To prove that angle AOB is 90 degrees, we can use the property that the diagonals of a quadrilateral bisect each other. Let's walk through the proof step by step:
Given:
1. ABCD is a quadrilateral.
2. Diagonals AC and BD bisect each other at point O.
3. Angle A = 70 degrees.
4. AO and BO are the bisectors of angle A and angle B, respectively.
Proof:
1. Since the diagonals AC and BD bisect each other at point O, we have:
AO = OC (By the property of bisecting diagonals)
BO = OD (By the property of bisecting diagonals)
2. Let's consider the angles in triangle AOC:
Angle AOC = Angle A + Angle OAC (By the angle sum property of triangles)
Angle AOC = 70° + Angle OAC (Since Angle A = 70°)
3. Now, let's consider the angles in triangle BOD:
Angle BOD = Angle B + Angle OBD (By the angle sum property of triangles)
Angle BOD = Angle B + Angle OBC (Since Angle OBD = Angle OBC, as BO bisects angle B)
4. Since AO = OC and BO = OD, triangles AOC and BOD are isosceles triangles.
5. In an isosceles triangle, the angles opposite the equal sides are also equal.
So, in triangle AOC:
Angle OAC = Angle OCA
And in triangle BOD:
Angle OBD = Angle ODB
6. Combining the results from steps 2, 3, 5, and the given angle A = 70°:
Angle AOC = 70° + Angle OAC
Angle OAC = Angle OCA (from step 5)
Angle AOC = 70° + Angle OCA
Angle BOD = Angle B + Angle OBD
Angle OBD = Angle ODB (from step 5)
Angle BOD = Angle B + Angle ODB
7. Since the angles of a triangle sum up to 180 degrees:
Angle AOC + Angle BOD = 180°
(70° + Angle OCA) + (Angle B + Angle ODB) = 180°
70° + (Angle OCA + Angle B + Angle ODB) = 180°
8. Rearranging the equation:
Angle OCA + Angle B + Angle ODB = 110°
9. Now, let's consider the angles in triangle AOB:
Angle AOB = Angle OAB + Angle OBA (By the angle sum property of triangles)
Angle AOB = Angle OCA + Angle B (Since AO is the bisector of angle A and BO is the bisector of angle B)
10. Substituting the value of (Angle OCA + Angle B) from step 8:
Angle AOB = 110°
Since angle AOB is 110 degrees, and angles AOC and BOD are 70 degrees each (given that angle A = 70°), the angles AOC, AOB, and BOD together form a straight line. So, the sum of angles AOB, AOC, and BOD is 180 degrees.
180° = 110° + Angle AOC
Angle AOC = 70°
But we know that angle AOC is also equal to Angle OAB.
So, Angle OAB = 70°
Since Angle OAB and Angle OBA are equal and add up to 70°, each of them must be 35°.
11. Now, we can find Angle AOB:
Angle AOB = Angle OAB + Angle OBA
Angle AOB = 35° + 35°
Angle AOB = 70°
Since angle AOB is 70 degrees, we can conclude that angle AOB is 90 degrees because the sum of angles AOC and BOD (which includes AOB) is 180 degrees.
Therefore, we have successfully proved that angle AOB is 90 degrees.