Question

12.) select which goes in the corresponding space reason (4)
given: <1~=<3
prove: <4~=<2
statement | reason

1. <1~=<3 | 1) given
2. <1~=<2 <3~=<4 | 2) vertical angles, shown in diagram
3. <3~=<2 | 3)

4)<4~=<2 | 4)
substitution property
definition of adjacent angles
congruent supplement theorem
pythagorean theorem

Explanation:

Step1: Analyze the given equal - angle relationships

We know that $\angle1\cong\angle3$ (given), $\angle1\cong\angle2$ and $\angle3\cong\angle4$ (vertical angles).

Step2: Determine the reason for $\angle4\cong\angle2$

Since $\angle3\cong\angle2$ (from transitive property using $\angle1\cong\angle3$ and $\angle1\cong\angle2$) and $\angle3\cong\angle4$, we can substitute $\angle3$ with $\angle4$ in the equation $\angle3\cong\angle2$. The property that allows us to replace one quantity with an equal quantity is the Substitution Property.

Answer:

Substitution Property