Question

question 1/1 complete the following proof using theorems, definitions, and/or properties to prove the vertical angles theorem. given: ∠11 and ∠13 are a linear pair; ∠13 and ∠12 are a linear pair prove: m∠11 = m∠12 1. ∠11 and ∠13 are a linear pair, ∠13 and ∠12 are a linear pair given 2. ∠11 and ∠13 are supplementary, ∠13 and ∠12 are supplementary linear pair theorem 3. m∠11 + m∠13 = 180°, m∠13 + m∠12 = 180° definition of supplementary angles 4. m∠11 + m∠13 = m∠13 + m∠12 substitution property 5. m∠11 = m∠12 subtraction property drag & drop the answer note: use ctrl+d to drag the option via keyboard (diagram shows intersecting lines with angles labeled 11, 12, 13, 14)

Explanation:

Step1: Fill in the first blank (Given)

The first statement is given, so we use "Given" for the first reason.

Step2: Fill in the second blank (Linear Pair Theorem)

For the statement "∠11 and ∠13 are supplementary", the reason is the Linear Pair Theorem (since a linear pair of angles is supplementary).

Step3: Fill in the third blank (Given)

The statement "∠13 and ∠12 are a linear pair" is given, so the reason is "Given".

Step4: Fill in the fourth blank (Linear Pair Theorem)

For the statement "m∠13 + m∠12 = 180°", the reason is the Linear Pair Theorem (linear pair implies supplementary, so their measures sum to 180°).

Step5: Fill in the fifth blank (Substitution Property)

For the statement "m∠11 + m∠13 = m∠13 + m∠12", we substitute the equal sums (from steps 2 and 4), so the reason is Substitution Property.

Step6: Fill in the sixth blank (Subtraction Property)

For the statement "m∠11 = m∠12", we subtract \( m\angle13 \) from both sides of the equation in step 5, so the reason is Subtraction Property.

Answer:

1. Reason: Given
2. Reason: Linear Pair Theorem
3. Reason: Given
4. Reason: Linear Pair Theorem
5. Reason: Substitution Property
6. Reason: Subtraction Property