Question

complete the two - column proof using the selections given. write the letter next to the number it corresponds with.
given: $overrightarrow{mp}$ bisects $angle lmn$
prove: $2mangle lmp = mangle lmn$
diagram: a triangle - like figure with vertex m, and points l, p, n on a line below m, with $overrightarrow{ml}$, $overrightarrow{mp}$, $overrightarrow{mn}$ as rays

statements	reasons
2. $angle lmp cong angle nmp$	2.
3.	3. definition of congruent angles
4. $mangle lmp + mangle nmp = mangle lmn$	4. angle addition postulate
5. $mangle lmp + mangle lmp = mangle lmn$	5. substitution property of equality
6. $2(mangle lmp) = mangle lmn$	6.

options:
a. $angle lmp cong angle nmp$
b. distributive property
c. multiplication prop. of equality
d. definition of congruent segments
e. definition of angle bisector

Explanation:

Step 1: Reason for Statement 2

The definition of an angle bisector states that if a ray bisects an angle, it divides the angle into two congruent angles. Since \(\overrightarrow{MP}\) bisects \(\angle LMN\), by the Definition of Angle Bisector (Option E), we have \(\angle LMP\cong\angle NMP\). So the reason for statement 2 is E.

Step 2: Statement for Reason 3

The Definition of Congruent Angles says that if two angles are congruent, then their measures are equal. So if \(\angle LMP\cong\angle NMP\), then \(m\angle LMP = m\angle NMP\). So statement 3 is \(m\angle LMP = m\angle NMP\).

Step 3: Reason for Statement 6

We have \(m\angle LMP + m\angle LMP = m\angle LMN\), which is \(2(m\angle LMP)=m\angle LMN\). This is using the Multiplication Property of Equality (or the Distributive Property in the sense of combining like terms, but more accurately the Multiplication Property of Equality which allows us to factor out the 2). The Multiplication Property of Equality (Option C) justifies this step as we are essentially multiplying \(m\angle LMP\) by 2 (since \(m\angle LMP + m\angle LMP = 2\times m\angle LMP\)). So the reason for statement 6 is C.

Answer:

1. E
2. \(m\angle LMP = m\angle NMP\)
3. C