Question

question 5 of 10
which of the following statements must be true about this diagram? check all that apply.
a. ( mangle4 ) is greater than ( mangle2 )
b. ( mangle4 ) is greater than ( mangle1 )
c. ( mangle3 ) is greater than ( mangle2 )
d. ( mangle1 + mangle2 = mangle3 )
e. ( mangle2 + mangle3 = mangle4 )
f. ( mangle1 + mangle2 = mangle4 )

Explanation:

Step1: Recall 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. Also, an exterior angle is greater than either of the non - adjacent interior angles.

Step2: Analyze each option

• Option A:

Since \(\angle4\) is an exterior angle and \(\angle2\) is a non - adjacent interior angle. By the exterior - angle theorem, \(m\angle4=m\angle1 + m\angle2\). So \(m\angle4>m\angle2\) (because \(m\angle1>0\)).

• Option B:

Since \(\angle4\) is an exterior angle and \(\angle1\) is a non - adjacent interior angle. By the exterior - angle theorem, \(m\angle4=m\angle1 + m\angle2\). So \(m\angle4>m\angle1\) (because \(m\angle2>0\)).

• Option C:

\(\angle3\) and \(\angle2\) are adjacent interior angles. There is no general rule from the basic triangle angle - sum or exterior - angle theorems that \(m\angle3>m\angle2\). For example, if the triangle is isosceles with \(\angle1=\angle2\), \(m\angle3 = 180-(m\angle1 + m\angle2)\) and we cannot say \(m\angle3>m\angle2\) in all cases.

• Option D:

By the triangle angle - sum theorem, \(m\angle1 + m\angle2+m\angle3=180^{\circ}\), so \(m\angle1 + m\angle2=180 - m\angle3\). This is not equal to \(m\angle3\) (unless \(m\angle3 = 90^{\circ}\), which is not always the case).

• Option E:

\(\angle2+\angle3\) is the sum of two adjacent interior angles. \(\angle4\) is an exterior angle. There is no relation \(m\angle2 + m\angle3=m\angle4\) from the standard angle theorems.

• Option F:

By the exterior - angle theorem, for \(\angle4\) (an exterior angle of the triangle), \(m\angle4=m\angle1 + m\angle2\)

Answer:

A. \(m\angle4\) is greater than \(m\angle2\), B. \(m\angle4\) is greater than \(m\angle1\), F. \(m\angle1 + m\angle2=m\angle4\)