Chapter 5 Relationships within Triangles Midsegments Definition of

Chapter 5 Relationships within Triangles

Midsegments • Definition of a midsegment: a segment connecting the _________ of two sides of the triangle. • Triangle Midsegment Theorem: if a segment joins the midpoints of a triangle, then the segment is _______ to the third side and is ______ as long.

Identifying Parallel Segments •

Using Properties of Midsegments • In the figure below, AD = 6 and DE = 7. 5. What are the lengths of: • DC = • AC = • EF = • AB =

Using Coordinate Geometry - Midsegments • The coordinates of the vertices of a triangle are E(1, 2), F(5, 6), and G(3, -2). • Find the coordinates of H, the midpoint of EG, and J, the midpoint of FG. • Show that HJ EF • Show that HJ = ½ EF.

Practice: pp. 289 – 290 #26, 38 -42 Challenge: p. 291 #48

Perpendicular Bisectors • Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is ___________ from the __________ of the segment. • Converse of the Perpendicular Bisector Theorem: If a point is _________ from the endpoints of a segment, then it is on the ____________ of the segment. • If PM is the perpendicular bisector of AB, then _____.

Using the Perpendicular Bisector Theorem • Find the length of QR.

Angle Bisectors • Angle Bisector Theorem: If a point is on the bisector of an angle, then the point is _________ from the sides of the angle. • 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 _________. • If QS is the angle bisector of PQR, then _____.

Using the Angle Bisector Theorem • What is the length of FB?

Practice: pp. 296 -297 #9 -11, 16, 17, 32, 33

Definitions • Concurrent: Three or more lines that ___________________________ • Point of Concurrency: The point at which three lines _______ • In the diagram, the three lines are __________ and ____ is the point of concurrency. A

Concurrency of Perpendicular Bisectors Theorem • The perpendicular bisectors of the sides of a triangle are _________ at a point equidistant from the _______. • P is the ___________ of the triangle. It is the center of the circle that is circumscribed about the triangle • In ∆ABC, _____

Location of the Circumcenter • Draw in the perpendicular bisectors of each triangle to find the location of the circumcenter for each type of triangle. Acute Triangle Right Triangle Obtuse Triangle

Using Coordinate Geometry – Perpendicular Bisectors • What are the coordinates of the circumcenter of the triangle with vertices P(0, 6), O(0, 0), and S(4, 0)?

Concurrency of Angle Bisectors Theorem • The angle bisectors of a triangle are _________ at a point equidistant from the ______________. • P is the _________ of the triangle. It is the center of the circle that is inscribed in the triangle. • In ∆ABC, _____

Identifying and Using the Incenter • QN = 5 x + 36 and QM = 2 x + 51. What is QO?

Practice: pp. 305 -306 # 11, 14, 16, 18

Medians • A median of a triangle is a segment whose endpoints are a ________ and the __________ of the opposite side.

Concurrency of Medians Theorem •

Finding the Length of a Median • In ∆XYZ, ZA = 9. • What is the length of CA? • What is the length of ZC?

Altitudes • An altitude of a triangle is the _________ segment from a ________ of the triangle to the line containing the ___________.

Concurrency of Altitudes Theorem • The lines that contain the altitudes of a triangle are concurrent. • H is the ________ of the triangle.

Location of the orthocenter • Draw in the altitudes of each triangle to find the location of the orthocenter for each type of triangle. Acute Triangle Right Triangle Obtuse Triangle

Using Coordinate Geometry - Altitudes • ∆DEF has vertices D(1, 2), E(1, 6), and F(4, 2). What are the coordinates of the orthocenter of ∆DEF?

Identifying Medians and Altitudes • Determine whether each line is a median, altitude, or neither. • EG • AD • CF

Naming Medians and Altitudes • A median in ∆ABC • An altitude for ∆ABC • A median in ∆AHC • An altitude for ∆AHB • An altitude for ∆AHG

Summary of Triangle Segments • Name the type of segments that are concurrent at the given point.

Practice: pp. 312 -313 #8 -10, 15, 17 -20, 24 -27

Indirect Proof • A proof involving _______ reasoning is an indirect proof (sometimes called proof by ________). • Step 1: Temporarily assume the _______ of what you want to prove (negation). • Step 2: Show that this temporary assumption leads to a _________. • Step 3: Conclude that the temporary assumption must be ______ and that what you want to prove must be true.

Writing an Indirect Proof • Given: ∆ABC is scalene (all sides have different lengths) • Prove: A, B, and C all have different measures • Step 1: Assume temporarily the opposite of what you want to prove. • Step 2: Show that this assumption leads to a contradiction • Step 3: Conclude that the temporary assumption must be false and that what you want o prove must be true.

Writing an Indirect Proof •

Writing an Indirect Proof • Given: ∆LMN • Prove: ∆LMN has at most one right angle

Practice: pp. 320 -322 #17, 23, 24, 29

Theorems • Theorem 5 -10: If two sides of a triangle are not congruent, then the _______ angle lies opposite the ______ side. • Theorem 5 -11: If two angles of a triangle are not congruent, then the ______ side lies opposite the ______ angle. • Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is ___________ the length of the third side.

Using Theorem 5 -10 and 5 -11 • In ∆SOX, OX = 4, OS = 6, and XS = 9. Order the angles from least to greatest. • In ∆SOX, m S = 24 and m O = 130. Which side of ∆SOX is the shortest side?

Using the Triangle Inequality Theorem • Can a triangle have sides with lengths 2 m, 6 m, and 9 m? Explain. • A triangle has side lengths of 4 in. and 7 in. What is the range of possible lengths for the third side?

Practice: pp. 329 -330 #12, 19, 25, 26, 30, 31, 37 -39

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. BC ______ YZ • Converse of the Hinge Theorem (SSS Inequality): If two sides of one triangle are congruent to two sides of another triangle, and the third sides are not congruent, then the larger included angle is opposite the longer third side. A ______ X

Using the Hinge Theorem • What inequality relates LN and OQ in the figure below.

Using the Converse of the Hinge Theorem • What is the range of possible values for x in the figure below?

Practice: pp. 336 -337 #6 -9, 11 -14, 16 -18