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 NameEquilateralIsoscelesScalene Looks Like Characteristics3 congruent sides At least 2 congruent sides No Congruent Sides

Classification of Triangles by Angles NameAcuteEquiangularRightObtuse Looks Like Characteristics3 acute angles 3 congruent angles 1 right angles 1 obtuse angle

Example 3.1 Classify the following triangles by their sides and their angles. Scalene Obtuse Scalene Right Isosceles Acute Equilateral Equiangular

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 180 o. A B C m  A + m  B + m  C = 180 o

Example 3.2 Solve for x and then classify the triangle based on its angles. 3x + 2x + 55 = 180 Triangle Sum Theorem 5x + 55 = 180 Simplify 5x = 125 SPOE x = 25 DPOE Acute 75 o 50 o

Example 3.3 Solve for x and classify each triangle by angle measure. 1. Right 2. Acute

Example 3.4 Draw a sketch of the triangle described. Mark the triangle with symbols to indicate the necessary information. 1.Acute Isosceles 2.Equilateral 3.Right Scalene

Example 3.5 Draw a sketch of the triangle described. Mark the triangle with specific angle measures, side lengths, or symbols to indicate the necessary information. 1.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. m  A +m  B = m  C A B C

Example 3.6 Solve for x 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. C A B m  A + m  B = 90 o

Example 3.7 Find the unknown angle measures VA If you don’t like the Exterior Angle Theorem, then find m  2 first using the Linear Pair Postulate. Then find m  1 using the Angle Sum Theorem. VA

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. Longest side Largest Angle 2 nd Longest Side 2 nd Largest Angle Smallest 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. Longest side Largest Angle 2 nd Longest Side 2 nd Largest Angle Smallest Side Smallest Angle

Example 3.8 Order the angles from largest to smallest

Example 3.9 Order the sides from largest to smallest o

Example 3.10 Order the angles from largest to smallest. 1.In  ABC AB = 12 BC = 11 AC = 5.8 Order the sides from largest to smallest. 2.In  XYZ m  X = 25 o m  Y = 33 o m  Z = 122 o

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. 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) > > > 10 YES! b)11, 16, 32 b) < 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 15 th

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 4x + 3 = 15 4x = 12 x = 3 Theorem 4.6 7x + 5 = x x + 5 = 47 6x = 42 x = 7

Equilateral Triangles Corollary to Theorem 4.6 –If a triangle is equilateral, then it is equiangular. Corollary to Theorem 4.7 –If a triangle is equiangular, then it is equilateral.

Example 11 Solve for x Corollary to Theorem 4.6 In order for a triangle to be equiangular, all angles must equal… 5x = 60 x = 12 Corollary to Theorem 4.6 It does not matter which two sides you set equal to each other, just pick the pair that looks the easiest! 2x + 3 = 4x = 2x = 2x x = 4

Homework 3.3 Lesson 3.3 – Isosceles, Equilateral, and Right Triangles –p9-11 Due Tomorrow Quiz Tomorrow –Friday, October 15 th

Lesson 5.3 Medians and Altitudes of Triangles

Lesson 5.3 Objectives Define a median of a triangleDefine a median of a triangle Identify a centroid of a triangleIdentify a centroid of a triangle Define the altitude of a triangleDefine the altitude of a triangle Identify the orthocenter of a triangleIdentify 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.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: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 C D

Centroid When all three medians are drawn in, they intersect to form the centroid of a triangle.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.In Physics, this is how we locate the center of mass. AcuteRight Obtuse Remember: All medians intersect the midpoint of the opposite side.

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 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 / 3 AE BP = 2 / 3 BF CP = 2 / 3 CD 2 / 3 AE 2 / 3 BF 2 / 3 CD

Example 6 S is the centroid of  RTW, RS = 4, VW = 6, and TV = 9. Find the following: a)RV a)6 b)RU b)6 4 is 2 / 3 of 64 is 2 / 3 of 6 Divide 4 by 2 and then muliply by 3. Works everytime!!Divide 4 by 2 and then muliply by 3. Works everytime!! c)SU c)2 d)RW d)12 e)TS e)6 6 is 2 / 3 of 96 is 2 / 3 of 9 f)SV f)3

Altitudes An altitude of a triangle is the perpendicular segment from a vertex to the opposite side.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.The altitude is often thought of as the height. While true, there are 3 altitudes in every triangle but only 1 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 three altitudes of a triangle intersect at a point that we call the orthocenter of the triangle. The orthocenter can be located:The orthocenter can be located: –inside the triangle –outside the triangle, or –on one side of the triangle Acute Right Obtuse 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.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! Median Altitude PerpendicularBisector 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 perimeter can be found by adding the three sides together. –P = a + b + c If the third side is unknown, use the Pythagorean Theorem to solve for the unknown side. –a 2 + b 2 = c 2 Where a,b are the two shortest sides and c is the longest side. The area of a triangle is half the length of the base times the height of the triangle. –The height of a triangle is the perpendicular length from the base to the opposite vertex of the triangle. –A = ½bh a b c h

Homework 3.5 Lesson 3.5 – Area and Perimeter of Triangles –p14-15 Due Tomorrow