Question

question 14. which statements must be true about these angles? choose all that are correct. a. ∠1≅∠4 b. m∠2 + m∠4 + m∠6 = 180° c. ∠3≅∠5 d. m∠2 + m∠3 + m∠4 = 180°

Explanation:

Step1: Recall vertical - angle property

Vertical angles are congruent. $\angle1$ and $\angle3$ are vertical angles, $\angle2$ and $\angle4$ are vertical angles, $\angle5$ and $\angle1$ are vertical angles, $\angle6$ and $\angle2$ are vertical angles.

Step2: Analyze option A

$\angle1$ and $\angle4$ are not vertical angles and there is no information to suggest $\angle1\cong\angle4$. So option A is incorrect.

Step3: Analyze option B

$\angle2$, $\angle4$, and $\angle6$ are non - adjacent angles that together form a full - circle (a complete rotation around a point). The sum of the measures of angles around a point is $360^{\circ}$, not $180^{\circ}$. So option B is incorrect.

Step4: Analyze option C

$\angle3$ and $\angle5$ are vertical angles. By the vertical - angle theorem, vertical angles are congruent. So $\angle3\cong\angle5$. Option C is correct.

Step5: Analyze option D

$\angle2$, $\angle3$, and $\angle4$ are adjacent angles that form a straight - line. The sum of the measures of angles on a straight - line is $180^{\circ}$. So $m\angle2 + m\angle3+m\angle4 = 180^{\circ}$. Option D is correct.

Answer:

C. $\angle3\cong\angle5$
D. $m\angle2 + m\angle3 + m\angle4=180^{\circ}$