Inequalities in One Triangle LESSON 5–3 Inequalities in One Triangle

Five-Minute Check (over Lesson 5–2) TEKS 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: Order Triangle Angle Measures Example 3: Order Triangle Side Lengths Example 4: Real-World Example: Angle-Side Relationships Lesson Menu

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

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. 5-Minute Check 2

In ΔRST, RU is an altitude and SV is a median In ΔRST, RU is an altitude and SV is a median. Find y if mRUS = 7y + 27. ___ A. 5 B. 7 C. 9 D. 11 5-Minute Check 3

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

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

of the Pythagorean Theorem, the sum of Targeted TEKS G.6(D) Verify theorems about the relationships in triangles, including proof of the Pythagorean Theorem, the sum of interior angles, base angles of isosceles triangles, midsegments, and medians, and apply these relationships to solve problems. Mathematical Processes G.1(E), G.1(F) TEKS

You found the relationship between the angle measures of a triangle. 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. Then/Now

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Use the Exterior Angle Inequality Theorem Example 1

Use the Exterior Angle Inequality Theorem 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. 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. 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. Example 1

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A. B. C. D. Example 1

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List the angles of ΔABC in order from smallest to largest. Order Triangle Angle Measures List the angles of ΔABC in order from smallest to largest. 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. Answer: C, A, B Example 2

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

List the sides of ΔABC in order from shortest to longest. Order Triangle Side Lengths List the sides of ΔABC in order from shortest to longest. 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. Answer: AC, AB, BC 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 3

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

Answer: So, Ebony should tie the ends marked Y and Z. 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. Example 4

KITE ASSEMBLY Tanya is following directions for making a kite 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? A. A and D B. B and F C. C and E D. A and B Example 4

Inequalities in One Triangle LESSON 5–3 Inequalities in One Triangle