Angles and Parallel Lines - Problems on Circles

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Q1. 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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Q2. 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.

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Q3. In triangle ABC, if angle A is 50 degrees and angle B is 60 degrees, what is the measure of angle C?

Solution:

The sum of angles in a triangle is 180 degrees. Therefore, angle C = 180 - (50 + 60) = 70 degrees.

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Q4. 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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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 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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Q7. In a circle, if the measure of an inscribed angle is 40 degrees, what is the measure of the arc it intercepts?

Solution:

The measure of the arc intercepted by an inscribed angle is twice the measure of the angle, so it is 40 * 2 = 80 degrees.

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Q8. If two angles are alternate exterior angles when two parallel lines are cut by a transversal, what can be concluded?

Solution:

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

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Q9. 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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Q10. 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.