In the figure, (AC) ̅≅(AB). Find the value of y in terms of x. If the vertex angle of an isosceles triangle is y and the exterior angle ABE is 3x+20.
Answer: y= 6x-140
Pls explain it to me how they get the answer.
ABE+ABC = 180, so ABC = 180-ABE
ABC = (180-y)/2
so,
180-(3x+20) = (180-y)/2
160-3x = 90 - y/2
70-3x = -y/2
y = 6x-140
To find the value of y in terms of x, we need to use the given information about the figure and apply relevant geometric properties. Let's break it down step by step:
1. Start by understanding the given information: In the figure, (AC) ̅ is congruent to (AB). This means that the line segment AC is equal in length to the line segment AB.
2. Identify the triangle: From the given information, we can conclude that triangle ABC is an isosceles triangle because two sides, AC and AB, are equal in length.
3. Understand the relationships in an isosceles triangle: In an isosceles triangle, the vertex angle is the angle opposite to the equal sides. So, in triangle ABC, the vertex angle is ∠C, which we need to find in terms of x.
4. Examine the exterior angle: We are also given that the exterior angle ∠ABE is equal to 3x + 20.
5. Apply the Exterior Angle Theorem: The exterior angle of a triangle is equal to the sum of its remote interior angles. In this case, ∠ABE is equal to the sum of angles ∠C and ∠B. Therefore, we can write the equation as: ∠ABE = ∠C + ∠B.
6. Substitute the values: We can substitute the known values into the equation:
3x + 20 = y + ∠B.
7. Identify the relationship between ∠B and ∠C: Since triangle ABC is an isosceles triangle, ∠B is equal to ∠C.
8. Make the substitution: Replace ∠B with ∠C in the equation obtained from step 6:
3x + 20 = y + ∠C.
9. Simplify and solve for y: Since ∠C is the vertex angle of an isosceles triangle and we want y in terms of x, we can express ∠C in terms of x. From the given answer, y = 6x - 140.