We want to find the measures of the numbered angles. Let's take a look at the given figure. Notice that the given angle is a right angle. This means that the measure of this angle is 90^(∘).
We can see that ∠ 2 and the 90^(∘) angle are formed on opposite sides of the point of intersection of two lines. This means that they are vertical angles. Vertical angles are congruent. Because of that, ∠ 2 and the 90^(∘) angle are congruent. Congruent angles have the same measure, so the measure of ∠ 2 is 90^(∘).
Next, note that the 90^(∘) angle and ∠ 1 form a straight line.
These angles are supplementary so the sum of their measures is 180^(∘). With this information, we can find the measure of ∠ 1.
∠ 1+ 90^(∘)=180^(∘)
⇕
∠ 1= 90^(∘)
The measure of ∠ 1 is 90^(∘). Let's now find the measure of ∠ 3. Because ∠ 3 and ∠ 1 are vertical angles, they are congruent. Congruent angles have the same measure, so the measure of ∠ 3 is also 90^(∘).
Let's summarize the measurements that we have for the first point of intersection.
rc
given= 90^(∘) & ∠ 1 = 90^(∘)
∠ 2 = 90^(∘) & ∠ 3 = 90^(∘)
Now we can think about the second point of intersection. Lines a and b are parallel and cut by the transversal t. This forms 4 pairs of angles that lie on the same side of the transversal in corresponding positions. These are called corresponding angles.
From the graph, we can see that the following pairs of angles are corresponding.
90^(∘) &and ∠ 4
90^(∘) &and ∠ 5
90^(∘) &and ∠ 6
90^(∘) &and ∠ 7
Corresponding angles are congruent when a transversal intersects parallel lines. Using this information, we can find the measures of the missing angles.
∠ 4 &= 90^(∘)
∠ 5 &= 90^(∘)
∠ 6 &= 90^(∘)
∠ 7 &= 90^(∘)
