Angles in Parallel Lines
Angles formed by parallel lines and a transversal follow special rules that make solving angle problems much easier. In this lesson, we will use these rules to find the value of x + y based on the angles shown in a diagram.
1. Theory of Angles in Parallel Lines
When two parallel lines are cut by a transversal, several types of angles are created. Two important types are corresponding angles and supplementary angles. Corresponding angles are equal in measure, while supplementary angles add up to 180 degrees.
2. Concept Explanation
Corresponding angles appear in the same relative position when a transversal crosses parallel lines, so they have the same measure. Supplementary angles form a straight line, which means their total measure is always 180 degrees. By identifying these relationships, we can solve for unknown angles step by step.
3. Steps or Formula
- Identify pairs of corresponding angles and set them equal.
- Solve for the unknown variable.
- Identify supplementary angles and write an equation.
- Substitute known values and solve.
- Add the final values to find x + y.
4. Example Problem
Given:
Two parallel lines are intersected by a transversal. One angle measures 110°, another angle is labeled 2y, and a third angle is labeled x.
Step 1: Use corresponding angles.
Since corresponding angles are equal:
$$ 2y = 110^\circ $$ $$ y = 55^\circ $$Step 2: Use supplementary angles.
Angles x and 2y form a straight line, so they are supplementary:
$$ x + 2y = 180^\circ $$Substitute the known value of 2y:
$$ x + 110^\circ = 180^\circ $$ $$ x = 70^\circ $$Step 3: Find the required sum.
$$ x + y = 70^\circ + 55^\circ = 125^\circ $$5. Real-World Connection
Understanding corresponding and supplementary angles is essential in real-life applications such as architecture, civil engineering, and navigation, where accurate angle measurement is critical.
Final Answer:
$$ x + y = 125^\circ $$
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