Angles and Parallel Lines - Problems on Circles

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Q1. If two angles are supplementary and one angle measures 40 degrees, what is the measure of the other angle?

Solution:

Supplementary angles add up to 180 degrees, so the other angle measures 180 - 40 = 140 degrees.

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Q2. What is the measure of each angle formed by two parallel lines cut by a transversal if one angle measures 75 degrees?

Solution:

The corresponding angle will measure 75 degrees, and the alternate interior angle will measure 105 degrees.

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Q3. In a circle, if a radius and a tangent meet at a point, what is the angle between them?

Solution:

The angle between a radius and a tangent at the point of contact is always 90 degrees.

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Q4. If a line segment is drawn from the center of a circle to the midpoint of a chord, what can be said about this line segment?

Solution:

A line segment drawn from the center of a circle to the midpoint of a chord is perpendicular to the chord.

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Q5. If a transversal intersects two parallel lines, what is the sum of the interior angles on the same side of the transversal?

Solution:

The interior angles on the same side of the transversal are supplementary, summing to 180 degrees.

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Q6. What is the measure of an angle formed by a tangent and a chord through the point of contact?

Solution:

The angle formed by a tangent and a chord through the point of contact is equal to the angle subtended by the chord at the circumference.

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Q7. If a circle has a radius of 5 cm, what is the length of a chord that is 4 cm away from the center?

Solution:

Using the Pythagorean theorem, the length of the chord is 2√(5^2 - 4^2) = 2√9 = 6 cm.

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Q8. What is the measure of an angle formed by two tangents drawn from a point outside a circle?

Solution:

The angle formed by two tangents from a point outside a circle is half the difference of the measures of the intercepted arcs.

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Q9. If two angles are corresponding angles when two parallel lines are cut by a transversal, what can be concluded?

Solution:

Corresponding angles are equal when two parallel lines are cut by a transversal.

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Q10. What is the sum of the interior angles of a triangle formed by two intersecting chords in a circle?

Solution:

The sum of the interior angles of any triangle is always 180 degrees.