We are given a diagram with a triangle and a transversal line intersecting one of its sides. We are given an exterior angle of $84^\circ$ formed by the transversal line and the extended side of the triangle. We need to find the value of the angle $k$. The triangle has two sides marked with the same symbol, which means it is an isosceles triangle. We are also given that two lines are parallel.

1. Problem Description

We are given a diagram with a triangle and a transversal line intersecting one of its sides. We are given an exterior angle of $84^\circ$ formed by the transversal line and the extended side of the triangle. We need to find the value of the angle $k$. The triangle has two sides marked with the same symbol, which means it is an isosceles triangle. We are also given that two lines are parallel.

2. Solution Steps

First, note that the given exterior angle of $84^\circ$ and the angle $k$ are supplementary angles. Supplementary angles add up to $180^\circ$. Thus,

$k + 84^\circ = 180^\circ$.

From this we get,

$k = 180^\circ - 84^\circ = 96^\circ$.

Also, since two sides of the triangle are equal, the angles opposite these sides are also equal. Let us call this angle $x$. Since the two lines are parallel, the angle inside the triangle adjacent to $k$ is also $x$ because these are corresponding angles.

Since we have $k=96^\circ$, the angle adjacent to $k$ inside the triangle is $180^\circ - k = 180^\circ - 96^\circ = 84^\circ$. Thus $x=84^\circ$.

Since the triangle is isosceles with two angles being $84^\circ$, let the third angle be $y$.

The sum of the angles in a triangle is $180^\circ$.

$x + x + y = 180^\circ$

$84^\circ + 84^\circ + y = 180^\circ$

$168^\circ + y = 180^\circ$

$y = 180^\circ - 168^\circ = 12^\circ$.

The angle $k$ and the $84^\circ$ angle are supplementary, thus we have

$k + 84^\circ = 180^\circ$.

$k = 180^\circ - 84^\circ = 96^\circ$.

3. Final Answer

$k = 96$