Math Problem Statement
Solution
Let's solve the problem step by step.
We are given:
From the diagram, angles and appear to be vertically opposite angles, which means they are congruent (equal). Therefore, we can set their measures equal to each other: By subtracting 40 from both sides: Dividing both sides by 2:
Now that we know , we can substitute this into the equation for :
Next, we know that the sum of the angles on a straight line is 180°. Thus: Substituting and , and knowing : Simplifying the equation: Subtracting 40 from both sides: Dividing by 5:
Now that we have , and , so , we can find the measures of the angles:
Thus, the measures of the angles are:
Would you like more details on any step or further explanations?
Here are five related questions to explore:
- How do you identify vertically opposite angles in any geometric figure?
- Can you solve a problem where angles are given in terms of different variables?
- How does the concept of supplementary angles help in solving linear pairs of angles?
- What are other ways to verify that and are equal in the diagram?
- What happens if we change the expressions for the angles? How does the solution process adjust?
Tip: In geometry, identifying angle relationships such as vertical angles, corresponding angles, and supplementary angles is key to solving many problems involving intersecting lines.
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Math Problem Analysis
Mathematical Concepts
Algebra
Geometry
Angle Relationships
Formulas
m∠1 = 2x + 40
m∠2 = 2y + 40
m∠3 = x + 2y
Theorems
Vertical Angles Theorem
Linear Pair Theorem
Supplementary Angles Theorem
Suitable Grade Level
Grades 9-11
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