Question

through c construct a parallel to ab.
through a construct a parallel to bc.
through b construct a parallel to ac.
extend these lines to form a new triangle.
what relationship exists between the
two triangles?

Explanation:

Step1: Recall the properties of similar triangles

If a line is drawn parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. Also, if two triangles have their corresponding angles equal (due to parallel - line - induced equal angles), the triangles are similar.
Let the original triangle be \(\triangle ABC\). When we construct lines parallel as described:

• When we construct a line through \(A\) parallel to \(BC\), through \(B\) parallel to \(AC\), and through \(C\) parallel to \(AB\) and extend them.
• For two triangles, say \(\triangle ABC\) and the new - formed triangle \(\triangle A'B'C'\) (the triangle formed by the extended parallel lines).
• By the property of parallel lines, \(\angle ABC=\angle B'C'A'\) (corresponding angles as \(AB\parallel C'A'\) and \(BC\) is a transversal), \(\angle BAC=\angle B'C'B'\) (corresponding angles as \(AC\parallel B'C'\) and \(AB\) is a transversal), and \(\angle ACB=\angle A'B'C'\) (corresponding angles as \(BC\parallel A'B'\) and \(AC\) is a transversal)

Step2: Apply the AA (angle - angle) similarity criterion

The AA (angle - angle) similarity criterion states that if two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar. Since all three pairs of corresponding angles of the two triangles (the original \(\triangle ABC\) and the new - formed triangle) are equal (because of the parallel lines which create equal corresponding and alternate interior angles), the two triangles are similar.

Answer:

The two triangles are similar.