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Menu Theorem 1 Vertically opposite angles are equal in measure. 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 Theorem 7 The measure of the angle at the centre of the circle is twice the measure of the angle at the circumference standing on the same arc. 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 Theorem 1: Vertically opposite angles are equal in measure Use mouse clicks to see proof 1 4 2 3 1 = 3 and To Prove: 2 = 4 1 + 2 = 1800 Proof: ………….. 2 + 3 = 1800 ………….. Straight line Straight line 1 + 2 = 2 + 3 1 = 3 Similarly 2 = 4 Q.E.D. Constructions Sketches Menu Quit Theorem 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 Theorem 7: The measure of the angle at the centre of the circle is twice the measure of the angle at the circumference standing on the same arc. Use mouse clicks to see proof a To Prove: | boc | = 2 | bac | Construction: Join a to o and extend to r Proof: 2 5 o In the triangle aob 3 | oa| = | ob | …… Radii 1 4 r | 2 | = | 3 | …… Theorem 4 c b | 1 | = | 2 | + | 3 | …… Theorem 3 | 1 | = | 2 | + | 2 | | 1 | = 2| 2 | Similarly | 4 | = 2| 5 | | boc | = 2 | bac | Constructions Q.E.D Sketches Menu Quit Theorem 8: A line through the centre of a circle perpendicular to a chord bisects the chord. L Use mouse clicks to see proof Given: A circle with o as centre and a line L perpendicular to ab. To Prove: o | ar | = | rb | Construction: Proof: a r Join a to o and o to b 90 o In the triangles aor and the triangle orb aro = orb …………. b 90 o |ao| = |ob| ………….. Radii. |or| = |or| ………….. Common Side. The triangle aor is congruent to the triangle orb ……… RSH = RSH. |ar| = |rb| Q.E.D Constructions Sketches Menu Quit Theorem 9: If two triangles are equiangular, the lengths of the corresponding sides are in proportion. Use mouse clicks to see proof Given: Two Triangles with equal angles |ab| To Prove: = |de| |ac| |df| = |bc| |ef| Construction: On ab mark off ax equal in length to de. On ac mark off ay equal in length to df x 4 a d 2 2 5 y e 1 1 = 4 Proof: 3 f [xy] is parallel to [bc] |ab| |ax| = |ab| b 1 3 |de| c Constructions |ac| As xy is parallel to bc |ay| = |ac| |df| Similarly = |bc| |ef| Q.E.D. Sketches Menu Quit 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. Use mouse clicks to see proof b a a c c c b a Given: Triangle abc To Prove: a2 + b2 = c2 Construction: Three right angled triangles as shown b Proof: Area of large sq. = area of small sq. + 4(area D) (a + b)2 = c2 + 4(½ab) a2 + 2ab +b2 = c2 + 2ab c a b Constructions a2 + b2 = c2 Q.E.D. Sketches Menu Quit