Inequalities and Triangles

Slides:



Advertisements
Similar presentations
Splash Screen.
Advertisements

NCTM Standards: 2, 3, 6, 8, 9, 10. (Only one is possible) These properties can also be applied to the measures of angles & segments.
5-3 Inequalities in One Triangle
Lesson 5-3 Inequalities in one Triangle
Section 5.3 Inequalities in One Triangle. The definition of inequality and the properties of inequalities can be applied to the measures of angles.
Draw the following: 1. acute triangle 2.right triangle 3.obtuse triangle 4. acute, scalene triangle 5.obtuse, isosceles triangle 6. right, scalene.
Triangle Inequality Theorems Sec 5.5 Goals: To determine the longest side and the largest angle of a triangle To use triangle inequality theorems.
5.5 Inequalities in Triangles
Welcome to Interactive Chalkboard Glencoe Geometry Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Developed by FSCreations, Inc.,
Introduction Think of all the different kinds of triangles you can create. What are the similarities among the triangles? What are the differences? Are.
5-2 Inequalities and Triangles
Chapter 5: Inequalities!
Lesson 5-2 InequalitiesandTriangles. Ohio Content Standards:
Triangle Inequalities
5.4 The Triangle Inequality
Welcome to Interactive Chalkboard
Welcome to Interactive Chalkboard Glencoe Geometry Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Developed by FSCreations, Inc.,
Unit 5.
Chapter 5 Relationships in Triangles. Warm - Up Textbook – Page – 11 (all) This will prepare you for today’s lesson.
5-6 Inequalities in One Triangle
The Triangle Inequality Inequalities in Multiple Triangles
Lesson 4-2 Angles of Triangles.
5.5Use Inequalities in a Triangle Theorem 5.10: If one side of a triangle is longer than the other side, then the angle opposite the longest side is _______.
5.6 Inequalities in One Triangle The angles and sides of a triangle have special relationships that involve inequalities. Comparison Property of Inequality.
Splash Screen.
5.2 Inequalities and Triangles. Objectives Recognize and apply properties of inequalities to the measures of angles in a triangle Recognize and apply.
Chapter 5 Review Segments in Triangles. Test Outline Multiple Choice –Be able to identify vocab (pick out from a picture) –Be able to apply SAS and SSS.
Triangle Inequalities What makes a triangle and what type of triangle.
GEOMETRY HELP Explain why m  4 > m  5. Substituting m  5 for m  2 in the inequality m  4 > m  2 produces the inequality m  4 > m  5.  4 is an.
Chapter 4 Section 4.1 Section 4.2 Section 4.3. Section 4.1 Angle Sum Conjecture The sum of the interior angles of a triangle add to 180.
Thursday, November 8, 2012 Agenda: TISK & No MM Lesson 5-5: Triangle Inequalities Homework: 5-5 Worksheet.
Chapter 5: Relationships Within Triangles 5.5 Inequalities in Triangles.
Triangle Inequality Theorem and Side Angle Relationship in Triangle
4.7 Triangle Inequalities
Splash Screen. Lesson Menu Five-Minute Check (over Lesson 5–2) NGSSS Then/Now Key Concept: Definition of Inequality Key Concept: Properties of Inequality.
5-3 Inequalities in One Triangle 5-4 Indirect proof 5-5 The triangle Inequality 5-6 Inequality in two triangles. Chapter 5.
Chapter 5, Section 1 Perpendiculars & Bisectors. Perpendicular Bisector A segment, ray, line or plane which is perpendicular to a segment at it’s midpoint.
Triangle Inequalities Objectives: 1.Discover inequalities among sides and angles in triangles.
5-3 Inequalities and Triangles The student will be able to: 1. Recognize and apply properties of inequalities to the measures of the angles of a triangle.
Sect. 5.5 Inequalities in One Triangle Goal 1 Comparing Measurements of a Triangle. Goal 2 Using the Triangle Inequality.
Inequalities in One Triangle LESSON 5–3. Lesson Menu Five-Minute Check (over Lesson 5–2) TEKS Then/Now Key Concept: Definition of Inequality Key Concept:
Splash Screen.
Introduction Think of all the different kinds of triangles you can create. What are the similarities among the triangles? What are the differences? Are.
Proving Lines Parallel
Chapter Inequalities in One Triangle 5-4 Indirect proof 5-5 The triangle Inequality 5-6 Inequality in two triangles.
Recognize and apply properties of inequalities to the measures of angles of a triangle. Recognize and apply properties of inequalities to the relationships.
Splash Screen.
5.2: Triangle Inequalities
Splash Screen.
5.2 HW ANSWERS Pg. 338 #5-10, # YJ = SJ =
Inequalities in One Triangle
Lesson 5-4 The Triangle Inequality
Inequalities in One Triangle
Inequalities and Triangles pp. 280 – 287 &
Inequalities in One Triangle
6.5 & 6.6 Inequalities in One and Two Triangle
Warm Up 5.2 Skills Check.
Inequalities and Triangles
5-2 Inequalities and Triangles
Entry Task *Use the geosticks as models for the different board lengths. You will need 2 green, 2 orange, 1 brown, 1 blue and 1 lime green to represent.
Triangle Theorems.
Class Greeting.
5.5 Inequalities in Triangles
Use Inequalities in a Triangle
Base Angles & Exterior Angles
Inequalities in Triangles
INEQUALITIES Sides/Angles of Triangles
5-2 Inequalities and Triangles
Inequalities for Sides and Angles of a Triangle
Five-Minute Check (over Lesson 5–2) Mathematical Practices Then/Now
Presentation transcript:

Inequalities and Triangles Lesson 5-2 Inequalities and Triangles

5-Minute Check on Lesson 5-1 Transparency 5-2 5-Minute Check on Lesson 5-1 In the figure, A is the circumcenter of LMN. 1. Find y if LO = 8y + 9 and ON = 12y – 11. 2. Find x if mAPM = 7x + 13. 3. Find r if AN = 4r – 8 and AM = 3(2r – 11). In RST, RU is an altitude and SV is a median. 4. Find y if mRUS = 7y + 27. 5. Find RV if RV = 6a + 3 and RT = 10a + 14.

5-Minute Check on Lesson 5-1 Transparency 5-2 5-Minute Check on Lesson 5-1 In the figure, A is the circumcenter of LMN. 1. Find y if LO = 8y + 9 and ON = 12y – 11. 5 2. Find x if mAPM = 7x + 13. 11 3. Find r if AN = 4r – 8 and AM = 3(2r – 11). 12.5 In RST, RU is an altitude and SV is a median. 4. Find y if mRUS = 7y + 27. 9 5. Find RV if RV = 6a + 3 and RT = 10a + 14. 27

Objectives Recognize and apply properties of inequalities to the measures of angles of a triangle Recognize and apply properties of inequalities to the relationships between angles and sides of a triangle

Vocabulary No new vocabulary words or symbols

Theorems Theorem 5.8, Exterior Angle Inequality Theorem – If an angle is an exterior angle of a triangle, then its measure is greater that the measure of either of it corresponding remote interior angles. Theorem 5.9 – If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side. Theorem 5.10 – If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.

Key Concept Step 1: Arrange sides or angles from smallest to largest or largest to smallest based on given information Step 2: Write out identifiers (letters) for the sides or angles in the same order as step 1 Step 3: Write out missing letter(s) to complete the relationship Step 4: Answer the question asked 19 > 14 > 7 WT > AW > AT A > T > W A W T 19 7 14

Determine which angle has the greatest measure. Explore Compare the measure of 1 to the measures of 2, 3, 4, and 5. Plan Use properties and theorems of real numbers to compare the angle measures. Solve Compare m3 to m1. By the Exterior Angle Theorem, m1 = m3 + m4. Since angle measures are positive numbers and from the definition of inequality, m1 > m3. Compare m4 to m1. By the Exterior Angle Theorem, m1 m3 m4. By the definition of inequality, m1 > m4.

Compare m5 to m1. Since all right angles are congruent, 4 5. By the definition of congruent angles, m4 m5. By substitution, m1 > m5. Compare m2 to m5. By the Exterior Angle Theorem, m5 m2 m3. By the definition of inequality, m5 > m2. Since we know that m1 > m5, by the Transitive Property, m1 > m2. Examine The results on the previous slides show that m1 > m2, m1 > m3, m1 > m4, and m1 > m5. Therefore, 1 has the greatest measure. Answer: 1 has the greatest measure.

EXAMPLE 2 Order the angles from greatest to least measure. Answer: 5 has the greatest measure; 1 and 2 have the same measure; 4, and 3 has the least measure.

EXAMPLE 3 Use the Exterior Angle Inequality Theorem to list all angles whose measures are less than m14. By the Exterior Angle Inequality Theorem, m14 > m4, m14 > m11, m14 > m2, and m14 > m4 + m3. Since 11 and 9 are vertical angles, they have equal measure, so m14 > m9. m9 > m6 and m9 > m7, so m14 > m6 and m14 > m7. Answer: Thus, the measures of 4, 11, 9,  3,  2, 6, and 7 are all less than m14 .

EXAMPLE 4 Use the Exterior Angle Inequality Theorem to list all angles whose measures are greater than m5. By the Exterior Angle Inequality Theorem, m10 > m5, and m16 > m10, so m16 > m5, m17 > m5 + m6, m15 > m12, and m12 > m5, so m15 > m5. Answer: Thus, the measures of 10, 16, 12, 15 and 17 are all greater than m5.

EXAMPLE 5 Use the Exterior Angle Inequality Theorem to list all of the angles that satisfy the stated condition. a. all angles whose measures are less than m4 b. all angles whose measures are greater than m8 Answer: 5, 2, 8, 7 Answer: 4, 9, 5

EXAMPLE 6 Determine the relationship between the measures of RSU and SUR. Answer: The side opposite RSU is longer than the side opposite SUR, so mRSU > mSUR.

EXAMPLE 7 Determine the relationship between the measures of TSV and STV. Answer: The side opposite TSV is shorter than the side opposite STV, so mTSV < mSTV.

EXAMPLE 8 Determine the relationship between the measures of RSV and RUV. mRSU > mSUR mUSV > mSUV mRSU + mUSV > mSUR + mSUV mRSV > mRUV Answer: mRSV > mRUV

EXAMPLE 9 Determine the relationship between the measures of the given angles. a. ABD, DAB b. AED, EAD c. EAB, EDB Answer: ABD > DAB Answer: AED > EAD Answer: EAB < EDB

Summary & Homework Summary: Homework: The largest angle in a triangle is opposite the longest side, and the smallest angle is opposite the shortest side The longest side in a triangle is opposite the largest angle, and the shortest side is opposite the smallest angle Homework: pg 251: (17-34, 46-50)