Question

1. proof complete the two - column proof of the third angles theorem by providing the reason for each statement.

given: \\( \angle p \cong \angle x \\) and \\( \angle q \cong \angle y \\)
prove: \\( \angle r \cong \angle z \\)

1. \\( \angle p \cong \angle x, \angle q \cong \angle y \\)
2. \\( m \angle p = m \angle x, m \angle q = m \angle y \\)
3. \\( m \angle p + m \angle q + m \angle r = 180 ^ { \circ } \\)

\\( 180 ^ { \circ } = m \angle x + m \angle y + m \angle z \\)

1. \\( m \angle p + m \angle q + m \angle r = m \angle x + m \angle y + m \angle z \\)
2. \\( m \angle x + m \angle y + m \angle r = m \angle x + m \angle y + m \angle z \\)
3. \\( m \angle r = m \angle z \\)
4. \\( \angle r \cong \angle z \\)

Explanation:

Step1: Given

The first statement is given in the problem.

Step2: Definition of congruent angles

If two angles are congruent, their measures are equal.

Step3: Triangle Angle - Sum Theorem

The sum of the interior angles of a triangle is \(180^{\circ}\).

Step4: Transitive Property of Equality

Since \(m\angle P + m\angle Q + m\angle R=180^{\circ}\) and \(180^{\circ}=m\angle X + m\angle Y + m\angle Z\), then \(m\angle P + m\angle Q + m\angle R=m\angle X + m\angle Y + m\angle Z\) by transitive property.

Step5: Substitution Property of Equality

Because \(m\angle P = m\angle X\) and \(m\angle Q = m\angle Y\), we substitute \(m\angle P\) with \(m\angle X\) and \(m\angle Q\) with \(m\angle Y\) in \(m\angle P + m\angle Q + m\angle R=m\angle X + m\angle Y + m\angle Z\).

Step6: Subtraction Property of Equality

Subtract \(m\angle X + m\angle Y\) from both sides of \(m\angle X + m\angle Y + m\angle R=m\angle X + m\angle Y + m\angle Z\).

Step7: Definition of congruent angles

If \(m\angle R = m\angle Z\), then \(\angle R\cong\angle Z\) by the definition of congruent angles.

Answer:

1. Given
2. Definition of congruent angles
3. Triangle Angle - Sum Theorem
4. Transitive Property of Equality
5. Substitution Property of Equality
6. Subtraction Property of Equality
7. Definition of congruent angles