Question

what is the sum of all the angle measures in the diagram?
$(mangle1 + mangle4)+(mangle2 + mangle5)+(mangle3 + mangle6)=$

Explanation:

Step1: Recall triangle angle sum and linear pairs

The sum of angles in a triangle is \(180^\circ\), so \(m\angle1 + m\angle2 + m\angle3 = 180^\circ\). Also, \(\angle4\) and \(\angle1\) form a linear pair, \(\angle5\) and \(\angle2\) form a linear pair, \(\angle6\) and \(\angle3\) form a linear pair. A linear pair sums to \(180^\circ\), so \(m\angle4 = 180^\circ - m\angle1\), \(m\angle5 = 180^\circ - m\angle2\), \(m\angle6 = 180^\circ - m\angle3\).

Step2: Substitute and sum

Substitute into the total sum: \((m\angle1 + m\angle4)+(m\angle2 + m\angle5)+(m\angle3 + m\angle6)\)
\(=(m\angle1 + 180^\circ - m\angle1)+(m\angle2 + 180^\circ - m\angle2)+(m\angle3 + 180^\circ - m\angle3)\)
Simplify each term: \(180^\circ + 180^\circ + 180^\circ\)

Step3: Calculate the total

\(180^\circ\times3 = 540^\circ\)

Answer:

\(540\)