Objectives: To use properties of midsegments to solve problems Chapter 5 Section 5.1 – Midsegments of Triangles

In triangle ABC above, LN is a triangle midsegment. Midsegment - > of a triangle is a segment connecting the midpoints of two sides. A C B L N

Theorem 5.1 -> Triangle Midsegment Theorem If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side, and is half its length

Ex: Finding the Length of Segments In triangle EFG, H, J, and K are midpoints. Find HJ, JK, and FG. F E G HJ K

Ex: AB = 10 and CD = 18. Find EB, BC, and AC. A E D B C

Ex: Identifying Parallel Segments In triangle DEF, A, B, and C are midpoints. Name three pairs of parallel segments. E A B D C F

Find the measure of <VUZ. Justify your answer. X U Z Y V 65

Homework #26 Due Friday (November 16) Page 262 – 263 #1 – 19 odd #22 – 25 all

Objectives: To use properties of perpendicular bisectors and angle bisectors Section 5.2 – Bisectors in Triangles

In the diagram below to the left, CD is the perpendicular bisector of AB. CD is perpendicular to AB at its midpoint. The diagram below to the right, CA and CB are drawn to complete the triangles CAD and CBD. A D B C C A DB

Theorem 5.2 – Perpendicular Bisector Theorem If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Theorem 5.3 – Converse of the Perpendicular Bisector Theorem If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.

Example: CD is the perpendicular bisector of AB. Find CA and DB. Explain your reasoning. C B D A 5 6

Distance from a point to a line -> the length of the perpendicular segment from the point to the line. In the diagram below, AD is the bisector of <CAB. If you measure the lengths of the perpendicular segments from D to the two sides of the angle, you will find that the lengths are equal. Therefore, D is equidistant from the sides. A C B D

Theorem 5.4 – Angle Bisector Theorem If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle. Theorem 5.5 – Converse of the Angle Bisector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the angle bisector.

Ex: Using the Angle Bisector Theorem What is the length of FD ? C B D E A 5x 2x + 24 F

A.According to the diagram, how far is K from EH? From ED? B.What can you conclude about EK? C.Find the value of x. D.Find m<DEH E K D H 10 2x x + 20

Homework # 27 Due Friday (November 16) Page 267 – 268 #1 – 31 odd

Objectives: To identify properties of perpendicular bisectors and angle bisectors. To identify properties of medians and altitudes of a triangle. Section 5.3 – Concurrent Lines, Medians, and Altitudes

Concurrent -> when three or more lines intersect in one point. Point of Concurrency -> the point at which three or more lines intersect.

Theorem 5.6 The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices. Theorem 5.7 The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides.

Circumcenter of the Triangle -> the point of concurrency of the perpendicular bisectors of a triangle. In triangle QRS (below), points Q, R, and S are equidistant from C, the circumcenter. The circle is circumscribed about the triangle Q S R C

Ex: Finding the Circumcenter Find the center of the circle that you can circumscribe about a triangle with vertices (0,0), (4,0), and (0,6). Find the center of the circle that you can circumscribe about the triangle with vertices (0,0), (-8,0), and (0,6). Why is it not necessary to find the third perpendicular bisector?

The figure below shows triangle UTV with the bisectors of its angles concurrent at J. The point of concurrency of the angle bisectors of a triangle is called the incenter of the triangle. Points X, Y, Z are equidistant from J, the incenter, The circle is inscribed in the triangle. T U V Y Z X J XJ = YJ = ZJ

Median of a Triangle -> a segment whose endpoints are a vertex and the midpoint of the opposite side. Theorem 5.8 – Triangle Medians Theorem The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side. D H E J F G C

Centroid -> the point of concurrency of the medians in a triangle. **Also called the center of gravity of a triangle because is the point where a triangular shape will balance.**

Example: Finding Lengths in Medians In triangle ABC below, D is the centroid and DE = 6. Find BE. Find BD. B A E C D

Altitude of a Triangle -> the perpendicular segment from a vertex to the line containing the opposite side. **Unlike angle bisectors and medians, an altitude of a triangle can be a side of a triangle or it may lie outside the triangle.** Acute TriangleRight TriangleObtuse Triangle Altitude is insideAltitude is a sideAltitude is outside

Theorem 5.9 The lines that contain the altitudes of a triangle are concurrent. Orthocenter of the Triangle -> the point at which the lines containing the altitudes of a triangle are concurrent

Homework #28 Due Monday (November 26) Page 275 – 276 # 1 – 22 all

Objectives: To write the negation of a statement and the inverse and contrapositive of a conditional statement To use indirect reasoning Section 5.4 – Inverses, Contrapositives, and Indirect Reasoning

Negation -> of a statement has the opposite truth value. Ex: Los Angeles is the capital of California (false) Los Angeles is not the capital of California (true)

Inverse -> negates both the hypothesis and the conclusion of a conditional statement. Contrapositive -> switches the hypothesis and the conclusion of a conditional statement and negates both. Equivalent Statements -> statements that have exactly the same truth value.

Summary -> Negation, Inverse, and Contrapositive Statements StatementExampleSymbolic FormYou Read It ConditionalIf an angle is a straight angle, then its measure is 180 p -> qIf p, then q. Negation (of p)An angle is not a straight angle. ~pNot p. InverseIf an angles is not a straight angle, then its measure is not 180. ~p -> ~qIf not p, then not q. ContrapositiveIf an angle’s measure is not 180, then it is not a straight angle. ~q -> ~pIf not q, then not p.

Examples: Write the negation of each statement. <ABC is obtuse.Lines m and n are not perpendicular. m 70.Today is not Tuesday. Write the inverse and the contrapositive of the conditional statement. If a figure is a square, then it is a rectangle. If you don’t stand for something, you’ll fall for anything.

Suppose your brother tells you, “Your girlfriend called a few minutes ago.” You think through these three steps: Step 1 – You have two girlfriends. Step 2 – You know that one of them is at cheerleading practice. Step 3 – You conclude that the other one must have been the one that called. This type of reasoning is called indirect reasoning. In this type of reasoning, all possibilities are considered and then all but one are proved false. Ergo, the remaining possibility must be true. A proof involving indirect reasoning is an indirect proof.

Summary -> Writing an Indirect Proof Step 1 – State as an assumption the opposite (negation) of what you want to prove. Step 2 – Show that this assumption leads to a contradiction. Step 3 – Conclude that the assumption must be false and that what you want to prove must be true.

Homework # 29 Due Tuesday (November 27) Page 283 #1 – 19 all

Objectives: To use inequalities involving angles of triangles To use inequalities involving sides of triangles Comparison Property of Inequality If a = b + c and c > 0, then a > b Section 5.5 – Inequalities in Triangles

Corollary to the Triangle Exterior Angle Theorem The measure of an exterior angle of a triangle is greater than the measure of each of its remote interior angles m<1 greater than m<2 m<1 greater than m<3

Theorem 5.10 If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side. X Y Z If XZ > XY, then m<Y greater than m<Z

Ex: List the angles of triangle ABC in order from smallest to largest. A B C 27 ft 21 ft 18 ft

Theorem 5.11 (Converse of 5.10) If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle. B A C If m AC

Ex: List the sides in order from shortest to longest. T U V 5862

Theorem 5.12 – Triangle Inequality Theorem The sum of the lengths of any two sides of a triangle is greater than the length of the third side. Y X Z XY + YZ > XZ YZ + ZX > YX ZX + XY > ZY

Ex: Can a triangle have sides with the given lengths? Explain. a.3ft, 7ft, 8ft b.3cm, 6cm, 10cm A triangle has sides of lengths 8cm and 10 cm. Describe the lengths possible for the third side.

Homework #30 Due Wednesday (Nov 28) Page 292 – 293 # 1 – 27 all Chapter 5 Test Thursday/Friday