Chapter 5: Relationships in Triangles

Bisectors, Medians, and Altitudes Lesson 5.1 Bisectors, Medians, and Altitudes

Foldable Frayer: Fold paper in half both width and length wise. While folded, fold the corner where all creases meet. Then open.

Perpendicular Bisector ( bisector) A line, segment, or ray that passes through the midpoint of the opposite side and is perpendicular to the opposite side Does not need to pass through a vertex Any point on a perpendicular bisector is equidistant from the endpoints Any point equidistant from the endpoints is on the perpendicular bisector Circumcenter: the point where 3 perpendicular bisectors meet Is equidistant from all vertices of the triangle

Perpendicular Bisector CF, BE, and AD are perpendicular bisectors OB, OC and OA are congruent There are always 3 perpendicular bisectors for a triangle F E O C B D

Median A segment that goes from a vertex of the triangle to the midpoint of the opposite side Centroid: is the point where three medians intersect The center of gravity for the triangle The small part of the median is 1/3 of the whole median 2 x small = big 1/3 median = small 2/3 median = big

Median L LQ, MP, and NO are medians of triangle LMN 2 x PX = MX 2 x OX = NX 2 x QX = LX There are always 3 medians per triangle O P X N M Q

Angle Bisector ( bisector) A line, segment or ray that passes through the middle of the angle it begins in and extends to the opposite side Any point on the angle bisector is equidistant from the sides of the triangle Any point equidistant from both sides is on the angle bisector Incenter: the point where three angle bisectors intersect Equidistant from all sides of the triangle

Angle bisector I L M A H J K X U T W Z V Y IK, HM, and JL are angle bisectors: Angles LIA and MIA, angles MJA and KJA, and angles KHA and LHA are congruent to their partner angle. LA, MA, and KA are congruent. ZW is an angle bisector TU and TV are congruent There are always 3 angle bisectors per triangle L M A H J K X U T W Z V Y

Altitude A segment that goes from a vertex of the triangle to the opposite side and is perpendicular to that side Orthocenter: the point where three altitudes intersect

Altitude SU, VW, and TY are altitudes of the triangle R is the orthocenter An altitude of a triangle could be outside of the triangle. There are always 3 altitudes per triangle S Y W R V T U

C. Find the measure of EH.

A. Find QS. B. Find WYZ.

In the figure, A is the circumcenter of ΔLMN. Find x if mAPM = 7x + 13.

In the figure, point D is the incenter of ΔABC In the figure, point D is the incenter of ΔABC. What segment is congruent to DG?

In ΔXYZ, P is the centroid and YV = 12. Find YP and PV.

In ΔLNP, R is the centroid and LO = 30. Find LR and RO.

Inequalities and Triangles Lesson 5.2 Inequalities and Triangles

Foldable Fold the paper into three sections (burrito fold) Then fold the top edge down about ½ an inch Unfold the paper and in the top small rectangles label each column…

Exterior Angle Inequality Inequality with Angles Inequality with Sides Inequality with Angles Exterior Angle = Remote Int. + Remote Int. The exterior angle is greater than either of the remote interior angles by themselves rem. Int. < ext. Ex: The biggest side is across from the biggest angle The smallest side is across from the smallest angle -The biggest angle is across from the biggest side/ the smallest angle is across from the smallest side

List the angles of ΔABC in order from smallest to largest.

List the sides of ΔABC in order from shortest to longest.

What is the relationship between the lengths of RS and ST? ___

What is the relationship between the measures of A and B?

Lesson 5.3 Indirect Proof

Steps to Completing an Indirect Proof: Assume that ______________ (the conclusion is false) Then _______________ (show that the assumption leads to a contradiction) This contradicts the given information that ________________. Therefore, __________________ (rewrite the conclusion) must be true.

The given info is also called the hypothesis or the if part of an if-then statement The info to prove is also called the conclusion or the then part of an if-then statement

B. State the assumption you would make to start an indirect proof for the statement 3x = 4y + 1.

Example Indirect Proof Given: 5x < 25 Prove: x < 5 Assume that x 5. Then, x = 6. So, 5(6) < 25 30 < 25 This contradicts the given information that 5x < 25. 3. Therefore, x < 5 must be true.

Example Indirect Proof Given: m is not parallel to n Prove: m 3 m 2 Assume that m 3 = m 2 . Then, angle 3 and angle 2 are alternate interior angles. When alternate interior angles are congruent then the lines that make them are parallel. That means m and n are parallel. This contradicts the given information that m is not parallel to n. 3. Therefore, m 3 m 2 must be true m 3 2 n

Write an indirect proof to show that if –2x + 11 < 7, then x > 2. Given: –2x + 11 < 7 Prove: x > 2

Given: ΔJKL with side lengths 5, 7, and 8 as shown. Prove: mK < mL Write an indirect proof.

The Triangle Inequality Lesson 5.4 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

Triangle Inequality Theorem Problems Determine if the measures given could be the sides of a triangle. 16, 17, 19 16 + 17 = 33 yes, the sum of the two smallest sides is larger than the third side 6, 9, 15 6 + 9 = 15 no, the sum of the two smallest sides is equal to the other side so it cannot be a triangle Find the range for the measure of the third side given the measures of two sides. 7.5 and 12.1 12.1- 7.5 < x < 12.1 + 7.5 4.6 < x < 19.6 9 and 41 41-9 < x < 41 + 9 32 < x < 50

Determine whether it is possible to form a triangle with side lengths 5, 7, and 8.

Is it possible to form a triangle with the given side lengths of 6 Is it possible to form a triangle with the given side lengths of 6.8, 7.2, 5.1? If not, explain why not.

Find the range for the measure of the third side of a triangle if two sides measure 4 and 13.

In ΔPQR, PQ = 7.2 and QR = 5.2. Which measure cannot be PR?

Inequalities Involving Two Triangles Lesson 5.5 Inequalities Involving Two Triangles

SAS Inequality Theorem SSS Inequality Theorem On the other side of the foldable from Lesson 2 (3 column chart) SAS Inequality Theorem SSS Inequality Theorem Examples: (Hinge Theorem) When 2 sides of a triangle are congruent to 2 sides of another triangle, and the included angle of one triangle is greater than the included angle of the other triangle… Then, the side opposite the larger angle is larger than the side opposite the smaller angle When 2 sides of a triangle are congruent to 2 sides of another triangle, and the 3rd side of a triangle is greater than the 3rd side of the other triangle… Then, the angle opposite the larger side is larger than the angle opposite the smaller side Ex:

A. Compare the measures AD and BD.

B. Compare the measures ABD and BDC.

ALGEBRA Find the range of possible values for a.

Find the range of possible values of n.