Question

In given figure, if $ \angle A = \angle C $ , then prove that $ \Delta AOB \sim \Delta COD $

Answer 1

Hint: Here we can use the concept of transversal and similarity of triangles.
Similarity of triangle: Two triangles are said to be similar if the corresponding angles are similar.

Two triangles ABC and XYZ are similar to each other if
 $
  \angle {\text{A = }}\angle {\text{X}} \\
  \angle {\text{B = }}\angle {\text{Y}} \\
  \angle {\text{C = }}\angle {\text{Z}} \\
  $
This type of similarity is called AAA Similarity.

Complete step-by-step answer:
Let's look at the below definitions to prove the similarity.
Parallel lines: Two lines are said to be parallel when the distance between them is the same and they never intersect.
Transversal: Transversal is a line that cuts two or more parallel lines.

From the above figure,
Lines A and B are parallel to each other.
Vertically opposite angles:
 $ \angle {\text{1 = }}\angle {\text{3}} $
\[
  \angle {\text{2 = }}\angle {\text{4}} \\
  \angle {\text{5 = }}\angle {\text{7}} \\
  \angle {\text{6 = }}\angle {\text{8}} \\
 \]
Corresponding Angles:
\[
  \angle {\text{1 = }}\angle {\text{5}} \\
  \angle {\text{2 = }}\angle {\text{6}} \\
  \angle {\text{3 = }}\angle {\text{7}} \\
  \angle {\text{4 = }}\angle {\text{8}} \\
 \]
Alternate interior angles:
\[
  \angle {\text{3 = }}\angle {\text{5}} \\
  \angle {\text{4 = }}\angle {\text{6}} \\
 \]
Alternate exterior angles:
\[
  \angle {\text{1 = }}\angle {\text{7}} \\
  \angle {\text{2 = }}\angle {\text{8}} \;
 \]
Given: $ \angle A = \angle C $
We need to prove that $\Delta AOB \sim\Delta COD $

From the above figure,
DB is a transversal to parallel lines AB and CD.
Comparing two triangles AOB and COD
 $ \angle {\text{A = }}\angle {\text{C = 9}}{{\text{0}}^{\text{o}}} $ …… (Given)
 $ \angle {\text{B = }}\angle {\text{D}} $ ………(Alternate interior angles)
 $ \angle {\text{1 = }}\angle {\text{2}} $ ………..(Vertically opposite angles)
As we know that corresponding angles in a triangle are equal,
Then by AAA Similarity we can say that two triangles AOB and COD are similar triangles
Hence proved.

Note: In this type of questions that involves the concept of transversal and similarity of triangles we need to have knowledge about the parallel lines and about how two triangles can be termed as similar triangles. Following the given data in the question by application of the appropriate concept involved helps us to prove the required.