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Find the Unknown Interior Angle x in a Triangle with Parallel Lines and Transversals
Mathematics
Grade 8 (Junior High School)
Question Content
Find the value of x in the given intersecting lines/angle diagram, where one exterior angle is 105° and another corresponding angle is 50°.
Correct Answer
55
Detailed Solution Steps
1
Step 1: Identify the supplementary angle to the 105° angle. Since they form a linear pair, this angle is 180° - 105° = 75°.
2
Step 2: Identify the alternate interior angle to the 50° angle. This angle inside the triangle is equal to 50° due to the alternate interior angles theorem for parallel lines cut by a transversal.
3
Step 3: Use the triangle angle sum theorem, which states the sum of interior angles of a triangle is 180°. Set up the equation: x + 50° + 75° = 180°.
4
Step 4: Calculate the value of x: x = 180° - 50° - 75° = 55°.
Knowledge Points Involved
1
Linear Pair (Supplementary Angles)
Two adjacent angles formed by intersecting lines that add up to 180°. They are always supplementary, used to find unknown angles that form a straight line with a given angle.
2
Alternate Interior Angles Theorem
When two parallel lines are cut by a transversal, the pairs of alternate interior angles are congruent. This applies to angles that lie between the parallel lines and on opposite sides of the transversal.
3
Triangle Angle Sum Theorem
The sum of the three interior angles of any triangle is always 180°. This is a fundamental theorem used to solve for unknown interior angles in a triangle when the other two are known.
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