Question

given: ( mangle abd = 25^circ )
prove: ( mangle cbd = 65^circ )

julie’s proof

statements	reasons
2. ( mangle abd = 25^circ )	2. given
3. ( mangle abd + mangle cbd = 90^circ )	3. angle addition postulate
4. ( 25^circ + mangle cbd = 90^circ )	4. substitution property of equality
5. ( mangle cbd = 65^circ )	5. subtraction property of equality

samuel’s proof

statements	reasons

| 1. ( mangle cbd
eq 65^circ ) | 1. assumption |

2. ( mangle abc = 90^circ )	2. given
3. ( mangle abd + mangle cbd = 90^circ )	3. angle addition postulate

| 4. ( mangle abd + 65^circ
eq 90^circ ) | 4. substitution property of equality |
| 5. ( mangle abd
eq 25^circ ) | 5. subtraction property of equality |

julie used an indirect proof / a direct proof
samuel used an indirect proof / a direct proof

Explanation:

To determine which proof (Julie’s or Samuel’s) is an indirect proof, we analyze the structure of each:

Julie’s Proof (Direct Proof):

Julie uses given information (\( m\angle ABC = 90^\circ \), \( m\angle ABD = 25^\circ \)) and the angle addition postulate (\( m\angle ABD + m\angle CBD = 90^\circ \)) to directly solve for \( m\angle CBD \) (via substitution and subtraction). This follows a direct logical chain from known facts to the conclusion.

Samuel’s Proof (Indirect Proof):

Samuel assumes the opposite of the conclusion (Step 1: \( m\angle CBD
eq 65^\circ \)) and then shows this assumption leads to a contradiction (e.g., \( m\angle ABD + 65^\circ
eq 90^\circ \), but \( m\angle ABD = 25^\circ \) implies \( 25^\circ + 65^\circ = 90^\circ \), a contradiction). Indirect proofs (proof by contradiction) start with the negation of the desired conclusion and derive a logical inconsistency, proving the original statement.

Answer:

Samuel used an indirect proof.