Question

which statements are always true regarding the diagram? check all that apply.
□ ( mangle3 + mangle4 = 180^circ )
□ ( mangle2 + mangle4 + mangle6 = 180^circ )
□ ( mangle2 + mangle4 = mangle5 )
□ ( mangle1 + mangle2 = 90^circ )
□ ( mangle4 + mangle6 = mangle2 )
□ ( mangle2 + mangle6 = mangle5 )

Explanation:

Step1: Analyze \( m\angle3 + m\angle4 = 180^\circ \)

\(\angle3\) and \(\angle4\) form a linear pair (they are adjacent and their non - common sides form a straight line). By the definition of a linear pair, the sum of their measures is \(180^\circ\). So \(m\angle3 + m\angle4=180^\circ\) is always true.

Step2: Analyze \( m\angle2 + m\angle4 + m\angle6 = 180^\circ \)

The sum of the interior angles of a triangle is \(180^\circ\). In the triangle formed by \(\angle2\), \(\angle4\), and \(\angle6\) (assuming the figure is a triangle with these angles as interior angles), the sum of these three angles should be \(180^\circ\). So \(m\angle2 + m\angle4 + m\angle6 = 180^\circ\) is always true.

Step3: Analyze \( m\angle2 + m\angle4 = m\angle5 \)

By the exterior - angle theorem of a triangle, the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. \(\angle5\) is an exterior angle of the triangle with interior angles \(\angle2\) and \(\angle4\) (non - adjacent to \(\angle5\)). So \(m\angle2 + m\angle4=m\angle5\) is always true.

Step4: Analyze \( m\angle1 + m\angle2 = 90^\circ \)

There is no information given that \(\angle1\) and \(\angle2\) are complementary (sum to \(90^\circ\)). They form a linear pair, so \(m\angle1 + m\angle2 = 180^\circ\) (if they are adjacent and form a straight line), not necessarily \(90^\circ\). So this statement is not always true.

Step5: Analyze \( m\angle4 + m\angle6 = m\angle2 \)

From the triangle angle - sum property, \(m\angle2+m\angle4 + m\angle6=180^\circ\), and there is no reason for \(m\angle4 + m\angle6=m\angle2\) to hold. In fact, \(m\angle2=180^\circ-(m\angle4 + m\angle6)\), so this statement is false.

Step6: Analyze \( m\angle2 + m\angle6 = m\angle5 \)

By the exterior - angle theorem, the exterior angle \(\angle5\) should be equal to the sum of the two non - adjacent interior angles, which are \(\angle2\) and \(\angle4\), not \(\angle2\) and \(\angle6\). So this statement is false.

Answer:

\(m\angle3 + m\angle4 = 180^\circ\), \(m\angle2 + m\angle4 + m\angle6 = 180^\circ\), \(m\angle2 + m\angle4 = m\angle5\)