Question

question 1 of 10
which of the following statements must be true about this diagram? check all that apply.

a. the degree measure of \\( \angle 3 \\) equals the sum of the degree measures of \\( \angle 1 \\) and \\( \angle 2 \\).
b. the degree measure of \\( \angle 4 \\) equals the sum of the degree measures of \\( \angle 1 \\) and \\( \angle 2 \\).
c. \\( m \angle 4 \\) is greater than \\( m \angle 1 \\).
d. \\( m \angle 3 \\) is greater than \\( m \angle 2 \\)
e. \\( m \angle 4 \\) is greater than \\( m \angle 2 \\).
f. the degree measure of \\( \angle 4 \\) equals the sum of the degree measures of \\( \angle 2 \\) and \\( \angle 3 \\).

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 of a triangle is greater than either of the non - adjacent interior angles.

Step2: Analyze each option

• Option A:

By the exterior - angle theorem, \(m\angle4=m\angle1 + m\angle2\), not \(m\angle3=m\angle1 + m\angle2\). So, option A is false.

• Option B:

Since \(m\angle4=m\angle1 + m\angle2\) (exterior - angle theorem), option B is true.

• Option C:

Because \(m\angle4=m\angle1 + m\angle2\) and \(m\angle2>0\), then \(m\angle4>m\angle1\). Option C is true.

• Option D:

There is no information or theorem to support \(m\angle3>m\angle2\). In fact, \(m\angle3 + m\angle4 = 180^{\circ}\) (linear pair) and \(m\angle4=m\angle1 + m\angle2\). There is no direct relationship given to compare \(\angle3\) and \(\angle2\) in terms of size. So, option D is false.

• Option E:

Since \(m\angle4=m\angle1 + m\angle2\) and \(m\angle1>0\), then \(m\angle4>m\angle2\). Option E is true.

• Option F:

\(\angle3\) and \(\angle4\) are a linear pair (\(m\angle3 + m\angle4=180^{\circ}\)), and \(m\angle1 + m\angle2 + m\angle3=180^{\circ}\) (angle - sum property of a triangle). So, \(m\angle4=m\angle1 + m\angle2
eq m\angle2 + m\angle3\). Option F is false.

Answer:

B. The degree measure of \(\angle4\) equals the sum of the degree measures of \(\angle1\) and \(\angle2\); C. \(m\angle4\) is greater than \(m\angle1\); E. \(m\angle4\) is greater than \(m\angle2\)