In trapezium ABCD AB Parallel to CD diagonal intersect at O(AC and BD)how can we prove that AOD and COD are similar
vertical angles at O
alternate interior angles where the diagonals meet the parallel sides
AAA
To prove that triangles AOD and COD are similar, we need to show that their corresponding angles are congruent and their corresponding sides are proportional.
Here's how you can prove it:
Step 1: Draw the trapezium ABCD, where AB is parallel to CD, and the diagonal AC intersects BD at point O.
Step 2: Consider triangles AOD and COD.
Step 3: By construction, we know that AD and BC are parallel sides of the trapezium. Therefore, angle A and angle C are corresponding angles.
Step 4: Since AC is a diagonal, it intersects both AB and CD, forming alternate interior angles. Therefore, angle AOC and angle COD are also corresponding angles.
Step 5: Now that we have established the corresponding angles, we need to prove that the corresponding sides are proportional.
Step 6: Notice that triangles AOC and COB are similar by the Angle-Angle (AA) criterion since angle AOC and angle COB are congruent, and angle OCA and angle OBC are congruent (both right angles).
Step 7: From triangle AOC, we have the proportion: AO/OC = AC/OC. Simplifying this equation gives AO/OC = AC/OC = AC/OC = 1.
Step 8: From triangle COB, we have the proportion: CO/OB = OC/OB = OC/OC = 1.
Step 9: Combining the ratios from steps 7 and 8, we get AO/OC = CO/OB.
Step 10: Using the Transitive Property of Equality, we can conclude that AO/OC = CO/OB, which implies that triangles AOD and COD are similar since their corresponding sides are proportional.
Therefore, triangles AOD and COD are similar based on the Angle-Angle (AA) criterion, which proves that the angles in these triangles are congruent, and their corresponding sides are proportional.