Question

in this coordinate plane, the slope of line a is equal to the slope of line b.
to prove that line a is parallel to line b, frank wants to first prove that δjkl is similar to δmnl. which similarity theorem can frank use in his proof?
side - side - side similarity theorem
side - angle - side similarity theorem
angle - angle similarity theorem

Explanation:

Step1: Recall the slope formula

The slope of a line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For line \(a\) passing through \(K(0,k)\) and \(J(j,0)\), the slope \(m_a=\frac{0 - k}{j-0}=-\frac{k}{j}\). For line \(b\) passing through \(N(0,n)\) and \(M(m,0)\), the slope \(m_b = \frac{0 - n}{m - 0}=-\frac{n}{m}\). Since \(m_a=m_b\), we have \(\frac{k}{j}=\frac{n}{m}\).

Step2: Analyze the angles

\(\angle JLK=\angle MNL = 90^{\circ}\). Also, \(\angle LJK\) and \(\angle LMN\) are equal because the slopes of the lines (which are related to the angles of inclination) are equal.

Step3: Apply the similarity theorem

In \(\triangle JKL\) and \(\triangle MNL\), we have two pairs of equal angles (\(\angle JLK=\angle MNL\) and \(\angle LJK=\angle LMN\)). According to the Angle - Angle (AA) similarity theorem, if two angles of one triangle are equal to two angles of another triangle, then the two triangles are similar.

Answer:

Angle - Angle similarity theorem