Question: Question 4 In the diagram below, \(\overline{BC} \cong \overline{EC}\) and \(m\angle…

Question 4

In the diagram below, \(\overline{BC} \cong \overline{EC}\) and \(m\angle ECD = 140^\circ\). Find \(m\angle ABE\).

Step	Angle	Reason
1	\(m\angle ECD = 140^\circ\)	Given
Try	\(m\angle EBC = 20^\circ\)	Alternate exterior angles

[Diagram]

• Points: A, B, C, D, E, F
• Lines: \(\overline{AB}\), \(\overline{BC}\), \(\overline{CD}\), \(\overline{DE}\), \(\overline{EF}\), \(\overline{FA}\)
• Angles: \(\angle ECD = 140^\circ\)

You may assume lines that appear straight are straight, but the figure is not otherwise drawn to scale.

Solution

In the diagram, \( BC \cong EC \) and \( m\angle ECD = 140^\circ \). We need to find \( m\angle ABE \). Since \( BC \cong EC \), triangle \( BCE \) is isosceles. In isosceles triangles, the base angles are equal. Since angle \( ECD = 140^\circ \), angle \( ECB \) will be supplementary to this on a straight line, i.e., \[ m\angle ECB = 180^\circ - 140^\circ = 40^\circ \] Since \( \triangle ECB \) is isosceles: \[ m\angle EBC = m\angle ECB = 40^\circ \] Next, use the fact that the sum of angles in a triangle is \( 180^\circ \): \[ m\angle ABE + m\angle EBC + m\angle ECB = 180^\circ \] Substitute the known values: \[ m\angle ABE + 40^\circ + 40^\circ = 180^\circ \] \[ m\angle ABE + 80^\circ = 180^\circ \] Subtract \( 80^\circ \) from both sides: \[ m\angle ABE = 100^\circ \] Thus, the measure of \( \angle ABE \) is \( 100^\circ \).