The math club of Ritzy High designed a pennant for the school yacht. The pennant was in the shape of an isosceles triangle. Two points, P and Q, are located so that AC= PQ= QB. Find the measure of angle B.
To find the measure of angle B, we need to use the properties of isosceles triangles.
Let's assume that the triangle is named ABC, where AB = AC and PQ = AC.
Since AC = PQ, we have:
AP + PQ = AP + AC
AP + PQ = AQ
The sum of the angles in a triangle is 180 degrees. So, we have:
∠APQ + ∠AQP + ∠PAQ = 180 degrees
Since PQ = AQ, we can replace AQ with PQ in the equation:
∠APQ + ∠AQP + ∠PAQ = 180 degrees
∠APQ + ∠AQP + ∠PQA = 180 degrees
∠APQ + ∠APQ + ∠PQA = 180 degrees
2∠APQ + ∠PQA = 180 degrees
We can see that angle ∠APQ is equal to angle ∠PQA because AP = AQ (isosceles triangle property). So, let's replace ∠APQ with x:
2x + x = 180 degrees
3x = 180 degrees
x = 180 degrees / 3
x = 60 degrees
Therefore, the measure of angle B, which is ∠APQ, is equal to x, which is 60 degrees.
To find the measure of angle B, we need to analyze the given information about the isosceles triangle.
In the given diagram, let's assume that A, B, and C are the vertices of the isosceles triangle, with AC as the base. Also, let P and Q be the two points on AC such that AC = PQ = QB.
Let's consider the triangle APQ. Since PQ = QB, we can conclude that triangle APQ is an isosceles triangle as well. Therefore, angles PAQ and QAP are congruent.
Similarly, considering the triangle BPQ, we can conclude that angles BPQ and QBP are congruent.
Now, we know that the sum of the angles in a triangle is 180 degrees. Let's use this information to find the measure of angle B.
Since angles PAQ and QAP are congruent, let's denote each of them as x. Similarly, since angles BPQ and QBP are congruent, let's denote each of them as y.
Now, we can write the following equations:
x + x + y = 180 (sum of angles in triangle APQ is 180 degrees)
y + y + x = 180 (sum of angles in triangle BPQ is 180 degrees)
Simplifying these equations, we get:
2x + y = 180 (equation 1)
2y + x = 180 (equation 2)
Now, let's solve this system of equations to find the values of x and y.
By subtracting equation 2 from equation 1, we get:
2x - 2y = 0
2(x - y) = 0
Dividing both sides by 2, we have:
x - y = 0
Adding y to both sides, we get:
x = y
Therefore, x and y are equal. Let's represent their common value as z.
Replacing x and y in equations 1 and 2 with z, we have:
2z + z = 180
3z = 180
Dividing both sides by 3, we get:
z = 60
So, each of the angles x and y, which are congruent to angle B, measures 60 degrees.
Therefore, the measure of angle B is 60 degrees.