# Properties of Triangles

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1 Properties of Triangles

2 Perpendiculars and isectors segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called a perpendicular bisector. point is equidistant from two points if its distance from each point is the same. Perpendicular isector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. If P is the perpendicular bisector of, then. P

3 onverse of the Perpendicular isector Theorem: If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment. If D D, then Dlies on the the perpendicular bisector of. P In the diagram shown, MN is the perpendicular bisector of ST. a. What segment lengths in the diagram are equal? b. Explain why Q is on MN. T D M N Q S

4 ngle isector Theorem: If a point is on the bisector of an angle, then it is equidistant from the two sides of the angle. D If m D m D, then D D onverse of the ngle isector 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. D

5 Given: D is on the bisector of Prove: D = D. D, D D Statements Reasons

6 isectors of a Triangle perpendicular bisector of a triangle is a line (or ray or segment) that is perpendicular to a side of the triangle at the midpoint of the side.

7 When three or more lines (or rays or segments) intersect in the same point, they are called concurrent lines (or rays or segments). The point of intersection of the lines is called the point of concurrency. The three perpendicular bisectors of a triangle are concurrent. The point of concurrency can be inside the triangle, on the triangle, or outside the triangle.

8 The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle. oncurrency of Perpendicular isectors of a Triangle: The perpendicular bisectors of a triangle intersect at a point that is equidistant from the vertices of the triangle.

9 company plans to build a distribution center that is convenient to three of its major clients. The planners start by roughly locating the three clients on a sketch and finding the circumcenter of the triangle formed. a) Explain why using the circumcenter as the location of a distribution center would be convenient for all the clients. b) Make a sketch of the triangle formed by the clients. Locate the circumcenter of the triangle. Tell what segments are congruent. F E G

10 n angle bisector of a triangle is a bisector of an angle of the triangle. The three angle bisectors are concurrent. The point of concurrency of the angle bisectors is called the incenter of the triangle, and it always lies inside the triangle. The incenter has a special property called the oncurrency of ngle isectors of a Triangle. The angle bisectors of a triangle intersect at a point that is equidistant from the sides of the triangle. PD = PE = PF

11 MNP The angle bisectors of meet at point L. a) What segments are congruent? b) Find LQ and LR

12 Medians and ltitudes of a Triangle median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. D The three medians of a triangle are concurrent. The point of concurrency is called the centroid of the triangle. cute Triangle Right Triangle Obtuse Triangle

13 oncurrency of Medians of a Triangle: The medians of a triangle intersect at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side. P is the centroid of QRS shown below and PT = 5. Find RT and RP.

14 n altitude of a triangle is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side. n altitude can lie inside, on, or outside the triangle. Every triangle has three altitudes. The lines containing the altitudes are concurrent and intersect at a point called the orthocenter of the triangle. Where is the orthocenter located in each type of triangle?

15 oncurrency of ltitudes of a Triangle: The lines containing the altitudes of a triangle are concurrent. If E, EF, and D are the altitudes of, then the lines E, F, and D intersect at some point H.

16 Midsegment Theorem midsegment of a triange is a segment that connects the midpoints of two sides of a triangle. Midsegment Theorem: The segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half as long. DE and DE 1 2 D E

17 UW and VW are midsegment s of RST. Find UW and RT. R U 12 V 8 T W S

18 D 24 J E G H 10.6 F 1. JH? 2.? DE EF? GH? DF? JH? Find the perimeter of GHJ

19 Inequalities in One Triangle Theorem: If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. 3 5 m m In and. What can you conclude about the angles in?

20 Theorem: If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. D E DF > EF F H Write the measurements of the triangles in order from least to greatest. J R Q G P

21 Exterior ngle Inequality Theorem: The measures of an exterior angle of a triangle is greater than the measure of either of the two nonadjacent interior angles. 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. + > + > + > triangle has one side of 10 centimeters and another of 14 centimeters. Describe the possible lengths of the third side.

22 Indirect Proof and Inequalities in Two Triangles n indirect proof is a proof in which you prove that a statement is true by first assuming that its opposite is true. This leads to an impossibility, then you have proved that the original statement is true. Example: Use an indirect proof to prove that a triangle cannot have more than one obtuse angle. egin by assuming that does have more than one obtuse angle.

23 Statement Reason m > 90 and m > 90 ssume has two obtuse angles m + m > 180 dd the two given inequalities You know, however, that the sum of the measures of all three angles is 180 m + m + m = 180 Triangle Sum Theorem m + m = m Subtraction property of equality So you can substitute m for m + m in m + m > m > 180 Substitution property of equality 0 > m Simplify This last statement is not possible; angle measures in triangles cannot be negative. So, you can conclude that the original assumption must be false. That is, cannot have more than one obtuse angle.

24 Guidelines for Writing an Indirect Proof Identify the statement that you want to prove is true egin by assuming the statement false; assume its opposite is true. Obtain statements that logically follow from you assumption. If you obtain a contradiction, then the original statement must be true. Hinge Theorem: If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first is larger than the included angle of the second, then the third side of the first is longer than the third side of the second. V R RT > VX 100 S T W 80 X

25 onverse of the Hinge Theorem: If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first is longer than the third side of the second, then the included angle of the first is larger than the included angle of the second. F m > m D 8 7 D E

26 You can use the Hinge Theorem and its converse to choose possible side lengths or angle measures from a given list. a) m DE, EF, 36, and m 80 Which of the following is a possible length for or 23 in? E 12 inches,. DF : 8 in., 10 in., 12 in., b) In RST and XYZ, XY 4.5cm,and m possible measurefor F RT Z 75 T : 60 XZ, ST. Which of, 75, 90 YZ, RS thefollowing is, or 105 S 3.7 cm,? X a D E R T Y Z

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