Geometry 5-1 isectors, Medians, and ltitudes. Special Segments 1. Perpendicular -the perpendicular bisector does what it sounds like, it is perpendicular to a segment and it bisects the segment. - DF is a perpendicular bisector of in D F a.) Theorem 5-1 ny point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment. b.) Theorem 5- ny point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment. c.) Since a triangle has three sides, there are three perpendicular bisectors in a triangle. The perpendicular bisectors of a triangle intersect at a common point. When three or more lines intersect at a common point, the lines are called concurrent lines. d.) Their point of intersection is called the point of concurrency. e.) The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter. f.) Theorem 5- ircumcenter Theorem - The circumcenter of a triangle is equidistant from the vertices of the triangle.. ngle isector - segment that bisects an angle of the triangle and has one endpoint at a vertex of the triangle and the other endpoint at another point on the triangle. - is an angle bisector of, therefore. a.) Theorem 5-4 - ny point on the bisector of an angle is from the sides of the angle. b.) Theorem 5-5 - ny point on or in the interior of an angle and equidistant from the sides of an angle lies on the bisector of the angle. c.) The three angle bisectors in any triangle are concurrent and meet at the incenter of the triangle. d.) Theorem 5-6 Incenter Theorem - The incenter of a triangle is equidistant from each side of the triangle
. Median - segment that connects the vertex of a triangle to the midpoint of the side opposite that vertex. -F is a median of. F a.) The medians of a triangle also intersect at a common point, the point of for the medians of a triangle is called a centroid. b.) Theorem 5-7 entroid Theorem - The centroid of a triangle is located twothirds of the distance from a vertex to the midpoint of the side opposite the vertex on a median. 4. ltitude - It has one endpoint at a vertex of a triangle and the other on the line so that the segment is perpendicular to this line. -D is an altitude of. D a.) The intersection point of the altitudes of a triangle is called the orthocenter. Ex 1: has vertices at (-, 10), (9, ), and (9, 15). a.) Determine the coordinates of point P on so that P is a median of. b.) Determine if P is an altitude of a < b. HW: Geometry 5-1 p. 4-45 7, 10, 1-0, 5-8, 41-54 Hon: 8, 9, 11, 1
Geometry 5- Inequalities and Triangles. ngle Inequalities 1. Properties a.) omparison Property a < b, a = b, or a > b b.) Property if a < b, and b < c, then a < c. c.) ddition and Properties if a < b, then a + c < b + c d.) Properties if c > 0 and a < b, then a c < b c. Theorem 5-8 - Exterior ngle Inequality Theorem If an angle is an exterior angle of a triangle, then its measure is greater than the measure of either of its corresponding remote interior angles. m > m m > m 1 Ex 1: Use the Exterior ngle Inequality Theorem to list all the angles that satisfy the given conditions. a.) ll angles whose measures are less than m 8 1 b.) ll angles whose measures are greater than m 4 7 6 8 5. ngle Side elationships 1. Theorem 5-9 - If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. If >, then
Ex : Determine the relationship between the measures of the given angles. a.) SU, SU b.) c.) TSV, STV SV, UV 5. 5..6 U S 5.1 4.8 6.6 V 4.4 T. Thoerem 5-10 - If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the shorter angle. If m > m, then HW: Geometry 5- p. 5-54 17- odd, 7-49 odd, 55-59, 64-68 Hon: 5
5-4 The Triangle Inequality. Theorem 5-1 Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. + > + > + > Example 1 Q. If 18, 45, 1, and 5 represent the length of segments, what is the probability that a triangle can be formed if three of these numbers are randomly chosen as the lengths of the sides?. -Make a list of the possible sets of three measures. a.) 18, 45, 1 b.) 18, 45, 5 c.) 18, 1, 5 d.) 45, 1, 5 Test the triangle for each case. Inequality test Sketch Triangle? a.) 18 + 1 > 45? b.) 18 + 45 > 5? c.) 18 + 1 > 5? d.) 1 + 45 > 5? So the probability that a triangle is formed is Example The length of two sides of a triangle are 4 cm and 16 cm, what is the possible range for the measures of the third side? 16 16 4 4 HW: Geometry 5-4 p. 64-66 15-4 odd, 5-5, 55-56, 57-59 odd, 60-6 Hon: 8
Geometry 5-5 Inequalities Involving Two Triangles. Theorem 5-1 SS Inequality (Hinge Theorem) If two sides of one triangle are congruent to two sides of another triangle, and the included angle in one triangle is greater than the included angle in the other, then the third side of the first triangle is longer than the third side in the second triangle. Example 1: Proof: Given: ST E = ST Prove: S > TE 1 E T Statements easons 1.) E = ST 1.) Given.) ES = ES.).) 1 is an exterior angle.) of EST 4.) m 1 > m 4.) 5.) S > TE 5.) S. Theorem 5-14 SSS Inequality If two sides of a triangle are congruent to two sides of another triangle and the third side in one triangle is longer than the third side in the other, then the angle between the pair of congruent sides in the first triangle is greater than the corresponding angle in the second triangle.
Example : Given: ST = PQ S = Q ST = SP Prove: m SP > m PQ T S Q P Statements easons 1. S = Q 1.. P = P.. ST = PQ. 4. ST = SP; SP > ST 4. 5. 5. 6. 6. Example : a.) Write an inequality relating m LDM to m MDN M b.) Find the range of values containing a. 18 141 (9a + 15) D 1 1 16 L N HW: Geometry 5-5 p. 71-7 10-1, 8-0, -6, 40, 41-45 odd Hon: