Inequalities Involving Two Triangles

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Presentation transcript:

Inequalities Involving Two Triangles Lesson 5-5 Inequalities Involving Two Triangles

Standardized Test Practice: Transparency 5-5 5-Minute Check on Lesson 5-4 Determine whether the given measures can be the lengths of the sides of a triangle. Write yes or no. 1. 5, 7, 8 2. 4.2, 4.2, 8.4 3. 3, 6, 10 Find the range for the measure of the third side of a triangle given the measures of two sides. 4. 4 and 13 5. 8.3 and 15.6 6. Which cannot be the measure of the third side of a triangle if two sides measure 8 and 17? Standardized Test Practice: A 15 B 19 C 22 D 26

Standardized Test Practice: Transparency 5-5 5-Minute Check on Lesson 5-4 Determine whether the given measures can be the lengths of the sides of a triangle. Write yes or no. 1. 5, 7, 8 yes 2. 4.2, 4.2, 8.4 no 3. 3, 6, 10 no Find the range for the measure of the third side of a triangle given the measures of two sides. 4. 4 and 13 9 < n < 17 5. 8.3 and 15.6 7.3 < n < 23.9 6. Which cannot be the measure of the third side of a triangle if two sides measure 8 and 17? Standardized Test Practice: A 15 B 19 C 22 D 26

Objectives Apply the SAS Inequality Apply the SSS Inequality

Vocabulary Included side – the side in common between two angles (the end points are the vertexes) Included angle – the angle formed by two sides sharing a common end point (or vertex)

Theorems Theorem 5.13, SAS Inequality, or Hinge Theorem – If two sides of a triangle are congruent to two sides of another triangle and the included angle in one triangle has a greater measure than the included angle in the other, then the third side of the first triangle is longer that the third side of the second triangle. Theorem 5.14, SSS Inequality – If two sides of a triangle are congruent to two sides of another triangle and the third side in one triangle is longer than the third side in the other, then the angle between the pair of congruent sides in the first triangle is greater than the corresponding angle in the second triangle.

Non Congruent Triangles Theorem 5.13: SAS Inequality or Hinge Theorem AB  XY and AC  XZ mA > mX, then BC > YZ C Z A X B Y Theorem 5.14: SSS Inequality: AB  XY and AC  XZ BC > YZ, then mA > mX

Write an inequality comparing mLDM and mMDN using the information in the figure. The SSS Inequality allows us to conclude that Answer:

Write an inequality finding the range of values containing a using the information in the figure. By the SSS Inequality, SSS Inequality Substitution Subtract 15 from each side. Divide each side by 9.

Also, recall that the measure of any angle is always greater than 0. Subtract 15 from each side. Divide each side by 9. The two inequalities can be written as the compound inequality Answer:

Write an inequality using the information in the figure. a. Compare mWYX and mZYW b. Find the range of values containing n. Answer: Answer: 6 < n < 25

HEALTH Doctors use a straight-leg-raising test to determine the amount of pain felt in a person’s back. The patient lies flat on the examining table, and the doctor raises each leg until the patient experiences pain in the back area. Nitan can tolerate the doctor raising his right leg 35° and his left leg 65° from the table. Which foot can Nitan raise higher above the table? Assume both of Nitan’s legs have the same measurement, the SAS Inequality tells us that the height of the left foot opposite the 65° angle is higher than the height of his right foot opposite the 35° angle. This means that his left foot is raised higher. Answer: his left foot

HEALTH Doctors use a straight-leg-raising test to determine the amount of pain felt in a person’s back. The patient lies flat on the examining table, and the doctor raises each leg until the patient experiences pain in the back area. Megan can lift her right foot 18 inches from the table and her left foot 13 inches from the table. Which leg makes the greater angle with the table? Answer: her right leg

Summary & Homework Summary: Homework: pg 270 – 273: 3-6, 13-16, 32 SAS Inequality: In two triangles, if two sides are congruent, then the measure of the included angle determines which triangle has the longer third side. SSS Inequality: In two triangles, if two sides are congruent, then the length of the third side determines which triangle has the included angle with the greatest measure. Homework: pg 270 – 273: 3-6, 13-16, 32