Question

using triangle theorems
m∠eal + m∠eaj = 180°
m∠h + m∠l + m∠j = 180°
m∠h + m∠l + m∠eal + m∠eaj + m∠j = 360°
m∠hea = m∠l + m∠eal
triangle exterior angle theorem
m∠h + m∠hea + m∠eaj + m∠j = 360°

Explanation:

Step1: Identify the given equations

We have \( m\angle EAL + m\angle EAJ = 180^\circ \) (linear pair), \( m\angle H + m\angle L + m\angle J = 180^\circ \) (triangle angle - sum), and \( m\angle HEA=m\angle L + m\angle EAL \) (exterior angle theorem). Also, \( m\angle H + m\angle L + m\angle EAL + m\angle EAJ + m\angle J=360^\circ \)

Step2: Substitute \( m\angle L + m\angle EAL \) with \( m\angle HEA \)

From the exterior angle theorem \( m\angle HEA=m\angle L + m\angle EAL \). Substitute this into the equation \( m\angle H + m\angle L + m\angle EAL + m\angle EAJ + m\angle J = 360^\circ \)

We get \( m\angle H + m\angle HEA+ m\angle EAJ + m\angle J=360^\circ \)

This shows the logical flow of using triangle - related theorems (angle - sum and exterior angle) and linear pair to derive the final equation.

Answer:

The diagram shows the derivation of the equation \( m\angle H + m\angle HEA+ m\angle EAJ + m\angle J = 360^\circ \) by using the triangle angle - sum theorem (\( m\angle H + m\angle L + m\angle J=180^\circ \)), the linear pair theorem (\( m\angle EAL + m\angle EAJ = 180^\circ \)) and the triangle exterior angle theorem (\( m\angle HEA=m\angle L + m\angle EAL \)) to combine the sums of angles and arrive at the result.