Ticket In the Door Write out each of the following: 1.SSS Postulate 2.SAS Postulate 3.ASA Postulate 4.AAS Postulate

Isosceles and Equilateral Triangles

Let’s Regroup Move to your new Assigned Groups.

Isosceles Triangle An Isosceles Triangle is a triangle that has at least 2 congruent sides. The congruent sides of an isosceles triangle are the legs. The third side is the base. The two congruent legs form the vertex angle. The other two angles are the base angles.

Theorem If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

Theorem 4-4 Converse of the Isosceles Triangle Theorem If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

Theorem If a line bisects the vertex angle of an isosceles triangle, then the line is also the perpendicular bisector of the base

Equilateral Triangle An equilateral triangle is a triangle that has three congruent sides.

Corollary A corollary is a theorem that can be proved easily using another theorem.

Corollary to Theorem 4-3 If a triangle is equilateral, then the triangle is equiangular.

Corollary to Theorem 4-4 If a triangle is equiangular, then the triangle is equilateral.

The 1 st Assignment Complete p. 107 #5 - 13

Definition of Midsegment A midsegment of a triangle is a segment connecting the midpoints of two sides of the triangle.

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 as long.

Ticket In the Door

Graded Homework for Workbook p. 121 #1 – 5 Workbook p. 123 Section 5.2 #1 – 8, 11 – 30

Section 5-2 Perpendicular and Angle Bisectors Objectives: To use properties of perpendicular bisectors and angles bisectors Essential Understanding: There is a special relationship between the points on the perpendicular bisector and the endpoints of the segment.

Definition of Equidistant A point is equidistant from two objects if it is the same distance from the objects.

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.

Examples: Guided Practice

Essential Understandings There is a special relationship between the points on the bisector of an angle and the sides of the angle. The distance from a point to a line is the length of the perpendicular segment from the point to the line. This distance is also the length of the shortest segment from the point to the line.

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.

Examples: Guided Practice

Questions?

Let’s review.

Now, it’s your turn. Textbook pp. 296 – 298 #1 – 3, 6 – 8, 12 – 22, 29 – 31 You have exactly 20 minutes to work on this.

Let’s talk about it.

5.3 Bisectors in Triangles Objectives: To identify properties of perpendicular bisectors and angle bisectors.

Concurrent Definition: When three or more lines intersect at one point, they are concurrent. The point at which they intersect is the point of concurrency.

Essential Understanding For any triangle, certain sets of lines are always concurrent. Two of these sets of lines are: – the perpendicular bisectors of the triangle’s three sides, and – the bisectors of the triangle’s three angles.

Theorem 5.6 Concurrency of Perpendicular Bisectors Theorem The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices The point of concurrency of the perpendicular bisectors of a triangle is called the circumcenter of the triangle.

Theorem 5.7 Concurrency of Angle Bisectors Theorem The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides of the triangle The point of concurrency of the angle bisectors of a triangle is called the incenter of the triangle.

Example: Guided Practice

Questions?

Let’s review.

Now, it’s your turn. Textbook pp #2, 15 – 18, 21, You have exactly 20 minutes to work on this.

Ticket In the Door (due in 10 minutes) Define each of the following terms: 1.Concurrent 2.Point of concurrency 3.Circumcenter of the triangle 4.Incenter of the triangle 5.Write out Theorem 5.6 Concurrency of Perpendicular Bisectors Theorem. Then, draw a figure related to the theorem. 6.Write out Theorem 5.7 Concurrency of Angle Bisectors Theorem. Then, draw a figure related to the theorem.

Agenda Ticket In the Door Announcement Section 5.4 Medians and Altitudes – Median – Centroid – Altitude – Orthocenter Section 5.6 Inequalities in One Triangle – Comparison Property of Inequality – Corollary to the Triangle Exterior Angle Theorem – Theorem 5.10 – Theorem 5.11 – Theorem 5.12 Triangle Inequality Theorem Section 5.7 Inequalities in Two Triangles – Theorem 5.14 Hinge Theorem (SAS Inequality Theorem) Quiz: 5.2 – 5.3

Announcement: The BIG TEST will be on Tuesday, October 9, It will cover chapter 4 AND 5. We will review on Monday, October 8. If you are not at school on Monday, you are still expected to take the test on Tuesday.

5.4 Medians and Altitudes Objective: To identify properties of medians and altitudes of a triangle

Definition of Median of a Triangle A median of a triangle is a segment whose endpoints are a vertex and the midpoint of the opposite side. Let’s draw a median of a triangle.

Essential Understanding A triangle’s three medians are always concurrent.

Theorem 5.8 Concurrency of 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 In a triangle, the point of concurrency of the medians is the centroid of the triangle. The point is also called the center of gravity of a triangle because it is the point where a triangular shape will balance. For any triangle, the centroid is always inside the triangle.

Let’s draw a figure to support Theorem 5.8

Definition of an Altitude of a Triangle. An altitude of a triangle is the perpendicular segment from a vertex of the triangle to the line containing the opposite side. An altitude of a triangle can be inside or outside the triangle, or it can be a side of the triangle.

Theorem 5.9 Concurrency of Altitudes Theorem The lines that contain the altitudes of a triangle are concurrent The lines that contain the altitudes of a triangle are concurrent at the orthocenter of the triangle. The orthocenter of a triangle can be inside, on, or outside the triangle.

Take Note We have discussed 4 key terms related to concurrency in triangles. They are: Circumcenter (Perpendicular Bisectors) Incenter (Angle Bisectors) Medians (Centroid) Orthocenter (Altitudes)

Your turn to do some work. Complete: pp. 312 – 313 #1 – 4, 8 – 13, 17 – 20 This is due in exactly 20 minutes. If you finish early, read pp324 – 335 in the textbook.

5.6 Inequalities in One Triangle Objective: To use inequalities involving angles and sides of triangles. Essential Understanding: The angles and sides of a triangle have special relationships that involve inequalities.

Comparison Property of Inequality If a = b + c and c > 0, then a >b.

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.

Theorem 5.10 If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side.

Theorem 5.11 If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle.

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.

Theorem 5.13 The Hinge Theorem (SAS Inequality Theorem) If two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger included angle.

Assignment Textbook – p. 329 #9 – 32 – Pp. 336 – 337 #1, 2, 6 – 9,

Announcement: The BIG TEST will be on Tuesday, October 9, It will cover chapter 4 AND 5. We will review on Monday, October 8. If you are not at school on Monday, you are still expected to take the test on Tuesday.

Ticket Out the Door Describe three things you learned today regarding triangles