Question

1. in the accompanying diagram, l || m and line t is a transversal. if m∠1 = 2x + 20 and m∠2 = 4x+10, what is the number of degrees ∠3? 8. in the diagram below, (overrightarrow{ab}) || (overrightarrow{cd}) m∠cab = 80 and m∠dcb = 40, what is m∠acb?

Explanation:

Step1: Find the value of x

Since $\angle1$ and $\angle2$ are supplementary (linear - pair of angles), we have $(2x + 20)+(4x + 10)=180$.
Combining like - terms: $2x+4x+20 + 10=180$, which simplifies to $6x+30 = 180$.
Subtract 30 from both sides: $6x=180 - 30=150$.
Divide both sides by 6: $x=\frac{150}{6}=25$.

Step2: Find the measure of $\angle1$

Substitute $x = 25$ into the expression for $\angle1$: $m\angle1=2x + 20=2\times25+20=50 + 20=70$.

Step3: Find the measure of $\angle3$

Since $\angle1$ and $\angle3$ are corresponding angles and $l\parallel m$, $m\angle3=m\angle1$. So $m\angle3 = 70$.

for question 8:
Since $\overrightarrow{AB}\parallel\overrightarrow{CD}$, $\angle CAB$ and the angle formed by extending $AC$ and $CD$ are alternate - interior angles.
The sum of the angles in the triangle formed by points $A$, $C$, and $B$ is $180^{\circ}$.
Let $\angle ACB=y$.
We know that the angle adjacent to $\angle DCB$ along the line $CD$ and $\angle CAB$ are related by the parallel lines.
Since $\overrightarrow{AB}\parallel\overrightarrow{CD}$, we use the fact that the sum of the angles around point $C$ and the parallel - line properties.
The sum of the angles $\angle CAB$, $\angle ACB$, and $\angle DCB$ (in the context of the parallel lines) gives us the relationship.
We know that $\angle CAB = 80^{\circ}$ and $\angle DCB=40^{\circ}$.
Since $\overrightarrow{AB}\parallel\overrightarrow{CD}$, we can consider the angles in the non - overlapping way.
The sum of the angles $\angle CAB$ and $\angle ACB$ and $\angle DCB$ (in the geometric setup) is such that $\angle CAB+\angle ACB+\angle DCB = 180^{\circ}$ (by the properties of parallel lines and angles in a plane).
So, $80 + y+40=180$.
Subtract 80 and 40 from both sides: $y=180-(80 + 40)=60$.

Answer:

$70$