Question

what is the missing statement? a $sm = mt$ b $sm+mt = st$ c $angle ums+angle umt=angle smt$ d $overleftrightarrow{um}$ is a perpendicular bisector of $overline{st}$. question 3 rivka is given the following scenario: given: $mangle dbg = 2(mangle dba)$ and $mangle ebg = 30^{circ}$ prove: $mangle ebc = 75^{circ}$ stating the given information, how might rivka continue her proof?

Explanation:

Step1: Analyze angle - relationships

Since $\angle ABG = 180^{\circ}$ (a straight - line angle), and $\angle EBG=30^{\circ}$, then $\angle ABE = 180^{\circ}-\angle EBG=150^{\circ}$.

Step2: Use the given angle - ratio

Let $x = m\angle DBA$. Then $m\angle DBG = 2x$. Also, $\angle ABE=\angle ABD+\angle DBE$. And $\angle DBE=\angle EBG + \angle DBG$.
We know that $\angle ABE = 150^{\circ}$ and $\angle EBG = 30^{\circ}$. So, $\angle ABD+\angle DBE=150^{\circ}$. Substituting $\angle DBE=\angle EBG+\angle DBG$ and $\angle DBG = 2\angle DBA$, we get $x+(30 + 2x)=150$.

Step3: Solve the equation for $x$

Combining like terms in $x+(30 + 2x)=150$, we have $3x+30 = 150$. Subtracting 30 from both sides gives $3x=120$, and dividing by 3 gives $x = 40$. So $m\angle DBG=80^{\circ}$.

Step4: Find $m\angle EBC$

Since $\angle EBC+\angle EBG+\angle GBC = 180^{\circ}$ and $\angle GBC=\angle DBA$ (vertically - opposite angles or linear - pair relationships in the context of the intersecting lines), and we know $\angle EBG = 30^{\circ}$. Also, since $\angle ABE = 150^{\circ}$ and $\angle ABC = 180^{\circ}$, we can also note that $\angle EBC=\frac{1}{2}\angle ABE+\angle EBG$. Substituting the values, $m\angle EBC=\frac{1}{2}(150 - 30)+30$. First, $\frac{1}{2}(120)=60$, then $60 + 15=75^{\circ}$.

Answer:

To continue her proof, Rivka can use angle - addition postulates and properties of linear pairs and vertical angles to solve for the unknown angles step - by - step as shown above. She can start by expressing the relationships between the given angles in equations and then solve those equations to find the measure of $\angle EBC$.