Question

proving a theorem complete the paragraph proof for the congruent complements theorem. given: \\( \angle 1 \\) and \\( \angle 2 \\) are complementary. \\( \angle 1 \\) and \\( \angle 3 \\) are complementary. prove: \\( \angle 2 \cong \angle 3 \\) \\( \angle 1 \\) and \\( \angle 2 \\) are complementary, and \\( \angle 1 \\) and \\( \angle 3 \\) are complementary. by the definition of angles, \\( m \angle 1 + m \angle 2 = 90 ^ { \circ } \\) and \\( = 90 ^ { \circ } \\). by the, \\( m \angle 1 + m \angle 2 = m \angle 1 + m \angle 3 \\). by the subtraction property of equality, so, \\( \angle 2 \cong \angle 3 \\) by the definition of.

Explanation:

Step1: Fill in the first blank

By the definition of complementary angles.

Step2: Fill in the second blank

Since \( \angle1\) and \( \angle3\) are complementary, \(m\angle1 + m\angle3=90^{\circ}\).

Step3: Fill in the third blank

Because \(m\angle1 + m\angle2 = 90^{\circ}\) and \(m\angle1 + m\angle3=90^{\circ}\), by the Transitive Property of Equality.

Step4: Fill in the fourth blank

Subtract \(m\angle1\) from both sides of \(m\angle1 + m\angle2 = m\angle1 + m\angle3\), we get \(m\angle2=m\angle3\).

Step5: Fill in the fifth blank

By the definition of congruent angles.

Answer:

\(\angle1\) and \(\angle2\) are complementary, and \(\angle1\) and \(\angle3\) are complementary. By the definition of \(\text{complementary}\) angles, \(m\angle1 + m\angle2 = 90^{\circ}\) and \(m\angle1 + m\angle3=90^{\circ}\). By the \(\text{Transitive Property of Equality}\), \(m\angle1 + m\angle2 = m\angle1 + m\angle3\). By the Subtraction Property of Equality, \(m\angle2 = m\angle3\). So, \(\angle2\cong\angle3\) by the definition of \(\text{congruent angles}\).