1. Menu Constructions Sketches Quit Select the proof required then click
mouse key to view proof.
Theorem Vertically opposite angles are equal in measure.
Theorem 2 The measure of the three angles of a triangle sum to
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 The opposite sides and opposite sides of a parallelogram are respectively equal in measure.
Theorem A diagonal bisects the area of a parallelogram
Theorem 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 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 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.
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2. 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 = ………… Straight line
 B = 1800 – X
A + X = ………… Straight line
 A = 1800 – X
 A = B Since both equal to 1800 – X
Q.E.D.
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3. 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 = Straight line
 A + B + C = Since D = A and E = C
Q.E.D.
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4. 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 = ………….. Straight line
 X = 1800 – C
Also A + B + C = Three angles in triangle sum to 1800
 A + B = 1800 – C
 X = A + B Since both are equal to 1800 – C
Q.E.D.
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5. 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.
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6. 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.
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7. 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.)
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8. 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.
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9. 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
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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
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11. 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.
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12. 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
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