Question

1. proving a theorem copy and complete the paragraph proof for the congruent complements theorem (theorem 2.5). then write a two - column proof. (see example 3.) given ∠1 and ∠2 are complementary. ∠1 and ∠3 are complementary. prove ∠2≅∠3 ∠1 and ∠2 are complementary, and ∠1 and ∠3 are complementary. by the definition of __________ angles, m∠1 + m∠2 = 90° and ________ = 90°. by the ________, m∠1 + m∠2 = m∠1 + m∠3. by the subtraction property of equality, ________. so, ∠2≅∠3 by the definition of __________.

Explanation:

Step1: Recall complementary - angle definition

By the definition of complementary angles, if two angles are complementary, the sum of their measures is 90 degrees. So, if $\angle1$ and $\angle2$ are complementary, $m\angle1 + m\angle2=90^{\circ}$, and if $\angle1$ and $\angle3$ are complementary, $m\angle1 + m\angle3 = 90^{\circ}$.

Step2: Use transitive property

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

Step3: Apply subtraction property

By the Subtraction Property of Equality, we can subtract $m\angle1$ from both sides of the equation $m\angle1 + m\angle2=m\angle1 + m\angle3$. So, $m\angle2=m\angle3$.

Step4: Use congruent - angle definition

Since $m\angle2 = m\angle3$, by the definition of congruent angles, $\angle2\cong\angle3$.

Answer:

$\angle1$ and $\angle2$ are complementary, and $\angle1$ and $\angle3$ are complementary. By the definition of complementary angles, $m\angle1 + m\angle2 = 90^{\circ}$ and $m\angle1 + m\angle3$ $= 90^{\circ}$. By the 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 congruent angles.