Question

1. proof copy and complete the two - column proof. then write a flowchart proof. (see example 2.) given ∠abd is a right angle. ∠cbe is a right angle. prove ∠abc≅∠dbe statements 1. ∠abd is a right angle. ∠cbe is a right angle. 2. ∠abc and ∠cbd are complementary. 3. ∠dbe and ∠cbd are complementary. 4. ∠abc≅∠dbe reasons 1. ______ 2. definition of complementary angles 3. ____ 4. ______

Explanation:

Step1: Given information

Given

Step2: Complementary - angle reasoning

Since \(\angle ABD = 90^{\circ}\), \(\angle ABC+\angle CBD=90^{\circ}\) (by the definition of a right - angle). Similarly, since \(\angle CBE = 90^{\circ}\), \(\angle DBE+\angle CBD = 90^{\circ}\) (by the definition of a right - angle).

Step3: Substitution property

If \(\angle ABC+\angle CBD = 90^{\circ}\) and \(\angle DBE+\angle CBD=90^{\circ}\), then \(\angle ABC+\angle CBD=\angle DBE+\angle CBD\) (both equal to \(90^{\circ}\)).

Step4: Subtraction property of equality

Subtracting \(\angle CBD\) from both sides of the equation \(\angle ABC+\angle CBD=\angle DBE+\angle CBD\), we get \(\angle ABC=\angle DBE\) (by the subtraction property of equality for angles). So \(\angle ABC\cong\angle DBE\) (congruent angles have equal measures).

Answer:

1. Given
2. Definition of complementary angles (because \(\angle CBE\) is a right - angle)
3. If two angles are complementary to the same angle, then they are congruent (or Subtraction property of equality for angles)

For the flowchart proof:

Statement	Reason
\(\angle ABC+\angle CBD = 90^{\circ}\), \(\angle DBE+\angle CBD=90^{\circ}\)	Definition of right - angle
\(\angle ABC+\angle CBD=\angle DBE+\angle CBD\)	Substitution (both equal \(90^{\circ}\))
\(\angle ABC=\angle DBE\)	Subtraction property of equality
\(\angle ABC\cong\angle DBE\)	Congruent angles have equal measures