Question

1. select the correct proof from the options listed. given: m∠1 = m∠2 prove: m∠3 = m∠4 a. statements 1. m∠1 = m∠2 2. m∠1 = m∠3 3. m∠2 = m∠4 4. m∠3 = m∠2 5. m∠3 = m∠4 reasons 1. given 2. vertical angles are equal. 3. vertical angles are equal. 4. transitive property 5. substitution property b. statements 1. m∠1 = m∠2 2. m∠1 = m∠3 3. m∠2 = m∠4 4. m∠3 = m∠2 5. m∠3 = m∠4 reasons 1. given 2. given 3. acute angles are equal. 4. transitive property 5. self - evident c. statements 1. m∠1 = m∠2 2. m∠1 = m∠3 3. m∠2 = m∠4 4. m∠3 = m∠2 5. m∠3 = m∠4 reasons 1. given 2. adjacent angles are equal. 3. adjacent angles are equal. 4. transitive property 5. substitution property d. statements 1. m∠1 = m∠2 2. m∠1 = m∠3 3. m∠2 = m∠4 4. m∠3 = m∠2 5. m∠3 = m∠4 reasons 1. given 2. transitive property 3. vertical angles are equal. 4. vertical angles are equal. 5. substitution property e. statements 1. m∠1 = m∠2 2. m∠1 = m∠3 3. m∠2 = m∠4 4. m∠3 = m∠2 5. m∠3 = m∠4 reasons 1. vertical angles are equal. 2. given 3. complementary angles are equal. 4. transitive property 5. substitution property

Explanation:

Step1: Analyze Option A

• Statement 1 is given.
• Statement 2: $\angle1$ and $\angle3$ are vertical - angles, so $m\angle1 = m\angle3$ because vertical angles are equal.
• Statement 3: $\angle2$ and $\angle4$ are vertical - angles, so $m\angle2 = m\angle4$ because vertical angles are equal.
• Statement 4: Since $m\angle1 = m\angle2$ (given) and $m\angle1 = m\angle3$, by the transitive property $m\angle3 = m\angle2$.
• Statement 5: Since $m\angle2 = m\angle4$ and $m\angle3 = m\angle2$, by the substitution property $m\angle3 = m\angle4$. This option is correct.

Step2: Analyze Option B

• Reason 3 “Acute angles are equal” is incorrect. Just because angles are acute does not mean they are equal. Also, reason 5 “Self - evident” is not a valid geometric reason. So, this option is incorrect.

Step3: Analyze Option C

• Reason 2 and 3 “Adjacent angles are equal” is incorrect. Adjacent angles are not necessarily equal. So, this option is incorrect.

Step4: Analyze Option D

• Reason 2 “Transitive Property” is misused. There is no basis for using the transitive property in the second step as given. So, this option is incorrect.

Step5: Analyze Option E

• Reason 1 “Vertical angles are equal” is misused as the first statement is given and not a vertical - angle relationship. Reason 3 “Complementary angles are equal” is incorrect as there is no indication of complementary angles in the problem. So, this option is incorrect.

Answer:

A. Statements: 1. $m\angle1 = m\angle2$; 2. $m\angle1 = m\angle3$; 3. $m\angle2 = m\angle4$; 4. $m\angle3 = m\angle2$; 5. $m\angle3 = m\angle4$. Reasons: 1. Given; 2. Vertical angles are equal; 3. Vertical angles are equal; 4. Transitive Property; 5. Substitution Property.