Menu Constructions Sketches Quit Select the proof required then click mouse key to view proof. Theorem 1 Vertically opposite angles are equal in measure. 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 7 If two sides of a triangle are equal in measure, then the angles opposite these sides are equal in measure. Theorem 4 The opposite sides and opposite sides of a parallelogram are respectively equal in measure. Theorem 5 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

Constructions Sketches Menu Quit Theorem 1: Vertically opposite angles are equal in measure 180 90 45 135 180 90 45 135 X A B Given: Two vertically opposite angles, A and B To Prove: A = B Proof: B + X = 1800 ………….. Straight line  B = 1800 – X A + X = 1800 ………….. Straight line  A = 1800 – X  A = B Since both equal to 1800 – X Q.E.D. Constructions Sketches Menu Quit

Theorem 2: The measure of the three angles of a triangle sum to 1800 . Given: Triangle with angles A, B, C. To Prove: A + B + C = 1800 Construction: Draw line through the upper vertex, parallel to the base A C B D E Proof: A = D Alternate angles C = E Alternate angles A C Now, D + B + E = 1800 Straight line  A + B + C = 1800 Since D = A and E = C Q.E.D. Constructions Sketches Menu Quit

Constructions Sketches Menu Quit Theorem 3: An exterior angle of a triangle equals the sum of the two interior opposite angles in measure. X C B A 180 90 45 135 Given: A triangle with interior angles A, B, C and with exterior angle X. To Prove: X = A + B Proof: X + C = 1800 ………….. Straight line  X = 1800 – C Also A + B + C = 1800 Three angles in triangle sum to 1800  A + B = 1800 – C  X = A + B Since both are equal to 1800 – C Q.E.D. Constructions Sketches Menu Quit

c b a d Constructions Sketches Menu Quit Theorem 4: The opposite sides and opposite angles of a parallelogram are respectively equal in measure. Given A Parallelogram abcd c b a d To Prove: (i) |ab| = |dc| and |bc| = |ad| (ii)Ðabc = Ðadc and bad = bcd Construction: Draw the diagonal |ac| Proof: abc is congurent to adc because ……… Ðbac = Ðacd  …….. Alternate angles |ac| = |ac| ……. Common Ðacb = Ðcad ……… Alternate angles ………… which establishes congruence by ASA  abc and adc have the same lengths, angles and area Þ |ab| = |dc| and |bc| = |ad| and also Ðabc = Ðadc Similarly, Ðbad = Ðbcd Q.E.D. Constructions Sketches Menu Quit

Theorem 5: A diagonal bisects the area of a parallelogram Given: A Parallelogram abcd To Prove: Area of  abc = Area of  acd Construction: Draw the diagonal ac  Proof: abc is congurent to adc because …….. |ab| = |dc| ………….. Opposite sides of parallelogram equal in measure |ac| = |ac| ………….. Common |bc| = |ad| ………….. Opposite sides of parallelogram equal in measure ………… which establishes congruence by SSS  Area abc = Area acd Þ The diagonal ac bisects the area of the parallelogram Q.E.D. Constructions Sketches Menu Quit

Theorem 6: The diagonals of a parallelogram bisect each other.  m ie. am= mc and bm = md (The proof of this theorem is not required.) Constructions Sketches Menu Quit

Constructions Sketches Menu Quit Theorem 7: If two sides of a triangle are equal in measure, then the angles opposite these sides are equal in measure. a b c Given: Isosceles triangle abc in which |ab| = |ac| d   To Prove: Ðabc = Ðacd Construction: Construct ad the bisector of Ðbac Proof: abd and acd are congurent because….. Ðbad = Ðcad ………. Since ad bisects Ðbac |ab| = |ac| ………….. Given |ad| = |ad| ………….. Common Side ………… which establishes congruence by SAS  Ðabc = Ðacd by congruence Q.E.D. Constructions Sketches Menu Quit

Constructions Sketches Menu 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 b c o To Prove: | Ðboc | = 2 | Ðbac | r 2 5 Construction: Join a to o and extend to r Proof: In the triangle aob 4 3 1 | oa| = | ob | …… Radii Þ | Ð2 | = | Ð3 | …… Theorem 4 | Ð1 | = | Ð2 | + | Ð3 | …… Theorem 3 Þ | Ð1 | = | Ð2 | + | Ð2 | Þ | Ð1 | = 2| Ð2 | Similarly | Ð4 | = 2| Ð5 | Þ | Ðboc | = 2 | Ðbac | Q.E.D Constructions Sketches Menu Quit

Constructions Sketches Menu Quit Theorem 8: A line through the centre of a circle perpendicular to a chord bisects the chord. Use mouse clicks to see proof L 90 o o a b r Given: A circle with o as centre and a line L perpendicular to ab. To Prove: | ar | = | rb | Construction: Join a to o and o to b Proof: In the triangles aor and the triangle orb Ðaro = Ðorb …………. 90 o |ao| = |ob| ………….. Radii. |or| = |or| ………….. Common Side. Þ The triangle aor is congruent to the triangle orb ……… RSH = RSH. Þ |ar| = |rb| Q.E.D Constructions Sketches Menu Quit

a c b d f e 1 2 3 x y 4 5 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 To Prove: |df| |ac| = |de| |ab| |ef| |bc| Construction: On ab mark off ax equal in length to de. On ac mark off ay equal in length to df a c b d f e 1 2 3 Proof: Ð1 = Ð4 Þ [xy] is parallel to [bc] |ay| |ac| = |ax| |ab| Þ As xy is parallel to bc x y 4 5 |df| |ac| = |de| |ab| Similarly |ef| |bc| Q.E.D. Constructions Sketches Menu Quit

a b c a b c a b c a b c Constructions 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 Given: Triangle abc To Prove: a2 + b2 = c2 a b c a b c Construction: Three right angled triangles as shown Proof: Area of large sq. = area of small sq. + 4(area D) (a + b)2 = c2 + 4(½ab) a2 + 2ab +b2 = c2 + 2ab a2 + b2 = c2 Q.E.D. a b c a b c Constructions Sketches Menu Quit