To solve the problem, let's analyze the given information step by step.
Step 1: Analyze Triangle ABC
Since \( \Delta ABC \) is equilateral:
- All angles are equal. The measure of each angle in an equilateral triangle is given by: \[ m\angle ABC = m\angle BCA = m\angle CAB = \frac{180^\circ}{3} = 60^\circ \]
So:
- \( m\angle ABC = 60^\circ \)
- \( m\angle BCA = 60^\circ \)
- \( m\angle CAB = 60^\circ \)
Step 2: Analyze Angles BCD and BDC
Given that \( \angle BCD \cong \angle BDC \), we can denote: \[ \angle BCD = \angle BDC = x \] From the triangle sum theorem: \[ m\angle ABC + m\angle BCD + m\angle BDC = 180^\circ \] Substituting the known values: \[ 60^\circ + x + x = 180^\circ \] \[ 60^\circ + 2x = 180^\circ \] \[ 2x = 120^\circ \] \[ x = 60^\circ \] Thus: 4. \( m\angle BCD = 60^\circ \) 5. \( m\angle BDC = 60^\circ \) 6. Since \( m\angle CBD = 180^\circ - m\angle ABC - m\angle BDC \): \[ m\angle CBD = 180^\circ - 60^\circ - 60^\circ = 60^\circ \]
Step 3: Analyze Triangle CDE and Triangle ECF
Regarding triangle \( CDE \): Given that \( DE \cong CE \cong EF \), triangle \( CDE \) is also equilateral. Therefore: 7. \( m\angle EDC = 60^\circ \) 8. \( m\angle ECD = 60^\circ \) 9. \( m\angle CED = 60^\circ \)
For triangle \( ECF \) (since \( ECF \) is not specified as equilateral, but given the congruency relationships and symmetry): 10. \( m\angle ECF = 60^\circ \) 11. \( m\angle EFC = 60^\circ \) 12. Similarly, \( m\angle CEF = 60^\circ \)
Summary of Angles
- \( m\angle ABC = 60^\circ \)
- \( m\angle BCA = 60^\circ \)
- \( m\angle CAB = 60^\circ \)
- \( m\angle BCD = 60^\circ \)
- \( m\angle BDC = 60^\circ \)
- \( m\angle CBD = 60^\circ \)
- \( m\angle EDC = 60^\circ \)
- \( m\angle ECD = 60^\circ \)
- \( m\angle CED = 60^\circ \)
- \( m\angle ECF = 60^\circ \)
- \( m\angle EFC = 60^\circ \)
- \( m\angle CEF = 60^\circ \)
All of the angles found in triangles \( ABC \), \( CDE \), and \( ECF \) are \( 60^\circ \) due to the properties of equilateral triangles and the given conditions of congruence.