line segment AB and CD bisect each other at O . join AC and BD . prove that tringle AOC congruent triangle BOD
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To prove that triangle AOC is congruent to triangle BOD, we can use the method of proving triangles congruent known as SAS (Side-Angle-Side). This method involves proving that the lengths of two sides and the measure of an included angle in one triangle are equal to the corresponding lengths and measure of an included angle in the other triangle.
Here's how we can apply the SAS method to prove triangle AOC congruent to triangle BOD, using the given information:
1. Line segment AB and CD bisect each other at point O.
2. Since AB and CD are bisectors, it means that AO = BO and CO = DO. (Definition of bisectors)
3. Join AC and BD to form line segments AC and BD.
4. Now, we need to prove that the included angles are congruent. We can prove that angle AOC is congruent to angle BOD.
- Since AB and CD bisect each other at O, angle AOB and angle COD are right angles. (Definition of bisectors)
- As adjacent angles formed by the intersection of two lines, angle AOC and angle BOD are vertical angles, which are congruent.
- Therefore, angle AOC is congruent to angle BOD.
5. Finally, we have AO = BO, OC = OD, and angle AOC = angle BOD.
- According to the SAS congruence postulate, if two sides and the included angle of one triangle are congruent to the corresponding sides and included angle of another triangle, the triangles are congruent.
- Hence, triangle AOC is congruent to triangle BOD.
By using the SAS congruence postulate, we have proved that triangle AOC is congruent to triangle BOD.