Tangents of circles problem (example 1) | Mathematics II | High School Math | Khan Academy

TL;DR

In this video, the measure of angle A, which is circumscribed about circle O, is found to be 88 degrees. Additionally, the measure of angle D, also circumscribed about circle O, is determined to be 50 degrees.

Transcript

• [Voiceover] So we're told that angle A is circumscribed about circle O. So this is angle A right over here, we're talking about this angle right over there. And when they say it's circumscribed about circle O that means that the two sides of the angle they're segments that would be part of tangent lines, so if we were to continue, so for example ... Read More

Key Insights

• ⭕ Angle A is circumscribed about circle O, with its sides formed by tangent lines and intersecting the circle at points B and C.
• 🔺 The measure of angle A is determined by utilizing the concept of a radius intersecting a tangent line, resulting in a right angle.
• 🔺 The sum of the angles in a quadrilateral is 360 degrees, which is used to set up an equation to find the measure of angle A.
• 🇩🇬 Angle D, another circumscribed angle about circle O, is found to have a measure of 50 degrees, intercepting an arc with a measure of 100 degrees.
• 🔺 The concept of inscribed angles is employed to determine the measure of angle D.
• 🫠 The measure of an inscribed angle that intercepts an arc is half the measure of the arc.
• 🔺 The sum of the interior angles of a quadrilateral can be divided into two triangles, each with interior angles adding up to 180 degrees.

Questions & Answers

Q: How is the measure of angle A determined?

The measure of angle A is determined by utilizing the fact that a radius intersecting a tangent line forms a right angle. By finding the sum of angles in a quadrilateral, we can set up an equation to solve for the measure of angle A, which is found to be 88 degrees.

Q: How is the measure of angle D determined?

The measure of angle D is found by using the concept of inscribed angles. Since angle D intercepts the same arc as the interior angle with a measure of 100 degrees, the measure of angle D is half that measure, resulting in 50 degrees.

Q: What does it mean for an angle to be circumscribed about a circle?

When an angle is circumscribed about a circle, it means that the sides of the angle are tangent lines to the circle, intersecting the circle at two points. In this case, angles A and D are circumscribed about circle O.

Q: How is the sum of interior angles in a quadrilateral related to finding the measures of angles A and D?

The sum of interior angles in a quadrilateral is 360 degrees. By setting up an equation with the known measures of angle A and other angles in the quadrilateral, we can solve for the unknown measure of angle A. Additionally, this concept is extended to find the measure of angle D.

Summary & Key Takeaways

• Angle A, which is circumscribed about circle O, has its sides formed by tangent lines and intersects the circle at points B and C.

• The measure of angle A can be found by realizing that a radius intersecting a tangent line creates a right angle and that the sum of angles in a quadrilateral is 360 degrees.

• Angle D, another circumscribed angle, is found by using the same logic and determining that it intercepts an arc with a measure of 100 degrees.