Q1. In a triangle, if one angle measures 50 degrees and the other two angles are equal, what is the measure of each of the equal angles?
Solution:
The sum of angles in a triangle is 180 degrees. Let x be the measure of each equal angle: 50 + 2x = 180, so 2x = 130, thus x = 65 degrees.
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Q2. If two parallel lines are represented by the equations y = 2x + 3 and y = 2x - 5, what is the distance between them?
Solution:
The distance between two parallel lines y = mx + b1 and y = mx + b2 is |b2 - b1| / √(1 + m^2). Here, |(-5) - 3| / √(1 + 2^2) = 8/√5.
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Q3. If two parallel lines are represented by the equations y = 3x + 1 and y = 3x - 4, what is the distance between them?
Solution:
The distance between two parallel lines y = mx + b1 and y = mx + b2 is |b2 - b1| / √(1 + m^2). Here, |(-4) - 1| / √(1 + 3^2) = 5 / √10 = 5/√10.
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Q4. If two lines are parallel and the transversal creates an angle of 40 degrees with one of the lines, what is the measure of the same-side interior angle?
Solution:
Same-side interior angles are supplementary, so 180 - 40 = 140 degrees.
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Q5. If two lines are parallel and one line has the equation y = 3x + 2, what is the equation of a line parallel to it that passes through the point (1,1)?
Solution:
The slope of the parallel line remains 3. Using point-slope form, y - 1 = 3(x - 1) gives y = 3x - 2.
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Q6. If two lines are parallel and the transversal creates an angle of 70 degrees with one of the lines, what is the measure of the corresponding angle on the other line?
Solution:
Corresponding angles are equal, so the angle on the other line is also 70 degrees.
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Q7. If a transversal intersects two parallel lines and creates an angle of 40 degrees, what is the measure of the vertically opposite angle?
Solution:
Vertically opposite angles are equal, so the angle is also 40 degrees.
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Q8. In a coordinate plane, if the coordinates of two points on a line are (2, 3) and (4, 7), what is the slope of the line?
Solution:
The slope is calculated as (y2 - y1) / (x2 - x1) = (7 - 3) / (4 - 2) = 4/2 = 2.
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Q9. What is the distance between the points (3, 4) and (3, 10) in the coordinate plane?
Solution:
The distance formula gives d = |y2 - y1| = |10 - 4| = 6.
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Q10. In a coordinate system, if line A has the equation y = -1/2x + 4, what is the slope of a line parallel to line A?
Solution:
Parallel lines have the same slope, so the slope of the parallel line is -1/2.
