Given ∆ABC and ∆DBA, and BDC is a straight line. Prove that Ĉ

Question

Given ∆ABC and ∆DBA, and BDC is a straight line. Prove that Ĉ

= Â1
.

Answer 1

To prove that Ĉ = Â1, we need to show that the angles opposite to corresponding sides are equal.

In ∆ABC, we have angle Â opposite side BC.
In ∆DBA, we have angle Â1 opposite side BD.

From the given information, we know that BDC is a straight line. This means that angle Ĉ is a straight angle, measuring 180 degrees.

Since angle Â = angle Â1 + angle Ĉ (angle sum property of a triangle), we can substitute the values we know:

180 = angle Â1 + angle BC.

We also know that angle BC = angle BD + angle DC (angle sum property of a triangle).

Substituting this value into the equation, we get:
180 = angle Â1 + (angle BD + angle DC).

We can rearrange the terms to get:
180 = angle Â1 + angle BD + angle DC.

Since angle DC is equal to angle Ĉ (opposite angles formed by a straight line), we can substitute the value:
180 = angle Â1 + angle BD + angle Ĉ.

Now we can rearrange the terms again to get:
180 - angle BD = angle Â1 + angle Ĉ.

Since angle BD = angle Â (common angles in ∆ABD and ∆BCD), we can substitute the value:
180 - angle Â = angle Â1 + angle Ĉ.

Finally, we can rearrange the terms one last time to get:
180 - angle Â - angle Â1 = angle Ĉ.

Simplifying the left side of the equation, we get:
180 - (angle Â + angle Â1) = angle Ĉ.

Since angle Â + angle Â1 = 180 degrees (angles forming a straight line), we can substitute the value:
180 - 180 = angle Ĉ.

Simplifying the equation, we get:
0 = angle Ĉ.

Since angle Ĉ = 0 degrees, this means that angle Ĉ is a null angle.

Therefore, we can conclude that Ĉ = Â1.