Question

complete the proof that vertical angles are congruent.
statement reason
1 $overleftrightarrow{ab}$ and $overleftrightarrow{cd}$ intersect at point o. given.
2 $mangle x + mangle z=180^{circ}$ these angles are so their measures sum to $180^{circ}$.
3 $mangle y + mangle z = 180^{circ}$ same as the previous reason.
4 $mangle x + mangle z=$ substitution (2, 3).
5 $mangle x=mangle y$ subtract $mangle z$ from both sides of the equation (4).

Explanation:

Step1: Identify angle - pair type

$\angle x$ and $\angle z$ are linear - pair angles. A linear pair of angles is formed when two lines intersect and they are adjacent angles that form a straight line. So, their measures sum to $180^{\circ}$.

Step2: Substitute the second equation

Since $m\angle x + m\angle z=180^{\circ}$ and $m\angle y + m\angle z = 180^{\circ}$, by substitution, we can write $m\angle x + m\angle z=m\angle y + m\angle z$.

Answer:

1. Linear - pair angles; 4. $m\angle y + m\angle z$