Special Segments in TrianglesPerpendicular bisector: A line or line segment that passes through the midpoint of a side of a triangle and is perpendicular to that side.

Theorem 5-1-: Any point on the perpendicular bisector of a segment is equidistant from the endpoints of the segment.

Theorem 5-2-: Any point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment.

Median: A segment that connects a vertex of a triangle to the midpoint of the side opposite to that vertex. Every triangle has three medians.

Altitude: A segment that has an endpoint ata vertex of a triangle and the other on the lineopposite to that vertex, so that the segment is perpendicular to this line. Do example 1, page 239

Altitudes ofa right triangle

Altitudes ofan obtusetriangle

Median and Altitude

Examples

2) SGB [S(4,7), G(6,2), and B(12,-1)]: a) Determine the coordinates of point J

on GB so that SJ is a median of SGB b) Point M(8,3). Is GM an altitude of SGB ?

1) ABC [A(-3,10), B(9,2), and C(9,15)]: a) Determine the coordinates of point P on

AB so that CP is a median of ABC.b) Determine if CP is an altitude of ABC

Angle bisector of a triangle: A segment that bisects an angle of a triangle and has one endpoint at a vertex of the triangle and the other endpoint at another point on the triangle.

Theorem 5-3: Any point on the bisector of an angle is equidistant from the sides of the angle.

Theorem 5-4: Any point on or in the interior of an angle and equidistant from the sides of an angle, lies on the bisector of the angle.

Angle Bisector of a Triangle

Exaples to do

1. Do example 2, p. 240

2. Prove that if a triangle is equilateral, then an angle bisector is also a median.

3. Do example 3, p. 241

A

B C

R

S T

AB RSBC ST

IF

<B <S (both are right angles)Then, ABC RST : SAS

Theorem 5-5 : If the legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent.(LL)

Right Triangles - LL

Right Triangles - HA

Theorem 5-6 : If the hypotenuse and an acute angle of a right triangle are congruent to the hypotenuse and corresponding acute angle of another right triangle, then the two triangles are congruent.(HA) Prove Th.5-6

A

B C

R

S T

<A <R AB RS

Theorem 5-7 : If one leg and an acute angle of one right triangle are congruent to the corresponding leg and acute angle of another right triangle, then the tri-angles are congruent.(LA) (Do example 2,p.247)

Complete the two-column proof (paper)

Case 1 Case 2<C <T AB RS

Right Triangles - LA

Postulate 5-1 : If the hypotenuse and theleg of a right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the two triangles are congruent.(HL) Example 3, p248

Right Triangles - HL

Indirect Proof (Indirect reasoning)

Steps for writing and indirect proof: 1. Assume that the conclusion is false2. Assume that the assumption leads to a

contradiction of the hypothesis or some other fact, such as a postulate, theorem, corollary.

3. Point out that the assumption must be false and, therefore, the conclusion must be true. (Ex.1p252)

State the assumption you would make to start an indirect proof of each statement: AB bisects <A, XTZ is isosceles, m<1 < m<2

INEQUALITY

Theorem 5-8 ( Exterior Angle 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.

Definition of inequality: For any real numbers a and b, a>b if and only if there is a positive number c such that a = b + c

Inequalities for Sides and Angles of a

TriangleTheorem 5-9: If one side of a triangle is longer than another side, then the angle opposite to the longer sidehas a greater measure than the angle opposite to the shorter side

Theorem 5-10: If one angle of a triangle has a greatermeasure than another angle, then the side oppositeto the greater angle is longer than the side opposite tothe lesser angle.

Theorem 5-11: The perpendicular segment from a point to a line is the shortest segment from the point to the line.

Corollary 5.1: The perpendicular segment from a point to a plane is the shortest segment from the point to the plane.

Cont...Inequalities for Sides and Angles of a

Triangle

The Triangle Inequality

Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side.Ex. 1 p.267 Example 1: If 18, 45, 21 and 52 are the lengths of segments, what is the probability that a triangle can be formed if three of these numbers are chosen at random as lengths of the sides? (Ex.2 and 3-students do)

SAS 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.

SSS Inequality (Theorem): If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first 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.

Inequalities Involving Two Triangles