Question

lines $overleftrightarrow{ab}$ and $overleftrightarrow{cd}$ intersect at point $o$.
complete the proof that vertical angles are congruent.
statement\treason
1 $overleftrightarrow{ab}$ and $overleftrightarrow{cd}$ intersect at point $o$.\tgiven.
2 $mangle x + mangle z=180^{circ}$\tthese angles are so their measures sum to $180^{circ}$.
3 $mangle y + mangle z = 180^{circ}$\tsame as the previous reason.
4 $mangle x + mangle z=square$\tsubstitution (2, 3).
5 $mangle x = mangle y$\tsubtract $mangle z$ from both sides of the equation (4).

Explanation:

Step1: Identify linear - pair angles

$\angle x$ and $\angle z$ form a linear - pair, and $\angle y$ and $\angle z$ form a linear - pair. Linear - pair angles are supplementary, so $m\angle x + m\angle z=180^{\circ}$ and $m\angle y + m\angle z = 180^{\circ}$.

Step2: Use substitution

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

Step3: Solve for angle equality

Subtract $m\angle z$ from both sides of the equation $m\angle x + m\angle z=m\angle y + m\angle z$. We get $m\angle x=m\angle y$.

Answer:

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