Question

question 1 of 10 which of the following statements must be true about this diagram? check all that apply. a. $mangle3$ is greater than $mangle2$ b. the degree measure of $angle3$ equals the sum of the degree measures of $angle1$ and $angle2$. c. the degree measure of $angle4$ equals the sum of the degree measures of $angle1$ and $angle2$. d. $mangle4$ is greater than $mangle1$. e. the degree measure of $angle4$ equals the sum of the degree measures of $angle2$ and $angle3$. f. $mangle4$ is greater than $mangle2$.

Explanation:

Step1: Recall exterior - angle theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. Also, an exterior angle is greater than either of the non - adjacent interior angles.
Let's assume the triangle has interior angles $\angle1$, $\angle2$, and the third interior angle (not labeled), and exterior angles $\angle3$ and $\angle4$.

Step2: Analyze option A

$\angle3$ and $\angle2$ are not related by the exterior - angle property in a way that $m\angle3>m\angle2$ must be true. There is no information to support this.

Step3: Analyze option B

There is no such relationship that $m\angle3=m\angle1 + m\angle2$.

Step4: Analyze option C

By the exterior - angle theorem, if $\angle4$ is an exterior angle of the triangle with non - adjacent interior angles $\angle1$ and $\angle2$, then $m\angle4=m\angle1 + m\angle2$. This is true.

Step5: Analyze option D

Since $\angle4$ is an exterior angle and $\angle1$ is a non - adjacent interior angle, by the exterior - angle property, $m\angle4>m\angle1$. This is true.

Step6: Analyze option E

There is no such relationship that $m\angle4=m\angle2 + m\angle3$.

Step7: Analyze option F

Since $\angle4$ is an exterior angle and $\angle2$ is a non - adjacent interior angle, by the exterior - angle property, $m\angle4>m\angle2$. This is true.

Answer:

C. The degree measure of $\angle4$ equals the sum of the degree measures of $\angle1$ and $\angle2$.
D. $m\angle4$ is greater than $m\angle1$.
F. $m\angle4$ is greater than $m\angle2$.