Splash Screen. Lesson Menu Five-Minute Check (over Lesson 5–2) NGSSS Then/Now Key Concept: Definition of Inequality Key Concept: Properties of Inequality.

Slides:

Advertisements

Similar presentations

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.

Median ~ Hinge Theorem.

5-3 Inequalities in One Triangle

Concept. Example 1 Use the Exterior Angle Inequality Theorem.

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.

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

Anna Chang T2. Angle-Side Relationships in Triangles The side that is opposite to the smallest angle will be always the shortest side and the side that.

5-2 Inequalities and Triangles

Lesson 5-2 InequalitiesandTriangles. Ohio Content Standards:

Geometry Chapter 5 Benedict. Vocabulary Perpendicular Bisector- Segment, ray, line or plane that is perpendicular to a segment at its midpoint. Equidistant-

Triangle Inequalities

A B C 12 We know ∠B = ∠C S TU 1214 We could write a proof to show ∠T ≠∠U *We could also prove that m ∠T > m ∠U, BUT theorem 1 tells us that!

Welcome to Interactive Chalkboard Glencoe Geometry Interactive Chalkboard Copyright © by The McGraw-Hill Companies, Inc. Developed by FSCreations, Inc.,

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 8–2) NGSSS Then/Now Theorem 8.8: 45°-45°-90° Triangle Theorem Example 1:Find the Hypotenuse.

Warm-up HAIR ACCESSORIES Ebony is following directions for folding a handkerchief to make a bandana for her hair. After she folds the handkerchief in half,

Splash Screen. Lesson Menu Five-Minute Check (over Chapter 4) Then/Now New Vocabulary Theorems: Perpendicular Bisectors Example 1: Use the Perpendicular.

Medians and Altitudes of Triangles And Inequalities in One Triangle

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 4–7) NGSSS Then/Now New Vocabulary Example 1: Position and Label a Triangle Key Concept: Placing.

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 10–2) NGSSS Then/Now Theorem 10.2 Proof: Theorem 10.2 (part 1) Example 1: Real-World Example:

Splash Screen. Lesson Menu Five-Minute Check (over Chapter 7) NGSSS Then/Now New Vocabulary Key Concept: Geometric Mean Example 1:Geometric Mean Theorem.

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 10–3) NGSSS Then/Now New Vocabulary Theorem 10.6: Inscribed Angle Theorem Proof: Inscribed Angle.

The Triangle Inequality Inequalities in Multiple Triangles

Inequalities and Triangles

Splash Screen. Lesson Menu Five-Minute Check (over Chapter 7) Then/Now New Vocabulary Key Concept: Geometric Mean Example 1:Geometric Mean Theorem 8.1.

Geometry Today: 5.5 Instruction 5.5 Instruction Practice Practice Only the educated are free. Epictetus.

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 3–5) NGSSS Then/Now New Vocabulary Key Concept: Distance Between a Point and a Line Postulate.

5.2 Inequalities and Triangles. Objectives Recognize and apply properties of inequalities to the measures of angles in a triangle Recognize and apply.

4.7 Triangle Inequalities. Theorem 4.10 If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than.

4.7 Triangle Inequalities. In any triangle…  The LARGEST SIDE lies opposite the LARGEST ANGLE.  The SMALLEST SIDE lies opposite the SMALLEST ANGLE.

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 5–3) CCSS Then/Now New Vocabulary Key Concept: How to Write an Indirect Proof Example 1:State.

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 10–5) Then/Now Example 1: Find the Hypotenuse in a 45°–45°–90° Triangle Key Concept: 45°–45°–90°

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 10–4) NGSSS Then/Now New Vocabulary Example 1:Identify Common Tangents Theorem Example.

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 4–4) NGSSS Then/Now New Vocabulary Postulate 4.3: Angle-Side-Angle (ASA) Congruence Example.

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 5–4) NGSSS Then/Now Theorem 5.11: Triangle Inequality Theorem Example 1: Identify Possible Triangles.

Objective: 5.3 & Inequalities in One/Two Triangle(s) _________& The Triangle Inequality Warm Up: Solve the inequality: 1. x + 3 < > 10.

Geometry Section 5.5 Use Inequalities in a Triangle.

4.7 Triangle Inequalities

Lesson 5.5 Use Inequalities in a Triangle. Theorem 5.10 A B C 8 5 IF AB > BC, THEN C > A The angle opposite the longest side is the largest angle; pattern.

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 4–1) Then/Now New Vocabulary Theorem 4.1: Triangle Angle-Sum Theorem Proof: Triangle Angle-Sum.

Splash Screen. Lesson Menu Five-Minute Check (over Lesson 8–2) Then/Now Theorem 8.8: 45°-45°-90° Triangle Theorem Example 1:Find the Hypotenuse Length.

Chapter 5, Section 1 Perpendiculars & Bisectors. Perpendicular Bisector A segment, ray, line or plane which is perpendicular to a segment at it’s midpoint.

Chapter 5 Relationships in Triangles 5.1 Bisectors, Medians, and Altitudes 5.2 Inequalities and Triangles 5.4 The Triangle Inequality 5.5 Inequalities.

Chapter 4-3 Inequalities in One Triangle Inequalities in Two Triangles.

Splash Screen. Lesson Menu Five-Minute Check (over Chapter 10) NGSSS Then/Now New Vocabulary Postulate 11.1: Area Addition Postulate Key Concept: Area.

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:

Recognize and apply properties of inequalities to the measures of angles of a triangle. Recognize and apply properties of inequalities to the relationships.

Table of Contents Date: Topic: Description: Page:.

5.6 and 5.7 Triangle Inequalities You found the relationship between the angle measures of a triangle. Recognize and apply properties of inequalities.

You found the relationship between the angle measures of a triangle. Recognize and apply properties of inequalities to the measures of the angles.

Inequalities in One Triangle

Inequalities in One Triangle

6.5 & 6.6 Inequalities in One and Two Triangle

Warm Up 5.2 Skills Check.

Inequalities and Triangles

5.5 Inequalities in Triangles

Y. Davis Geometry Notes Chapter 5.

Inequalities in Triangles

Have your homework out when the bell rings.

T H E O R M S TRIANGLES CONCEPT MAP Prove Triangles are Congruent

Five-Minute Check (over Lesson 5–2) Mathematical Practices Then/Now

Presentation transcript:

Splash Screen

Lesson Menu Five-Minute Check (over Lesson 5–2) NGSSS Then/Now Key Concept: Definition of Inequality Key Concept: Properties of Inequality for Real Numbers Theorem 5.8: Exterior Angle Inequality Example 1: Use the Exterior Angle Inequality Theorem Theorems: Angle-Side Relationships in Triangles Example 2: Identify Arithmetic Sequences Example 3: Order Triangle Side Lengths Example 4: Real-World Example: Angle-Side Relationships

Over Lesson 5–2 A.A B.B C.C D.D 5-Minute Check 1 A.(–4, 5) B.(–3, 4) C.(–2, 5) D.(–1, 4) Find the coordinates of the centroid of the triangle with vertices D(–2, 9), E(3, 6), and F(–7, 0).

Over Lesson 5–2 A.A B.B C.C D.D 5-Minute Check 2 Find the coordinates of the orthocenter of the triangle with vertices F(–1, 5), G(4, 4), and H(1, 1). A. B. C.(2, 3) D.

Over Lesson 5–2 A.A B.B C.C D.D 5-Minute Check 3 A.5 B.7 C.9 D.11 In ΔRST, RU is an altitude and SV is a median. Find y if m  RUS = 7y ___

Over Lesson 5–2 A.A B.B C.C D.D 5-Minute Check 4 A.3 B.4 C.21 D.27 In ΔRST, RU is an altitude and SV is a median. ___ Find RV if RV = 6a + 3 and RT = 10a + 14.

Over Lesson 5–2 A.A B.B C.C D.D 5-Minute Check 5 A.centroid B.circumcenter C.incenter D.orthocenter Which of the following points is the center of gravity of a triangle?

NGSSS LA The student will relate new vocabulary to familiar words. MA.912.G.4.7 Apply the inequality theorems: triangle inequality, inequality in one triangle, and the Hinge Theorem.

Then/Now You found the relationship between the angle measures of a triangle. (Lesson 4–2) Recognize and apply properties of inequalities to the measures of the angles of a triangle. Recognize and apply properties of inequalities to the relationships between the angles and sides of a triangle.

Concept

Example 1 Use the Exterior Angle Inequality Theorem

Example 1 Use the Exterior Angle Inequality Theorem 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. By the Exterior Angle Inequality Theorem, m  14 > m  4 and m  14 > m  11. In addition, m  14 > m  2 and m  14 > m  4 + m  3, so m  14 > m  4 and m  14 > m  3.

Example 1 Use the Exterior Angle Inequality Theorem

Example 1 Use the Exterior Angle Inequality Theorem By the Exterior Angle Inequality Theorem, m  10 > m  5 and m  16 > m  10, so m  16 > m  5. Since  10 and  12 are vertical angles, m  12 > m  5. m  15 > m  12, so m  15 > m  5. In addition, m  17 > m  5 + m  6, so m  17 > m  5.

A.A B.B C.C D.D Example 1 A. B. C. D.

A.A B.B C.C D.D Example 1 A. B. C. D.

Concept

Example 2 Identify Arithmetic Sequence List the angles of ΔABC in order from smallest to largest. Answer:  C,  A,  B The sides from the shortest to longest are AB, BC, and AC. The angles opposite these sides are  C,  A, and  B respectively. So, according to the Angle-Side Relationship, the angles from smallest to largest are  C,  A,  B.

A.A B.B C.C D.D Example 2 A.  X,  T,  V B.  X,  V,  T C.  V,  T,  X D.  T,  V,  X List the angles of ΔTVX in order from smallest to largest.

Example 3 Order Triangle Side Lengths List the sides of ΔABC in order from shortest to longest. Answer: AC, AB, BC The angles from smallest to largest are  B,  C, and  A. The sides opposite these angles are AC, AB, and BC, respectively. So, the sides from shortest to longest are AC, AB, BC.

A.A B.B C.C D.D Example 3 List the sides of ΔRST in order from shortest to longest. A.RS, RT, ST B.RT, RS, ST C.ST, RS, RT D.RS, ST, RT

Example 4 Angle-Side Relationships HAIR ACCESSORIES Ebony is following directions for folding a handkerchief to make a bandana for her hair. After she folds the handkerchief in half, the directions tell her to tie the two smaller angles of the triangle under her hair. If she folds the handkerchief with the dimensions shown, which two ends should she tie?

Example 4 Angle-Side Relationships Theorem 5.10 states that 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. Since  X is opposite the longest side it has the greatest measure. Answer: So, Ebony should tie the ends marked Y and Z.

A.A B.B C.C D.D Example 4 A.  A and  D B.  B and  F C.  C and  E D.  A and  B KITE ASSEMBLY Tanya is following directions for making a kite. She has two congruent triangular pieces of fabric that need to be sewn together along their longest side. The directions say to begin sewing the two pieces of fabric together at their smallest angles. At which two angles should she begin sewing?

End of the Lesson