Question

complete the proof by choosing the correct
eason\.
given: ( mangle cye = mangle fxh )
( mangle 2 = mangle 4 )
prove: ( mangle 1 = mangle 3 )

statement	reason
2 ( mangle fxh = mangle 3 + mangle 4 )	reason
3 ( mangle cye = mangle fxh )	given
4 ( mangle 1 + mangle 2 = mangle 3 + mangle )	reflexive property, transitive property, addition and subtraction properties, multiplication and division properties, substitution property, definition of congruent segments, segment addition property, definition of congruent angles, angle addition property
5 ( mangle 2 = mangle 4 )
6 ( mangle 1 + mangle 4 = mangle 3 + mangle )
7 ( mangle 1 = mangle 3 )

Explanation:

Step1: Match Statement 1 reason

Statement 1: $m\angle CYE = m\angle 1 + m\angle 2$
This uses the Angle Addition Property (the measure of an angle formed by two adjacent angles is the sum of their measures).

Step2: Match Statement 2 reason

Statement 2: $m\angle FXH = m\angle 3 + m\angle 4$
This also uses the Angle Addition Property (same logic as Step 1, for the angle at point $X$).

Step3: Match Statement 4 reason

Statement 4: $m\angle 1 + m\angle 2 = m\angle 3 + m\angle 4$
Substitute $m\angle CYE = m\angle 1 + m\angle 2$ and $m\angle FXH = m\angle 3 + m\angle 4$ into the given $m\angle CYE = m\angle FXH$. This uses the Substitution Property.

Step4: Match Statement 5 reason

Statement 5: $m\angle 2 = m\angle 4$
This is given directly in the problem, so the reason is Given.

Step5: Match Statement 6 reason

Statement 6: $m\angle 1 + m\angle 4 = m\angle 3 + m\angle 4$
Substitute $m\angle 2 = m\angle 4$ into $m\angle 1 + m\angle 2 = m\angle 3 + m\angle 4$. This uses the Substitution Property.

Step6: Match Statement 7 reason

Statement 7: $m\angle 1 = m\angle 3$
Subtract $m\angle 4$ from both sides of $m\angle 1 + m\angle 4 = m\angle 3 + m\angle 4$. This uses the Addition and Subtraction Properties.

Answer:

Statement	Reason
2 $m\angle FXH = m\angle 3 + m\angle 4$	Angle Addition Property
3 $m\angle CYE = m\angle FXH$	Given
4 $m\angle 1 + m\angle 2 = m\angle 3 + m\angle 4$	Substitution Property
5 $m\angle 2 = m\angle 4$	Given
6 $m\angle 1 + m\angle 4 = m\angle 3 + m\angle 4$	Substitution Property
7 $m\angle 1 = m\angle 3$	Addition and Subtraction Properties