Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
History of geometry wikipedia , lookup
Multilateration wikipedia , lookup
Line (geometry) wikipedia , lookup
Noether's theorem wikipedia , lookup
Euler angles wikipedia , lookup
Rational trigonometry wikipedia , lookup
History of trigonometry wikipedia , lookup
Perceived visual angle wikipedia , lookup
Integer triangle wikipedia , lookup
Compass-and-straightedge construction wikipedia , lookup
Trigonometric functions wikipedia , lookup
Geometry 11/17/14 Bellwork Hint for #: 3 5.3 Use Angle Bisectors of Triangles Objectives Use properties of angle bisectors Locate the incenter Standards PS.1: Make sense of problems and persevere in solving them. G.PL.3: Prove and apply theorems about lines and angles, including the following: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and corresponding angles are congruent; when a transversal crosses parallel lines, same side interior angles are supplementary; and points on a perpendicular bisector of a line segment are exactly those equidistant from the endpoints of the segment. G.PL.5: Explain and justify the process used to construct, with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.), congruent segments and angles, angle bisectors, perpendicular bisectors, altitudes, medians, and parallel and perpendicular lines. Vocabulary • Angle bisector: ray that divides angle into 2 congruent angles Vocabulary • Point of concurrency: point of intersection of segments, lines, or rays • Incenter: point of concurrency of angle bisectors of a triangle Angle Bisector Theorems Theorem 5.5 - Angle Bisector Theorem: If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. Since AD is an bisector, E F then DE ≅ DF. Theorem 5.6 – Converse of the Angle Bisector 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. Example 1 EXAMPLE 1 Use the Angle Bisector Theorems SOLUTION Because DB BA and AC DC and DB = DC = 18, DA bisects BAC by the Converse of the Angle Bisector Theorem. So, mBAD = mCAD = 54°. With a partner, do #1-3 Example 2: of Triangles 5.3 – Use Angle Bisectors Three spotlights from two congruent angles. Is the actor closer to the spotlighted area on the right or on the left? Example 3 EXAMPLE 3 Use algebra to solve a problem For what value of x does P lie on the bisector of A? SOLUTION From the Converse of the Angle Bisector Theorem, you know that if P lies on the bisector of A then P is equidistant from the sides of A, so BP = CP. BP = CP x + 8 = 3x –6 7 =x Set segment lengths equal. Substitute expressions for segment lengths. Solve for x. Point P lies on the bisector of A when x = 7. GUIDED PRACTICE Example 4 Do you have enough information to conclude that QS bisects PQR? Explain. ANSWER No; you need to establish that SR QR and SP QP. Vocabulary and Another Theorem Theorem 5.7 Concurrency of Angle Bisectors of a Triangle: The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. The point of concurrency of the three angle bisectors of a triangle is called the incenter of the triangle. The incenter always lies inside the triangle. DP=EP=FP A circle can be drawn with P as It’s center and will just touch the Sides of the triangle. The circle Is said to be inscribed in the triangle. Example 5 EXAMPLE 4 Use the concurrency of angle bisectors In the diagram, G is the incenter of RST. Find GW. SOLUTION By the Concurrency of Angle Bisectors of a Triangle Theorem, the incenter G is equidistant from the sides of RST. So, to find GW, you can find GU in GUI. Use the Pythagorean Theorem, a2 + b2 = c2. Example 5 (continued): EXAMPLE 4 c 2 = a 2 + b2 2 13 2 = GU + 12 2 Pythagorean Theorem 2 169 = GU + 144 25 = 5= Substitute known values. Multiply. GU 2 Subtract 144 from each side. GU Take the positive square root of each side. Because GU = GW, GW = 5. Geometry 11/17/14 Homework Section 5.3: Pages 313-314: Exercises: 4-24 Even