Question

1. proof copy and complete the flowchart proof. then write a two - column proof. given ∠1 = ∠3 prove ∠2 ≅ ∠4 18. proof copy and complete the two - column proof. then write a flowchart proof. given ∠abd is a right angle. ∠cbe is a right angle. prove ∠abc = ∠dbe 19. proving a theorem copy and complete the paragraph proof for the congruent complements theorem (theorem 2.5). then write a two - column proof. 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:

$\angle1=\angle3$ (Given)

Step2: Use vertical - angles congruence

$\angle1\cong\angle2$ and $\angle3\cong\angle4$ (Vertical Angles Congruence Theorem)

Step3: Substitute equal angles

Since $\angle1 = \angle3$ and $\angle1\cong\angle2$, $\angle3\cong\angle4$, we can substitute. So $\angle2\cong\angle4$ (Transitive Property of Congruence)

Step1: Given information

$\angle ABD$ is a right - angle, $\angle CBE$ is a right - angle (Given)

Step2: Define complementary angles

$\angle ABC$ and $\angle CBD$ are complementary because $\angle ABD = 90^{\circ}$ (Definition of complementary angles: two angles are complementary if their sum is $90^{\circ}$)
$\angle DBE$ and $\angle CBD$ are complementary because $\angle CBE=90^{\circ}$ (Definition of complementary angles)

Step3: Use the Congruent Complements Theorem

Since $\angle ABC$ and $\angle DBE$ are both complements of $\angle CBD$, $\angle ABC\cong\angle DBE$ (Congruent Complements Theorem)

Step1: Use the definition of complementary angles

$\angle1$ and $\angle2$ are complementary, so $m\angle1 + m\angle2=90^{\circ}$; $\angle1$ and $\angle3$ are complementary, so $m\angle1 + m\angle3 = 90^{\circ}$ (Definition of complementary angles)

Step2: Set the two equations equal

Since $m\angle1 + m\angle2=90^{\circ}$ and $m\angle1 + m\angle3 = 90^{\circ}$, we have $m\angle1 + m\angle2=m\angle1 + m\angle3$ (Transitive Property of Equality)

Step3: Apply the Subtraction Property of Equality

Subtract $m\angle1$ from both sides of the equation $m\angle1 + m\angle2=m\angle1 + m\angle3$. We get $m\angle2=m\angle3$. So $\angle2\cong\angle3$ (Definition of congruent angles: two angles are congruent if they have the same measure)

Answer:

Statements	Reasons
2. $\angle1\cong\angle2,\angle3\cong\angle4$	2. Vertical Angles Congruence Theorem
3. $\angle2\cong\angle3$	3. Substitution (using $\angle1=\angle3$ and $\angle1\cong\angle2,\angle3\cong\angle4$)
4. $\angle2\cong\angle4$	4. Transitive Property of Congruence

18.