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Home » Circles » Tangent Lines » Tangents to a circle and inscribed angles

Tangents to a circle and inscribed angles

Last updated: Oct 1, 2019 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

Today we'll find the angles of a triangle formed by two tangent lines to a circle using inscribed angles, and theorems about the sum of angles in a polygon.

In this geometry problem, we will combine several theorems in order to arrive at the solution:

  • Two tangent theorem
  • Inscribed angle theorem
  • Sum of angles in a triangle theorem
  • Sum of angles in a quadrilateral theorem

Problem

A triangle ΔBCD is inscribed in a circle such that m∠BCD=75° and m∠CBD=60°. Show that the triangle ΔABC formed by two tangent lines from point A outside the circle to points B and C is a 45-45-90 Right Triangle.

Geometry drawing of a circle with a triangle and tangents.

Strategy

We are given two of the three angles of triangle ΔBCD, so it is easy and usually useful to get the third angle, using the sum of angles in a triangle theorem. We do that to find that m∠BDC=45°.

We've also previously said that most problems involving a polygon inscribed in a circle will make use of the inscribed angle theorem, so let's draw the corresponding central angle to ∠BDC, ∠BOC, and using the inscribed angle theorem know it measures 90°.

Geometry drawing: circle with triangle and tangents and center.

Since AB and AC are tangent lines to the circle, they are perpendicular to the radii OB and OC at the points B and C, so in quadrilateral ABOC we have three angles that measure 90° each, and the remaining angle ∠BAC must be 90° as well since the sum of angles in a quadrilateral is 360°.

Finally, from the two tangent theorem, we know that AB=AC, so ΔABC is an isosceles triangle, and by the base angle theorem, m∠ABC=m∠ACB, so they each must be 45°.

Solution

(1) m∠BCD=75° //Given
(2) m∠CBD=60° //Given
(3) m∠BDC=45° //(1) , (2) , Sum of angles in a triangle
(4) m∠BOC=90° //(3), inscribed angle theorem
(5) m∠ABO=90° //Given, AB is tangent to O
(6) m∠ACO=90° //Given, AC is tangent to O
(7) m∠BAC=90° //(4) ,(5), (6), Sum of angles in a quadrilateral
(8) AB=AC //Two tangent theorem
(9) m∠ABC=m∠ACB //(8), base angle theorem
(10) m∠ABC=m∠ACB=45° //(7),(9), Sum of angles in a triangle

And so we have found the angles of a triangle formed by two tangent lines to a circle using inscribed angles.

« Perimeter of a Polygon
Intersecting Secants Theorem »

About the Author

Ido Sarig is a high-tech executive with a BSc degree in Computer Engineering. His goal is to help you develop a better way to approach and solve geometry problems. You can contact him at [email protected]

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About

Welcome to Geometry Help! I'm Ido Sarig, a high-tech executive with a BSc degree in Computer Engineering and an MBA degree in Management of Technology. I'm here to tell you that geometry doesn't have to be so hard! My goal with this website is to help you develop a better way to approach and solve geometry problems, even if spatial awareness is not your strongest quality. Read More…

Geometry Topics

  • Area of Geometric Shapes
  • Circles
    • Arcs, Angles, and Sectors
    • Chords
    • Inscribed Shapes
    • Tangent Lines
  • Lines and Angles
    • Intersecting Lines and Angles
    • Parallel Lines
    • Perpendicular lines
  • Pentagons and Hexagons
  • Perimeter of Geometric Shapes
  • Polygons
  • Quadrangles
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    • Parallelograms
    • Rectangles
    • Rhombus
    • Squares
    • Trapezoids
  • Triangles
    • Congruent Triangles
    • Equilateral Triangles
    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

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