Question

question #10
select all the true statements given the figure below.
a ( m angle 4=m angle 2 )
b ( m angle 4=m angle 1+m angle 2 )
c ( m angle 4=180-m angle 3 )
d ( m angle 4=m angle 1+m angle 2+m angle 3 )
e ( m angle 4=m angle 3+m angle 1 )

Explanation:

Step1: Use the exterior angle theorem

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. In the given triangle, \(\angle4\) is an exterior angle, and \(\angle1\) and \(\angle2\) are the two non - adjacent interior angles. So, \(m\angle4=m\angle1 + m\angle2\) (Option B).

Step2: Use the linear pair property

\(\angle3\) and \(\angle4\) form a linear pair. By the linear pair property, \(m\angle3+m\angle4 = 180^{\circ}\), which can be rewritten as \(m\angle4=180 - m\angle3\) (Option C).

Step3: Use the triangle angle - sum property

The sum of the interior angles of a triangle is \(180^{\circ}\), so \(m\angle1+m\angle2+m\angle3 = 180^{\circ}\). Since \(m\angle4=m\angle1 + m\angle2\), substituting \(m\angle1 + m\angle2\) with \(m\angle4\) gives \(m\angle4+m\angle3 = 180^{\circ}\), or \(m\angle4=180-(m\angle3)\). Also, from the exterior angle theorem \(m\angle4=m\angle1 + m\angle2
eq m\angle2\) (Option A is wrong), \(m\angle4
eq m\angle3 + m\angle1\) (Option E is wrong), and \(m\angle4
eq m\angle1 + m\angle2+m\angle3\) (Option D is wrong)

Answer:

B. \(m\angle4=m\angle1 + m\angle2\), C. \(m\angle4=180 - m\angle3\)