Question

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Explanation:

Step1: Fill theorem blank

From the Third Angles Theorem, if $\angle A \cong \angle X$ and $\angle B \cong \angle Y$, then $\boldsymbol{\angle C \cong \angle Z}$.

Step2: Reason 2: Define congruent angles

Congruent angles have equal measures.
Reason 2: Definition of congruent angles

Step3: Statement 3: Triangle angle sum

The sum of angles in a triangle is $180^\circ$.
$m\angle A + m\angle B + m\angle C = 180^\circ$,
$m\angle X + m\angle Y + m\angle Z = 180^\circ$
Reason 3: Triangle Angle Sum Theorem

Step4: Reason 4: Transitive property

Equal quantities are equal to each other.
Reason 4: Transitive Property of Equality

Step5: Reason 5: Substitution property

Replace $m\angle A$ with $m\angle X$, $m\angle B$ with $m\angle Y$.
Reason 5: Substitution Property of Equality

Step6: Reason 6: Subtraction property

Subtract $m\angle X + m\angle Y$ from both sides.
Reason 6: Subtraction Property of Equality

Step7: Reason 7: Define congruent angles

Angles with equal measures are congruent.
Reason 7: Definition of congruent angles

Answer:

Third Angles Theorem Blank:

$\angle C \cong \angle Z$

Proof Completion:

Statements	Reasons
2) $m\angle A = m\angle X, m\angle B = m\angle Y$	2) Definition of $\boldsymbol{congruent\ angles}$.
3) $m\angle A + m\angle B + m\angle C = \boldsymbol{180^\circ}$, $m\angle X + m\angle Y + m\angle Z = \boldsymbol{180^\circ}$	3) $\boldsymbol{Triangle\ Angle\ Sum\ Theorem}$
4) $m\angle A + m\angle B + m\angle C = m\angle X + m\angle Y + m\angle Z$	4) $\boldsymbol{Transitive\ Property\ of\ Equality}$
5) $m\angle X + m\angle Y + m\angle C = m\angle X + m\angle Y + m\angle Z$	5) $\boldsymbol{Substitution\ Property\ of\ Equality}$
6) $m\angle C = m\angle Z$	6) $\boldsymbol{Subtraction}$ Property of Equality
7) $\angle C \cong \angle Z$	7) Definition of $\boldsymbol{congruent\ angles}$.