Dein cliceáil ar do luch chun gach céim a fheiscint Le Cruthú: | boc | = 2 | bac | Tógáil: Ceangail a le o agus sínigh é amach go r a 2 5 o Cruthú: Sa triantán aob 3 | oa| = | ob | …… Is gathanna iad araon | 2 | = | 3 | …… Triantán comhchosach 1 4 r c b | 1 | = | 2 | + | 3 | …… Uillinn sheachtrach | 1 | = | 2 | + | 2 | | 1 | = 2| 2 | Mar an gcéanna sa triantán eile | 4 | = 2| 5 | | boc | = 2 | bac | Constructions Q.E.D Sketches Menu Quit Teoirim 8: Líne trí lár ciorcail atá ingearach le corda , (T&T 2 ltch 268) déroinneann sé an corda sin. Cliceáil ar do luch chun gach céim a fheiscint Tugtha: Ciorcal le lár o agus líne L ingearach le ab. Le cruthú : Tógáil a o | ar | = | rb | r Ceangail a le o agus b go o 90 o Cruthú: Sa dhá thriantán aor agus orb aro = orb Tá L …………. b 90 o ( Dronuillin ) |ao| = |ob| ………….. Gathanna. (T aobhagáin) |or| = |or| ………….. Slios comónta ( Slios) an triantán aor iomchuí leis an triantán orb |ar| = |rb| ……… Rhs = Rhs. DTS = DTS Q.E.D Constructions Sketches Menu Quit Teoirim 9: Má tá dhá thriantán comhuilleach (comhchosúil) tá (T&T 2 ltch 282) na sleasa comhfhreagracha I gcomhréir (sa chóimheas céanna) Dein cliceáil ar do luch Tugtha: Dhá thriantán le huillinneacha ar comhéid |ab| Le Cruthú: |de| = |ac| |df| |bc| = |ef| Ar ab marcáil ax ar comhfhad le de. Tógáil: Ar ac marcáil ay ar comhfhad le df x 4 a d 2 2 5 y e 1 1 = 4 Cruthú 3 f [xy] comhthreomhar le [bc] |ab| |ax| |ab| b 1 3 |de| = c Constructions = |ac| |ay| Mar xy comhtreomhar le bc |ac| Ar an modh céanna = |df| |bc| |ef| Q.E.D. Sketches Menu Quit Teoirim 10: I dtriantán dronuilleach tá fad an tsleasa os comhair na dronuillinne cothrom le suim fhad cearnaithe an dá shlios eile. Úsáid do luch chun cliceáil T&T 2 ltch 288 b a a c c c b a c b a b Constructions Tugtha : An Triantán abc Le Cruthú: a2 + b2 = c2 Tógáil 3 Thriantán dronuilleach mar léirithe Cruthú: Achar na móire. = achar na bige + 4(achar D) (a + b)2 = c2 a2 + 2ab +b2 = c2 + 2ab + 4(½ab) a2 + b2 = c2 Q.E.D. Sketches Menu Quit