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Exploring Triangle Inequalities and Angle Relationships

This activity focuses on triangle inequalities, emphasizing the relationship between the lengths of triangle sides and their opposite angles. Students learn to defend conjectures, construct geometric arguments, and apply theorems to solve problems involving triangles. Through engaging examples, students can order angles and sides based on given lengths, reinforcing the fundamental properties of triangles, such as the sum of side lengths and the SAS and SSS inequalities. This activity enhances understanding of geometric relationships and strengthens problem-solving skills.

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Exploring Triangle Inequalities and Angle Relationships

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  1. -Triangle Inequalities 2.1: Triangle Properties Spingboard activity GSE’s M(G&M)–10–2 Makes and defends conjectures, constructs geometric arguments, uses geometric properties, or uses theorems to solve problems involving angles, lines, polygons, circles, or right triangle ratios(sine, cosine, tangent) within mathematics or across disciplines or contexts

  2. Th. 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. 1st identify the sides B Largest angle Largest side: 5 Smalles angle Smallest side: A 11 7 C

  3. Example 1 • In RGY, RG = 14, GY = 12, and RY = 20. List the angles in ascending order.

  4. Example 2 List the angles in descending order 1st- Find the side measures SG= SB = BG = 2nd- Use the theorem to order the angles Largest, middle, smallest angle

  5. Th. 5-10 • If one angle in a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the smaller angle. B 1st, identify the angles 110 Largest side: Largest Angle: C Smallest Angle: Smallest Side 2nd, use the theorem to generalize the measures of the sides 20 A

  6. Example List the sides of ABC from Greatest to least

  7. Ex. 2 Find the value of x and list the sides of ROT in order from least to greatest. O 54 62 64 T R

  8. Th. 5.12 Spingboard activity • The sum of the lengths of any two sides of a triangle is greater than the length of the third side

  9. Th. 5.13- SAS inequality • If 2 sides of 1 triangle are congruent to 2 sides of a 2nd triangle, and the ___________ of one triangle has a ______ measure than the included angle of the 2nd triangle, then the 3rd side of the 1 Triangle is _____ than the 3rd side of the 2nd triangle. D Q 80 4 in 4 in 20 6 in 6 in E G F A

  10. Examples

  11. Th. 5.14: SSS Inequality If 2 sides of 1 triangle are congruent to 2 sides of a 2nd triangle and the 3rd side in one triangle is _______ than the 3rd side in the 2nd triangle, then the ___________ in one triangle is ______ than the included angle in the 2nd. I P 5 cm 8 cm 5 cm 8 cm E 12 cm M W 10 cm Q

  12. Examples

  13. Assignments

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