Question

lesson 9.4 checkpoint
complete the previous problems, check your solutions, then complete the lesson checkpoint below.
complete the lesson reflection above by circling your current understanding of the learning goal(s).
a12) use the reason bank to write a two-column proof for the transitive property of angle congruence.
given: ∠a ≅ ∠b and ∠b ≅ ∠c
prove: ∠a ≅ ∠c

statements | reasons

1. ∠a ≅ ∠b | 1. given
2. m∠a = m∠b | 2.
3. ∠b ≅ ∠c | 3.
4. m∠b = m∠c | 4.
5. m∠a = m∠c | 5.
6. ∠a ≅ ∠c | 6.

reasons bank
definition of congruent angles
given
transitive property of equality

Explanation:

Step1: Reason for Statement 2

If two angles are congruent, their measures are equal (Definition of congruent angles). So for $\angle A \cong \angle B$, we have $m\angle A = m\angle B$.
Reason: Definition of congruent angles

Step2: Reason for Statement 3

This is given in the problem as $\angle B \cong \angle C$.
Reason: Given

Step3: Reason for Statement 4

Similar to Step 1, using the Definition of congruent angles for $\angle B \cong \angle C$, we get $m\angle B = m\angle C$.
Reason: Definition of congruent angles

Step4: Reason for Statement 5

We know $m\angle A = m\angle B$ (from Step 2) and $m\angle B = m\angle C$ (from Step 4). By the Transitive Property of Equality (if $a = b$ and $b = c$, then $a = c$), we have $m\angle A = m\angle C$.
Reason: Transitive Property of Equality

Step5: Reason for Statement 6

If the measures of two angles are equal, then the angles are congruent (Definition of congruent angles). Since $m\angle A = m\angle C$, we have $\angle A \cong \angle C$.
Reason: Definition of congruent angles

Filling the Two - Column Proof:

STATEMENTS	REASONS
2. $m\angle A = m\angle B$	2. Definition of congruent angles
3. $\angle B \cong \angle C$	3. Given
4. $m\angle B = m\angle C$	4. Definition of congruent angles
5. $m\angle A = m\angle C$	5. Transitive Property of Equality
6. $\angle A \cong \angle C$	6. Definition of congruent angles

Answer:

The two - column proof is filled as shown in the above table. The reasons for each statement are: 2. Definition of congruent angles; 3. Given; 4. Definition of congruent angles; 5. Transitive Property of Equality; 6. Definition of congruent angles.