Transcript Slide 1
Triangle Inequalities § 7.1 Segments, Angles, and Inequalities § 7.2 Exterior Angle Theorem § 7.3 Inequalities Within a Triangle § 7.4 Triangle Inequality Theorem Segments, Angles, and Inequalities You will learn to apply inequalities to segment and angle measures. Inequalities 1) Inequality Segments, Angles, and Inequalities The Comparison Property of Numbers is used to compare two line segments of unequal measures. The property states that given two unequal numbers a and b, either: a < b or a > b T V The length of 2 cm U 4 cm TU is less than the length W of VW, or TU < VW The same property is also used to compare angles of unequal measures. Segments, Angles, and Inequalities 60° 133° J K The measure of J is greater than the measure of K. inequalities because The statements TU > VW and J > K are called __________ they contain the symbol < or >. Postulate 7–1 Comparison Property For any two real numbers, a and b, exactly one of the following statements is true. a<b a=b a>b Segments, Angles, and Inequalities D S 0 -2 Replace 2 N 4 6 with <, >, or = to make a true statement. SN 6 – (- 1) 7 > DN 6–2 > 4 Lesson 2-1 Finding Distance on a number line. Segments, Angles, and Inequalities Theorem 7–1 If point C is between points A and B, and A, C, and B are AB > CB collinear, then ________ AB > AC and ________. A C B A similar theorem for comparing angle measures is stated below. This theorem is based on the Angle Addition Postulate. Segments, Angles, and Inequalities A similar theorem for comparing angle measures is stated below. This theorem is based on the Angle Addition Postulate. If EP is between ED and EF, then mDEF mDEP and mDEF mPEF D Theorem 7–2 P E F Segments, Angles, and Inequalities Use theorem 7 – 2 to solve the following problem. Replace with <, >, or = to make a true statement. C mBDA < mCDA 18° 40° Since DB is between DC and DA, then mBDA mCDA 149° D B 45° Check: mBDA mCDA 40° + 45° 45° 45° < 85° 108° A Segments, Angles, and Inequalities Property For any numbers a, b, and c, Transitive Property 1) if a < b and b < c, then a < c. if 5 < 8 and 8 < 9, then 5 < 9. 2) if a > b and b > c, then a > c. if 7 > 6 and 6 > 3, then 7 > 3. Segments, Angles, and Inequalities Property For any numbers a, b, and c, Addition and Subtraction Properties 1) if a < b, then a + c < b + c and a – c < b – c. 1<3 1+5<3+5 6<8 2) if a > b, then a + c > b + c and a – c > b – c. For any numbers a, b, and c, 1) If c 0 and a b, then Multiplication and Division Properties a b ac bc and c c 2) If c 0 and a b, then ac bc and a b c c 12 18 12 2 18 2 24 36 12 18 12 18 2 2 69 Exterior Angle Theorem You will learn to identify exterior angles and remote interior angles of a triangle and use the Exterior Angle Theorem. 1) Interior angle 2) Exterior angle 3) Remote interior angle Exterior Angle Theorem interior angles of In the triangle below, recall that 1, 2, and 3 are _______ ΔPQR. Angle 4 is called an exterior _______ angle of ΔPQR. linear pair with one of An exterior angle of a triangle is an angle that forms a _________ the angles of the triangle. In ΔPQR, 4 is an exterior angle at R because it forms a linear pair with 3. Remote interior angles of a triangle are the two angles that do not form ____________________ a linear pair with the exterior angle. In ΔPQR, 1, and 2 are the remote interior angles with respect to 4. P 1 Q 2 3 4 R Exterior Angle Theorem In the figure below, 2 and 3 are remote interior angles with respect to what angle? 5 1 2 3 4 5 Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to sum remote interior angles of the measures of its ___________________. X Theorem 7 – 3 Exterior Angle Theorem 1 2 3 Y 4 Z m4 = m1 + m2 Exterior Angle Theorem Exterior Angle Theorem The measure of an exterior angle of a triangle is greater than remote interior angles the measures of either of its two ____________________. X Theorem 7 – 4 Exterior Angle Inequality Theorem 1 2 3 Y 4 Z m4 > m1 m4 > m2 Exterior Angle Theorem Name two angles in the triangle below that have measures less than 74°. 1 and 3 74° 1 2 3 If a triangle has one right angle, then the other two angles Theorem 7 – 5 must be _____. acute Exterior Angle Theorem 1 and 3 Exterior Angle Theorem The feather–shaped leaf is called a pinnatifid. In the figure, does x = y? Explain. 28° ? x = y __ 28 + 81 = 32 + 78 109 = 110 No! x does not equal y Inequalities Within a Triangle You will learn to identify the relationships between the _____ sides and _____ angles of a triangle. Nothing New! Inequalities Within a Triangle If the measures of three sides of a triangle are unequal, then the measures of the angles opposite those sides in the same order are unequal ________________. P 11 Theorem 7 – 6 M 8 13 L LP < PM < ML mM < mL < mP Inequalities Within a Triangle If the measures of three angles of a triangle are unequal, then the measures of the sides opposite those angles in the same order are unequal ________________. W 45° 75° Theorem 7 – 7 60° J mW < mJ < mK JK < KW < WJ K Inequalities Within a Triangle In a right triangle, the hypotenuse is the side with the greatest measure ________________. W Theorem 7 – 8 5 3 X 4 WY > XW WY > XY Y Inequalities Within a Triangle The longest side is BC So, the largest angle is The largest angle is A L So, the longest side is MN Triangle Inequality Theorem You will learn to identify and use the Triangle Inequality Theorem. Nothing New! Triangle Inequality Theorem The sum of the measures of any two sides of a triangle is greater than the measure of the third side. _______ Theorem 7 – 9 Triangle Inequality Theorem b a+b>c a a+c>b c b+c>a Triangle Inequality Theorem Can 16, 10, and 5 be the measures of the sides of a triangle? No! 16 + 10 > 5 16 + 5 > 10 However, 10 + 5 > 16