Question

which statements are always true regarding the diagram? choose three correct answers.

$mangle2 + mangle3 + mangle5 = 180^{circ}$

$mangle3 + mangle4 + mangle5 = 180^{circ}$

$mangle5 + mangle3 = mangle4$

$mangle2 + mangle3 = mangle6$

$mangle5 + mangle6 = 180^{circ}$

Explanation:

Step1: Analyze the first option

By the triangle angle - sum property, in a triangle, the sum of interior angles is \(180^{\circ}\). Here, \(\angle2+\angle3+\angle5 = 180^{\circ}\) (sum of interior angles of a triangle).

Step2: Analyze the second option

\(\angle3+\angle4+\angle5
eq180^{\circ}\) because \(\angle4\) is an exterior angle. The sum of angles in a triangle is for interior angles.

Step3: Analyze the third option

By the exterior - angle property of a triangle, an exterior angle (\(\angle4\)) is equal to the sum of the two non - adjacent interior angles (\(\angle2\) and \(\angle5\)), not \(\angle3+\angle5\).

Step4: Analyze the fourth option

By the exterior - angle property of a triangle, an exterior angle (\(\angle6\)) is equal to the sum of the two non - adjacent interior angles (\(\angle2\) and \(\angle3\)). So, \(m\angle2 + m\angle3=m\angle6\).

Step5: Analyze the fifth option

Since \(\angle5\) and \(\angle6\) are adjacent angles forming a linear pair, \(m\angle5 + m\angle6=180^{\circ}\) (linear pair postulate).

Answer:

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