We want to find the measures of the numbered angles. Let's look at the given figure.
We can see that ∠ 2 and the 59^(∘) angle are formed on opposite sides of a point of intersection between two lines, so they are vertical angles. Vertical angles are congruent. Because of that, ∠ 2 and the 59^(∘) angle are congruent.
Congruent angles have the same measure, so the measure of ∠ 2 is 59^(∘). Next, note that the 59^(∘) angle and ∠ 1 form a straight line.
These angles are supplementary and the sum of their measures is 180^(∘). With this information, we can find the measure of ∠ 1.
∠ 1+ 59^(∘)=180^(∘)
⇕
∠ 1= 121^(∘)
Let's now find the measure of ∠ 3. Because ∠ 3 and ∠ 1 are also vertical angles, they are also congruent.
Congruent angles have the same measure, so the measure of ∠ 3 is 121^(∘). Let's summarize the measurements that we have for the first point of intersection.
rc
given= 59^(∘) & ∠ 1 = 121^(∘)
∠ 2 = 59^(∘) & ∠ 3 = 121^(∘)
Now we can think about the second point of intersection. Lines l and m are parallel and cut by a transversal t. This creates 4 pairs of corresponding angles. Corresponding angles are congruent. We will use this fact to find the measures of the remaining angles.
From the graph we can see that the following pairs of angles are corresponding.
121^(∘) &and ∠ 4
59^(∘) &and ∠ 6
121^(∘) &and ∠ 7
59^(∘) &and ∠ 5
Finally, using the fact that corresponding angles are congruent, we can find the measures of the missing angles.
∠ 4 &= 121^(∘)
∠ 5 &= 59^(∘)
∠ 6 &= 59^(∘)
∠ 7 &= 121^(∘)
