Question

try it! write a proof using a theorem

1. write a two - column proof.

given: (mangle4 = 35), (mangle1=mangle2 + mangle4)
prove: (mangle3 = 70)

Explanation:

Step1: Substitute the value of $\angle4$

Given $m\angle4 = 35$ and $m\angle1=m\angle2 + m\angle4$. Since vertical - angles are equal, $\angle1$ and $\angle3$ are vertical angles, so $m\angle1=m\angle3$. Also, assume $\angle2=\angle4$ (vertically - opposite angles). Substitute $m\angle4 = 35$ into $m\angle1=m\angle2 + m\angle4$.

Step2: Calculate $m\angle1$

If $\angle2=\angle4 = 35$, then $m\angle1=m\angle2 + m\angle4=35 + 35=70$.

Step3: Use vertical - angle property

Since $\angle1$ and $\angle3$ are vertical angles, $m\angle3=m\angle1$. So $m\angle3 = 70$.

Statements	Reasons
$m\angle1=m\angle2 + m\angle4$	Given
$\angle2=\angle4$ (vertically - opposite angles)	Vertical - angle theorem
$m\angle1=m\angle2 + 35$	Substitution ($m\angle4 = 35$)
$m\angle2 = 35$	Vertical - angle property
$m\angle1=35 + 35=70$	Substitution
$\angle1$ and $\angle3$ are vertical angles	Definition of vertical angles
$m\angle3=m\angle1$	Vertical - angle theorem
$m\angle3 = 70$	Substitution ($m\angle1 = 70$)

Answer:

$m\angle3 = 70$