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Home » Triangles » Congruent Triangles » Congruent Triangles in a Circle

Congruent Triangles in a Circle

Last updated: Mar 27, 2021 by Ido Sarig · This website generates income via ads and uses cookies · Terms of use · Privacy policy

Using the known properties of circles, like the fact that their radii are all equal, it is easy to solve geometry problems that require proof that triangles in a circle are congruent. Let's take a look at one such problem, involving two intersecting circles and the triangles formed by their radii.

Problem

Two circles, with centers O and Q, intersect at points A and B. Prove that △OAQ≅△OBQ.

Congruent triangles in a circle

Strategy

One of the sides of these two triangles, the line segment OQ which connects both centers, is common to both triangles. We are not given any information about the angles formed by the radii, so the two postulates of congruent triangles that require angles (Angle-Side-Angle or Side-Angle-Side) seem like less likely options at this point. But what about the third postulate- Side-Side-Side?

The hint as to which other parts we should use to show the congruency is that these triangles were formed by the intersection of two circles- and the other two sides of both triangles are radii of the circles, which are equal to each other, and so we have the needed addiotnal two sides, and the triangles are congruent using the Side-Side-Side postulate.

Proof

(1) OQ=OQ //Common side, reflexive property of equality
(2) QA=QB //Both are radii of circle Q, a circle's radii are equal to each other
(3) OA=OB //Both are radii of circle O, a circle's radii are equal to each other
(4) △OAQ≅△OBQ //Side-Side-Side postulate.

« Perimeter of a Rectangle
Tangent Line to a Circle »

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
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  • Triangles
    • Congruent Triangles
    • Equilateral Triangles
    • Isosceles Triangles
    • Pythagorean Theorem
    • Right Triangles
    • Similar Triangles
    • Triangle Inequalities

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