Question

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

Explanation:

Step1: Recall vertical - angle property

Vertical angles are congruent. $\angle1$ and $\angle4$ are vertical angles, so $\angle1\cong\angle4$.

Step2: Recall angle - sum property around a point

The sum of angles around a point is $360^{\circ}$. Also, since $\angle2+\angle3+\angle4+\angle5+\angle6+\angle1 = 360^{\circ}$ and $\angle1\cong\angle4$, we know that $\angle2+\angle3+\angle5+\angle6 = 180^{\circ}$ (because $\angle1$ and $\angle4$ are equal).
$\angle3$ and $\angle5$ are vertical angles, so $\angle3\cong\angle5$.

Step3: Analyze angle - sum in a half - rotation

Angles in a half - rotation (a straight - line) sum to $180^{\circ}$. $\angle2+\angle3+\angle4$ form a straight - line, so $m\angle2 + m\angle3+m\angle4=180^{\circ}$.
Since $\angle2+\angle3+\angle4 = 180^{\circ}$ and $\angle1\cong\angle4$, $\angle3\cong\angle5$, we can rewrite the equation for the sum of angles around a point to show that $m\angle2 + m\angle3=m\angle5 + m\angle6$.

Answer:

$\angle1\cong\angle4$
$m\angle2 + m\angle4 + m\angle6=180^{\circ}$
$\angle3\cong\angle5$
$m\angle2 + m\angle3 + m\angle4=180^{\circ}$
$m\angle2 + m\angle3=m\angle5 + m\angle6$