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Geometry

Learn how to identify and use perpendicular and angle bisectors in triangles. Discover their theorems, properties, and applications in finding segment lengths and angle measures.

bernardgilbert
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Geometry

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  1. Geometry Lesson 5 – 1 Bisectors of Triangles Objective: Identify and use perpendicular bisectors in triangles. Identify and use angle bisectors in triangles.

  2. Perpendicular Bisector • Perpendicular bisector • Any segment, line, or plane that intersects a segment at its midpoint forming a right angle.

  3. Theorems • Perpendicular Bisector Theorem • If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. • Converse of Perpendicular Bisector Theorem • If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

  4. Find AB AB = 4.1

  5. Find WY WY = 3

  6. Find RT RQ = RT 2x + 3 = 4x – 7 3 = 2x – 7 10 = 2x 5 = x RT = 4x – 7 = 4(5) – 7 = 20 – 7 = 13

  7. Concurrent lines • 3 or more lines intersect at a point • Point of Concurrency • The point where 3 or more lines intersect.

  8. Perpendicular Bisectors Right: On the Triangle Acute: Interior Obtuse: Exterior

  9. Circumcenter • Circumcenter • The point of concurrency of the perpendicular bisectors • Circumcenter Theorem • The perpendicular bisectors of a triangle intersect at a point called the circumcenter that is equidistant from the vertices of the triangle.

  10. Angle Bisector • Angle bisector • A line, segment, or ray that cuts an angle into 2 congruent parts.

  11. Theorem • Angle Bisector Theorem • If a point is on the bisector of an angle, then it is equidistant from the sides of that angle. • Converse Angle Bisector Theorem • If a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle.

  12. Find XY XY = 7

  13. Find the measure of angle JKL 37

  14. Find SP 3x + 5 = 6x – 7 5 = 3x – 7 12 = 3x 4 = x SP = 6x – 7 = 6(4) – 7 = 17 SP = 17

  15. Angle bisectors of a triangle Notice all angle bisectors go through a vertex and Intersect in the interior of the triangle.

  16. Incenter • Incenter • The point of concurrency of the angle bisectors of a triangle. • Incenter Theorem • The angle bisectors of a triangle intersect at a point called the incenter that is equidistant from each side of the triangle.

  17. Find each measure if J is the incenter. JF = JE How can we find JE? • JF (JE)2 + 122 = 152 (JE)2 + 144 = 225 (JE)2 = 81 JE = 9 JF = 9

  18. If P is the incenter find the following. PK = PJ (PJ)2 + 122 = 202 (PJ)2 + 144 = 400 (PJ)2 = 256 PJ = 16 • PK 62 31 27 54 PK = 16

  19. Homework • Pg. 327 1 – 8 all, 10 – 14 E, 18 – 34 E, 60 – 64 E

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