5.3 Use Angle Bisectors of

Transcription

1 5.3 Use ngle isectors of Triangles Goal p Use angle bisectors to find distance relationships. Your Notes VOURY Incenter THEOREM 5.5: NGE ISETOR THEOREM In geometry, distance means the shortest length between two objects. If a point is on the bisector of an angle, then it is equidistant from the two of the angle. If ###$ bisects and } ###$ and } ###$, then 5. THEOREM 5.6: ONVERSE O THE NGE ISETOR THEOREM If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the of the angle. If } ###$ and } ###$ and 5, then ###$. Example 1 Use the ngle isector Theorems ind the measure of E. ecause } E and } E and E 5 E 5 21, ###$ E bisects by the. So, m E 5 m E opyright Holt Mcougal. ll rights reserved. esson 5.3 Geometry Notetaking Guide 129

2 5.3 Use ngle isectors of Triangles Goal p Use angle bisectors to find distance relationships. Your Notes VOURY Incenter The point of concurrency of the three angle bisectors of a triangle is called the incenter of the triangle. THEOREM 5.5: NGE ISETOR THEOREM In geometry, distance means the shortest length between two objects. If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. If ###$ bisects and } ###$ and } ###$, then 5. THEOREM 5.6: ONVERSE O THE NGE ISETOR THEOREM If a point is in the interior of an angle and is equidistant from the sides of the angle, then it lies on the bisector of the angle. If } ###$ and } ###$ and 5, then ###$ bisects. Example 1 Use the ngle isector Theorems ind the measure of E. ecause } E ###$ and } E ###$ and E 5 E 5 21, ###$ E bisects by the onverse of the ngle isector Theorem. So, m E 5 m E E opyright Holt Mcougal. ll rights reserved. esson 5.3 Geometry Notetaking Guide 129

3 Example 2 Solve a real-world problem Web spider s position on its web relative to an approaching fly and the opposite sides of the web forms congruent angles, as shown. Will the spider have to move farther to reach a fly toward the right edge or the left edge? R The congruent angles tell you that the spider is on the of R. y the, the spider is equidistant from ###$ and ###$ R. So, the spider must move the each edge. to reach Example 3 Use algebra to solve a problem or what value of x does lie on the bisector of? rom the onverse of the ngle isector Theorem, you know that lies on the bisector of if is equidistant from the sides of, so when 5. 5 Set segment lengths equal. x Substitute expressions for segment lengths. 5 x Solve for x. oint lies on the bisector of when x 5. 2x 2 5 THEOREM 5.7: ONURRENY O NGE ISETORS O TRINGE The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. E If }, }, and } are angle bisectors of n, then esson 5.3 Geometry Notetaking Guide opyright Holt Mcougal. ll rights reserved.

4 Example 2 Solve a real-world problem Web spider s position on its web relative to an approaching fly and the opposite sides of the web forms congruent angles, as shown. Will the spider have to move farther to reach a fly toward the right edge or the left edge? R The congruent angles tell you that the spider is on the bisector of R. y the ngle isector Theorem, the spider is equidistant from ###$ and ###$ R. So, the spider must move the same distance to reach each edge. Example 3 or what value of x does lie on the bisector of? rom the onverse of the ngle isector Theorem, you know that lies on the bisector of if is equidistant from the sides of, so when 5. 5 x x 2 5 Use algebra to solve a problem Set segment lengths equal. x 1 1 Substitute expressions for segment lengths. 6 5 x Solve for x. oint lies on the bisector of when x x 2 5 THEOREM 5.7: ONURRENY O NGE ISETORS O TRINGE The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. E If }, }, and } are angle bisectors of n, then 5 E esson 5.3 Geometry Notetaking Guide opyright Holt Mcougal. ll rights reserved.

5 Example 4 Use the concurrency of angle bisectors In the diagram, is the incenter of nh. ind. y the oncurrency of ngle isectors of a Triangle Theorem, the incenter is from the sides of nh. So, to find, you can find in nhi. Use the ythagorean Theorem. 5 ythagorean Theorem 5 Substitute known values. 5 Simplify. 15 H 5 Take the positive square root of each side. ecause 5, 5. G 12 I heckpoint In Exercises 1 and 2, find the value of x. 1. (3x 2 5)8 2. 7x 1 3 (2x 1 5)8 8x 3. o you have enough information to conclude that ###$ bisects? Explain. Homework 4. In Example 4, suppose you are not given H or HI, but you are given that 5 25 and I ind. opyright Holt Mcougal. ll rights reserved. esson 5.3 Geometry Notetaking Guide 131

6 Example 4 In the diagram, is the incenter of nh. ind. y the oncurrency of ngle isectors of a Triangle Theorem, the incenter is equidistant from the sides of nh. So, to find, you can find I in nhi. Use the ythagorean Theorem. c 2 5 a 2 1 b 2 Use the concurrency of angle bisectors ythagorean Theorem I Substitute known values. 15 H 81 5 I 2 Simplify. 95 I Take the positive square root of each side. ecause I 5, 5 9. G 12 I heckpoint In Exercises 1 and 2, find the value of x. 1. (3x 2 5)8 2. 7x 1 3 (2x 1 5)8 8x x 5 10 x 5 3 Homework 3. o you have enough information to conclude that ###$ bisects? Explain. No, you must know that m 5 m before you can conclude that ###$ bisects. 4. In Example 4, suppose you are not given H or HI, but you are given that 5 25 and I ind opyright Holt Mcougal. ll rights reserved. esson 5.3 Geometry Notetaking Guide 131