Answer:
∠a = 80°
∠b = 80°
∠c = 160°
∠d = 100°
Step-by-step explanation:
So to start off with, we know that the angle adjacent to the 20° angle will also be 20 degrees due to the rules of congruent lines. Because of this, we can automatically know two more things. First of all, we know that the other two angles in this central angle cluster (∠c and the angle adjacent to it) will also be congruent. Second, we know that the sum of all these angles will be 360 degrees total.
We can begin to solve for ∠c by adding up the angles we already know. We will represent angle c and its adjacent/congruent angle as 2c.
20 + 20 + 2c = 360
40 + 2c = 360
2c = 320
c = 160
Both ∠c and its adjacent angle are equal to 160°. Next, take a look at the tiny upside down triangle near the top. It's an isosceles triangle (meaning two sides are congruent in length) meaning that two of its angles will be congruent. This is another important fact we can deduce from the fact that only two lines intersect at the intersection point. Thus, we know that since one angle is 20 degrees, and the other two are congruent, we can solve for the two unknown degrees in hopes we can eventually solve for ∠d.
(Just a reminder we are discussing the angles of the top smaller triangle. Not the bottom one.)
We will represent the two congruent angles in the smaller triangle as 2x.
20 + 2x = 180
*Remember: 180 in a triangle.
Subtract 20 from both sides.
2x = 160
x = 80
Both of these angles are equal to 80 degrees. This means that the angle next to d is equal to 80°. Both the 80° angle and ∠d are situated right next to each other on a straight line. Straight lines also have a grand total of 180 degrees. We can solve for ∠d by subtracting 80 from 180.
180 - 80 = 100
∠d = 100°
Next, we can easily solve for angles a and b. Again, since only two lines intersect at the intersection point AND lie on a set of parallel lines (l₁ and l₂), we can know that the larger triangle is an isosceles triangle with two equal sides and thus two equal angles. In this case, the larger isosceles triangle's two equal angles are ∠a and ∠b. With this knowledge, and the knowledge that a triangle has 180 degrees, we can begin to solve for these angles.
Since ∠a and ∠b are congruent, they will be represented as 2a. (It doesn't make a difference if the variable was represented as 2a or 2b, the number 2 will represent the amount of congruent angles, and the letter variable will just represent the angles we are trying to find in this case.)
20 + 2a = 180
Subtract 20 from 180
2a = 160
Divide by 2
a = 80
Thus since ∠a equals 80°, then its congruent angle ∠b will also be 80°.
