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Transcript
Unit 3 Triangles Chapter Objectives • • • • • • • Classification of Triangles by Sides Classification of Triangles by Angles Exterior Angle Theorem Triangle Sum Theorem Adjacent Sides and Angles Parts of Specific Triangles 5 Congruence Theorems for Triangles Lesson 3.1 Classifying Triangles Lesson 3.1 Objectives • Classify triangles according to their side lengths. (G1.2.1) • Classify triangles according to their angle measures. (G1.2.1) • Find a missing angle using the Triangle Sum Theorem. (G1.2.2) • Find a missing angle using the Exterior Angle Theorem. (G1.2.2) Classification of Triangles by Sides Name Equilateral Isosceles Scalene 3 congruent sides At least 2 congruent sides No Congruent Sides Looks Like Characteristics Classification of Triangles by Angles Name Acute Equiangular Right Obtuse 3 acute angles 3 congruent angles 1 right angles 1 obtuse angle Looks Like Characteristics Example 3.1 Classify the following triangles by their sides and their angles. Scalene Obtuse Scalene Right Equilateral Equiangular Isosceles Acute Vertex • The vertex of a triangle is any point at which two sides are joined. – It is a corner of a triangle. • There are 3 in every triangle Adjacent Sides and Adjacent Angles • Adjacent sides are those sides that intersect at a common vertex of a polygon. – These are said to be adjacent to an angle. • Adjacent angles are those angles that are right next to each other as you move inside a polygon. – These are said to be adjacent to a specific side. More Parts of Triangles • If you were to extend the sides you will see that more angles would be formed. • So we need to keep them separate – The three angles are called interior angles because they are inside the triangle. – The three new angles are called exterior angles because they lie outside the triangle. Theorem 4.1: Triangle Sum Theorem • The sum of the measures of the interior angles of a triangle is 180o. B mA + mB + mC = 180o C A Example 3.2 Solve for x and then classify the triangle based on its angles. Acute 75o 50o 3x + 2x + 55 = 180 Triangle Sum Theorem 5x + 55 = 180 Simplify 5x = 125 SPOE x = 25 DPOE Example 3.3 Solve for x and classify each triangle by angle measure. 1. mA ( x 30) o ( x 30) x ( x 60) 180 mB x o mC ( x 60) o mA 60o mB 30o mC 90 2. mA (6 x 11)o o 3x 90 180 3x 90 x 30 Right (6 x 11) (3x 2) (5 x 1) 180 mB (3 x 2)o mC (5 x 1)o mA 83o mB 34o mC 59o 14x 12 180 14x 168 x 12 Acute Example 3.4 Draw a sketch of the triangle described. 1. Mark the triangle with symbols to indicate the necessary information. Acute Isosceles 2. Equilateral 3. Right Scalene Example 3.5 Draw a sketch of the triangle described. 1. Mark the triangle with specific angle measures, side lengths, or symbols to indicate the necessary information. Obtuse Scalene 2. Right Isosceles 3. Right Equilateral (Not Possible) Theorem 4.2: Exterior Angle Theorem • The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. B A C m A +m B = m C Example 3.6 Solve for x 6 x 7 2 x (103 x) 6x 7 x 103 5x 7 103 5x 110 x 22 Exterior Angles Theorem Combine Like Terms Subtraction Property Addition Property Division Property Corollary to the Triangle Sum Theorem • A corollary to a theorem is a statement that can be proved easily using the original theorem itself. – This is treated just like a theorem or a postulate in proofs. • The acute angles in a right triangle are complementary. A mA + mB = 90o B C Example 3.7 If you don’t like the Exterior Angle Theorem, then find m2 first using the Linear Pair Postulate. Find the unknown angle measures. 1. 90o 53o m1 180o 3. 143o m1 180o 102o m2 180o m1 37 o m2 78o 2. 90 42 m1 180 o o VA o 132o m1 180o m1 48o 90 33 m2 180 o o 123o m2 180o m2 57o VA m2 m3 122o m1 34 o 78o 68o m1 180o 68o 34o m2 180o o 4. 68o m1 102o Then find m1 using the Angle Sum Theorem. 58o m2 180o m2 122o 102o m2 180o 146o m1 180o m1 34o m2 78o 122o 22o m1 180o 122o 20o m4 180o 144o m1 180o m1 36o 142o m4 180o m4 38o Homework 3.1 • Lesson 3.1 – All Sections – p1-6 • Due Tomorrow Lesson 3.2 Inequalities in One Triangle Lesson 3.2 Objectives • Order the angles in a triangle from smallest to largest based on given side lengths. (G1.2.2) • Order the side lengths of a triangle from smallest to largest based on given angle measures. (G1.2.2) • Utilize the Triangle Inequality Theorem. Theorem 5.10: Side Lengths of a Triangle Theorem • If one side of a triangle is longer than another side, then the angle opposite the longer side is larger than the angle opposite the shorter side. – Basically, the larger the side, the larger the angle opposite that side. 2nd Largest Angle Smallest Side Largest Angle 2nd Longest Side Smallest Angle Theorem 5.11: Angle Measures of a Triangle Theorem • If one angle of a triangle is larger than another angle, then the side opposite the larger angle is longer than the side opposite the smaller angle. – Basically, the larger the angle, the larger the side opposite that angle. 2nd Largest Angle Smallest Side Largest Angle 2nd Longest Side Smallest Angle Example 3.8 Order the angles from largest to smallest. 2. 1. B, A, C Q, P, R 3. A, C , B Example 3.9 Order the sides from largest to smallest. 1. ST , RS , RT 2. DE , EF , DF 33o Example 3.10 Order the angles from largest to smallest. 1. In ABC AB = 12 BC = 11 C , A, B AC = 5.8 Order the sides from largest to smallest. 2. In XYZ mX = 25o XY , XZ , YZ mY = 33o mZ = 122o Theorem 5.13: Triangle Inequality • The sum of the lengths of any two sides of a triangle is greater than the length of the third side. 4 3 1 3 6 6 4 2 6 Add each combination of two sides to make sure that they are longer than the third remaining side. Example 12 Determine whether the following could be lengths of a triangle. a) 6, 10, 15 a) 6 + 10 > 15 10 + 15 > 6 6 + 15 > 10 YES! b) 11, 16, 32 b) 11 + 16 < 32 NO! Hint: A shortcut is to make sure that the sum of the two smallest sides is bigger than the third side. The other sums will always work. Homework 3.2 • Lesson 3.2 – Inequalities in One Triangle – p7-8 • Due Tomorrow • Quiz Friday, October 15th Lesson 3.3 Isosceles, Equilateral, and Right Triangles Lesson 3.3 Objectives • Utilize the Base Angles Theorem to solve for angle measures. (G1.2.2) • Utilize the Converse of the Base Angles Theorem to solve for side lengths. (G1.2.2) • Identify properties of equilateral triangles to solve for side lengths and angle measures. (G1.2.2) Isosceles Triangle Theorems •Theorem 4.6: Base Angles Theorem –If two sides of a triangle are congruent, then the angles opposite them are congruent. •Theorem 4.7: Converse of Base Angles Theorem –If two angles of a triangle are congruent, then the sides opposite them are congruent. Example 10 Solve for x Theorem 4.7 Theorem 4.6 4x + 3 = 15 7x + 5 = x + 47 4x = 12 x=3 6x + 5 = 47 6x = 42 x=7 Equilateral Triangles •Corollary to Theorem 4.6 •Corollary to Theorem 4.7 –If a triangle is equilateral, then it is equiangular. –If a triangle is equiangular, then it is equilateral. Example 11 Solve for x Corollary to Theorem 4.6 Corollary to Theorem 4.6 In order for a triangle to be equiangular, all angles must equal… It does not matter which two sides you set equal to each other, just pick the pair that looks the easiest! 2x + 3 = 4x - 5 3 = 2x - 5 5x = 60 8 = 2x x = 12 x=4 Homework 3.3 • Lesson 3.3 – Isosceles, Equilateral, and Right Triangles – p9-11 • Due Tomorrow • Quiz Tomorrow – Friday, October 15th Lesson 5.3 Medians and Altitudes of Triangles Lesson 5.3 Objectives • • • • Define a median of a triangle Identify a centroid of a triangle Define the altitude of a triangle Identify the orthocenter of a triangle Perpendicular Bisector • A segment, ray, line, or plane that is perpendicular to a segment at its midpoint is called the perpendicular bisector. Triangle Medians • A median of a triangle is a segment that does the following: – Contains one endpoint at a vertex of the triangle, and – Contains the other endpoint at the midpoint of the opposite side of the triangle. A B D C Centroid Remember: All medians intersect the midpoint of the opposite side. • When all three medians are drawn in, they intersect to form the centroid of a triangle. – This special point of concurrency is the balance point for any evenly distributed triangle. • In Physics, this is how we locate the center of mass. Obtuse Acute Right Theorem 5.7: Concurrency of Medians of a Triangle • The medians of a triangle intersect at a point that is two-thirds of the distance from each vertex to the midpoint of the opposite side. – The centroid is 2/3 the distance from any vertex to the opposite side. AP = 2/3AE BP = 2/3BF CP = 2/3CD Example 6 S is the centroid of RTW, RS = 4, VW = 6, and TV = 9. Find the following: a) RV a) b) 6 RU b) 6 • 4 is 2/3 of 6 • c) Divide 4 by 2 and then muliply by 3. Works everytime!! SU c) d) 2 RW d) e) 12 TS e) 6 • f) 6 is 2/3 of 9 SV f) 3 Altitudes • An altitude of a triangle is the perpendicular segment from a vertex to the opposite side. – It does not bisect the angle. – It does not bisect the side. • The altitude is often thought of as the height. • While true, there are 3 altitudes in every triangle but only 1 height! Orthocenter • The three altitudes of a triangle intersect at a point that we call the orthocenter of the triangle. • The orthocenter can be located: – inside the triangle – outside the triangle, or – on one side of the triangle Obtuse Right Acute The orthocenter of a right triangle will always be located at the vertex that forms the right angle. Theorem 5.8: Concurrency of Altitudes of a Triangle • The lines containing the altitudes of a triangle are concurrent. Example 7 Is segment BD a median, altitude, or perpendicular bisector of ABC? Hint: It could be more than one! Perpendicular Bisector Altitude Median None Homework 3.4 • Lesson 3.4 – Altitudes and Medians – p12-13 • Due Tomorrow Lesson 1.7 Intro to Perimeter, Circumference and Area Lesson 1.7 Objectives • Find the perimeter and area of common plane figures. • Establish a plan for problem solving. Perimeter and Area of a Triangle • The area of a triangle is half the length of the base • The perimeter can be times the height of found by adding the the triangle. three sides together. a of a – The height – P=a+b+b c triangle is the h • If the third side is perpendicular unknown, use the length from the Pythagorean Theorembase to the c opposite vertex of to solve for the unknown side. the triangle. – a2 + b 2 = c 2 – A = ½bh • Where a,b are the two shortest sides and c is the longest side. Homework 3.5 • Lesson 3.5 – Area and Perimeter of Triangles – p14-15 • Due Tomorrow