Find the value of angle using elementary geometry rules

Question

In $\triangle ABC$ with base $AC$, $\angle C$ = $46^\circ$ and $AC$ is extended to point $D$.

$E$ is a point on $AB$ and $DE$ is joined.

Given that $AB=AD=DE$. Find $\angle ABC$.

Answer 1

Hints: you will need to use the following:

• The sum of the angles on a straight line equal $180^\circ$.
• The sum of the angles in a triangle is $180^\circ$.
• In a isosceles triangle, the two base angles are the same.

For example, find $\angle BCD$ from the first hint, and $\angle BAC$ from the second. Continue writing all angles in terms of $x$ and $46^\circ$. At some point, you will get an equation relating the two.

Answer 2

As stated, C lies between A and D, and E lies between A and B. But that is impossible. If you allow D to lie between A and C, or E to lie outside the segment AB, then $\angle ABC$ is not determined.