Question

proof - complete the two - column proof. (see example 2.)

1. given ∠abd is a right angle.

∠cbe is a right angle.
prove ∠abc ≅ ∠dbe
two - column proof
statements reasons

1. ∠abd is a right angle.

∠cbe is a right angle. 1.

1. ∠abd are complementary. 2.
2. ∠cbe are complementary. 3.
3. ∠abc ≅ ∠dbe 4.

reasons bank: definition of complementary angles (×2), congruent complements theorem, given

proof - complete the paragraph proof for the congruent complements theorem. then write a two - column proof. (see example 3.)

1. given ∠1 and ∠2 are complementary.

∠1 and ∠3 are complementary.
prove ∠2 ≅ ∠3
paragraph proof
∠1 and ∠2 are complementary, and ∠1 and ∠3 are complementary.
by definition of ____ angles, m∠1 + m∠2 = 90° and ____ = 90°.
by the ______ property of equality, m∠1 + m∠2 = m∠1 + m∠3.
by the subtraction property of equality, m∠2 = ______.
so, ∠2 ≅ ∠3 by the definition of ______ angles.
bank: m∠3, m∠1 + m∠3, transitive, complementary, congruent

Explanation:

Problem 18 (Two-Column Proof)

Step1: State given info

1. $\angle ABD$ is a right angle.

$\angle CBE$ is a right angle.
Reason: Given

Step2: Define complementary pairs

1. $\angle ABC$ and $\angle CBD$ are complementary.

Reason: Definition of complementary angles ($\times 2$)

Step3: Define second complementary pair

1. $\angle DBE$ and $\angle CBD$ are complementary.

Reason: Definition of complementary angles ($\times 2$)

Step4: Apply congruent complements theorem

1. $\angle ABC \cong \angle DBE$

Reason: Congruent Complements Theorem

Step1: Fill paragraph proof blanks

Complete the paragraph using the bank terms:
$\angle 1$ and $\angle 2$ are complementary, and $\angle 1$ and $\angle 3$ are complementary.
By definition of complementary angles, $m\angle 1 + m\angle 2 = 90^\circ$ and $\boldsymbol{m\angle 1 + m\angle 3}=90^\circ$.
By the Transitive Property of Equality, $m\angle 1 + m\angle 2 = m\angle 1 + m\angle 3$.
By the Subtraction Property of Equality, $m\angle 2 = \boldsymbol{m\angle 3}$.
So, $\angle 2 \cong \angle 3$ by the definition of congruent angles.

Step2: Build two-column proof

Translate the paragraph into structured proof:

Answer:

STATEMENTS	REASONS
2. $\angle ABC$ and $\angle CBD$ are complementary.	2. Definition of complementary angles ($\times 2$)
3. $\angle DBE$ and $\angle CBD$ are complementary.	3. Definition of complementary angles ($\times 2$)
4. $\angle ABC \cong \angle DBE$	4. Congruent Complements Theorem

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Problem 19 (Paragraph Proof + Two-Column Proof)