We are given a rhombus $ABCD$ with $m\angle CBD = 67^\circ$. We need to find the measure of $\angle BAD$.

1. Problem Description

We are given a rhombus $ABCD$ with $m\angle CBD = 67^\circ$. We need to find the measure of $\angle BAD$.

2. Solution Steps

Since $ABCD$ is a rhombus, all sides are equal, i.e., $BC = CD$. This means that triangle $BCD$ is an isosceles triangle.

In triangle $BCD$, since $BC=CD$, we have $\angle CBD = \angle CDB$.

Thus, $\angle CDB = 67^\circ$.

The sum of the angles in triangle $BCD$ is $180^\circ$. Therefore,

$\angle BCD + \angle CBD + \angle CDB = 180^\circ$.

$\angle BCD + 67^\circ + 67^\circ = 180^\circ$.

$\angle BCD = 180^\circ - 67^\circ - 67^\circ = 180^\circ - 134^\circ = 46^\circ$.

In a rhombus, opposite angles are equal. Thus, $\angle BAD = \angle BCD$.

Therefore, $\angle BAD = 46^\circ$.

3. Final Answer

The measure of $\angle BAD$ is $46^\circ$.