Unit 2 B1 Triangle Fundamentals Lesson 3 1

Unit 2 B-1 Triangle Fundamentals Lesson 3 -1: Triangle Fundamentals 1

Naming Triangles are named by using its vertices. For example, we can call the following triangle: ∆ABC ∆ACB ∆BAC ∆BCA ∆CAB ∆CBA B C A Lesson 3 -1: Triangle Fundamentals 2

Opposite Sides and Angles Opposite Sides: Side opposite to A : Side opposite to B : Side opposite to C : Opposite Angles: Angle opposite to : A Angle opposite to : B Angle opposite to : C Lesson 3 -1: Triangle Fundamentals 3

Classifying Triangles by Sides cm 47 3. = B A cm =3. 02 = AB C C m 5 c 3. 1 B A A A = AC cm Scalene: A triangle in which all 3 sides are different lengths. B BC = 3. 55 cm C BC = 5. 16 cm Isosceles: A triangle in which at least 2 sides are equal. GI G . 70 =3 Equilateral: A triangle in which all 3 sides are equal. H Lesson 3 -1: Triangle Fundamentals cm GH = 3. 70 cm HI = 3. 70 cm 4 I

Classifying Triangles by Angles Acute: A triangle in which all 3 angles are less than 90˚. G 76° 57° 47° H Obtuse: A A triangle in which one and only one angle is greater than 90˚& less than 180˚ B Lesson 3 -1: Triangle Fundamentals 44° 28° 108° C 5 I

Classifying Triangles by Angles Right: A triangle in which one and only one angle is 90˚ Equiangular: A triangle in which all 3 angles are the same measure. Lesson 3 -1: Triangle Fundamentals 6

Classification by Sides with Flow Charts & Venn Diagrams polygons Polygon triangles Triangle scalene Scalene Isosceles isosceles equilateral Equilateral Lesson 3 -1: Triangle Fundamentals 7

Classification by Angles with Flow Charts & Venn Diagrams Polygon polygons triangles Triangle right acute Right Obtuse Acute equiangular obtuse Equiangular Lesson 3 -1: Triangle Fundamentals 8

Theorems & Corollaries Triangle Sum Theorem: The sum of the interior angles in a triangle is 180˚. Third Angle Theorem: If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent. Corollary 1: Each angle in an equiangular triangle is 60˚. Corollary 2: Acute angles in a right triangle are complementary. Corollary 3: There can be at most one right or obtuse angle in a triangle. Lesson 3 -1: Triangle Fundamentals 9

Exterior Angle Theorem The measure of the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles. Remote Interior Angles Example: Find the m A. B 3 x - 22 = x + 80 3 x – x = 80 + 22 A Exterior Angle D C m A = x = 51° 2 x = 102 Lesson 3 -1: Triangle Fundamentals 10

Median - Special Segment of Triangle Definition: A segment from the vertex of the triangle to the midpoint of the opposite side. Since there are three vertices, there are three medians. C A B F E D In the figure C, E and F are the midpoints of the sides of the triangle. Lesson 3 -1: Triangle Fundamentals 11

Altitude - Special Segment of Triangle Definition: The perpendicular segment from a vertex of the triangle to the segment that contains the opposite side. B In a right triangle, two of the altitudes of are the legs of the triangle. F B F I A A D D In an obtuse triangle, two of the altitudes are outside of the triangle. K Lesson 3 -1: Triangle Fundamentals 12

Perpendicular Bisector – Special Segment of a triangle Definition: A line (or ray or segment) that is perpendicular to a segment at its midpoint. The perpendicular bisector does not have to start from a vertex! P Example: E M A A C B D B In the scalene ∆CDE, is the perpendicular bisector. Q O L N In the right ∆MLN, is the perpendicular bisector. Lesson 3 -1: Triangle Fundamentals R In the isosceles ∆POQ, is the perpendicular bisector. 13

Unit 2 B-1 Isosceles Triangles Lesson 3 -2: Isosceles Triangle 14

Parts of an Isosceles Triangle l l An isosceles triangle is a triangle with two congruent sides. The congruent sides are called legs and the third side is called the base. 3 Leg 1 and 2 are base angles 3 is the vertex angle 1 2 Base Lesson 3 -2: Isosceles Triangle 15

Isosceles Triangle Theorems If two sides of a triangle are congruent, then the angles opposite those sides are congruent. A B C Example: Find the value of x. By the Isosceles Triangle Theorem, the third angle must also be x. Therefore, x + 50 = 180 50° 2 x + 50 = 180 2 x = 130 x° x = 65 Lesson 3 -2: Isosceles Triangle 16

Isosceles Triangle Theorems If two angles of a triangle are congruent, then the sides opposite those angles are congruent. A B C Example: Find the value of x. Since two angles are congruent, the A sides opposite these angles must be congruent. 3 x - 7 x+15 3 x – 7 = x + 15 2 x = 22 ° ° 50 50 B C X = 11 Lesson 3 -2: Isosceles Triangle 17

Unit 2 B-3 Triangle Inequalities Lesson 3 -3: Triangle Inequalities 18

Triangle Inequality The smallest side is across from the smallest angle. l The largest angle is across from the largest side. B BC = 89° AB A Lesson 3 -3: Triangle Inequalities 37° 3. 2 54° =4 . 3 c m l = C A c 3. 5 19 cm m C

Triangle Inequality – examples… For the triangle, list the angles in order from least to greatest measure. m 6 c 4 cm B A 5 cm Lesson 3 -3: Triangle Inequalities 20 C

Triangle Inequality – examples… For the triangle, list the sides in order from shortest to longest measure. (7 x + 8) ° + (7 x + 6 ) ° + (8 x – 10 ) ° = 180° B 8 x-10 22 x + 4 = 180 ° 22 x = 176 m<C = 7 x + 8 = 64 ° X=8 m<A = 7 x + 6 = 62 ° m<B = 8 x – 10 = 54 ° 7 x+6 A Lesson 3 -3: Triangle Inequalities 7 x+8 62 ° 64 ° 21 C

Converse Theorem & Corollaries Converse: If one angle of a triangle is larger than a second angle, then the side opposite the first angle is larger than the side opposite the second angle. Corollary 1: The perpendicular segment from a point to a line is the shortest segment from the point to the line. Corollary 2: The perpendicular segment from a point to a plane is the shortest segment from the point to the plane. Lesson 3 -3: Triangle Inequalities 22

Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side. a+b>c a+c>b b+c>a Example: Determine if it is possible to draw a triangle with side measures 12, 11, and 17. 12 + 11 > 17 Yes Therefore a triangle can be drawn. 11 + 17 > 12 Yes 12 + 17 > 11 Yes Lesson 3 -3: Triangle Inequalities 23

Finding the range of the third side: Since third side cannot be larger than the other two added together, we find the maximum value by adding the two sides. Since third side and the smallest side cannot be larger than the other side, we find the minimum value by subtracting the two sides. Example: Given a triangle with sides of length 3 and 8, find the range of possible values for the third side. The maximum value (if x is the largest The minimum value (if x is not that largest side of the triangle) 3+8<x side of the ∆) 8– 3>x 11 < x 5> x Range of the third side is 5 < x < 11. Lesson 3 -3: Triangle Inequalities 24

Unit 2 B-4 A Congruent Triangles Lesson 4 -2: Congruent Triangles 25

Congruent Figures Congruent figures are two figures that have the same size and shape. IF two figures are congruent THEN they have the same size and shape. IF two figures have the same size and shape THEN they are congruent. Two figures have the same size and shape IFF they are congruent. Lesson 4 -2: Congruent Triangles 26

Congruent Triangles - CPCTC: Corresponding Parts of Congruent Triangles are Congruent Two triangles are congruent IFF their corresponding parts (angles and sides) are congruent. A = │ A ↔ P; B ↔ Q; C ↔ R B ≡ Vertices of the 2 triangles correspond in the same order as the triangles are named. P = Lesson 4 -2: Congruent Triangles ≡ │ Corresponding sides and angles of the two congruent triangles: Q C 27 R

Congruent Triangles B C Z ____ Y _____ X ______ B │ C Z ≡ ≡ X = A │ = A Note: ∆ABC ∆ ZYX ______ ∆ABC ∆XYZ Lesson 4 -2: Congruent Triangles 28 Y

Example………… When referring to congruent triangles (or polygons), we must name corresponding vertices in the same order. R Y S R U A N Y A N U SUN RAY Also NUS YAR Also USN ARY S Lesson 4 -2: Congruent Triangles 29

Example ……… If these polygons are congruent, how do you name them ? O U P N M A T E S R 1. Pentagon MONTA Pentagon PERSU 2. Pentagon ATNOM Pentagon USREP 3. Etc. Lesson 4 -2: Congruent Triangles 30

Lesson 4 -3 Proving Triangles Congruent (SSS, SAS, ASA) Lesson 4 -3: SSS, SAS, ASA 31

Postulates SSS If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. A B D C E F Included Angle: In a triangle, the angle formed by two sides is the included angle for the two sides. Included Side: The side of a triangle that forms a side of two given angles. Lesson 4 -3: SSS, SAS, ASA 32

Included Angles & Sides Included Angle: * * Included Side: Lesson 4 -3: SSS, SAS, ASA * 33

Postulates ASA If two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle, then the triangles are congruent. A B SAS A D C E F B D C F E If two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent. Lesson 4 -3: SSS, SAS, ASA 34

Steps for Proving Triangles Congruent 1. Mark the Given. 2. Mark … Reflexive Sides / Vertical Angles 3. Choose a Method. (SSS , SAS, ASA) 4. List the Parts … in the order of the method. 5. Fill in the Reasons … why you marked the parts. 6. Is there more? Lesson 4 -3: SSS, SAS, ASA 35

Unit 2 B-4 B Proving Triangles Congruent (SSS, SAS, ASA) Lesson 4 -3: SSS, SAS, ASA 36

Postulates SSS If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. A B D C E F Included Angle: In a triangle, the angle formed by two sides is the included angle for the two sides. Included Side: The side of a triangle that forms a side of two given angles. Lesson 4 -3: SSS, SAS, ASA 37

Included Angles & Sides Included Angle: * * Included Side: Lesson 4 -3: SSS, SAS, ASA * 38

Postulates ASA If two angles and the included side of one triangle are congruent to the two angles and the included side of another triangle, then the triangles are congruent. A B SAS A D C E F B D C F E If two sides and the included angle of one triangle are congruent to the two sides and the included angle of another triangle, then the triangles are congruent. Lesson 4 -3: SSS, SAS, ASA 39

Steps for Proving Triangles Congruent 1. Mark the Given. 2. Mark … Reflexive Sides / Vertical Angles 3. Choose a Method. (SSS , SAS, ASA) 4. List the Parts … in the order of the method. 5. Fill in the Reasons … why you marked the parts. 6. Is there more? Lesson 4 -3: SSS, SAS, ASA 40

Problem 1 Step 1: Mark the Given Step 2: Mark reflexive sides Step 3: Choose a Method (SSS /SAS/ASA ) Step 4: List the Parts in the order of the method Step 5: Fill in the reasons Statements Step 6: Is there more? A B SSS Reasons Given Reflexive Property D SSS Postulate C Lesson 4 -3: SSS, SAS, ASA 41

Problem 2 Step 1: Mark the Given Step 2: Mark vertical angles Step 3: Choose a Method (SSS /SAS/ASA) Step 4: List the Parts in the order of the method Step 5: Fill in the reasons Statements Step 6: Is there more? SAS Reasons Given Vertical Angles. Given SAS Postulate Lesson 4 -3: SSS, SAS, ASA 42

Problem 3 Step 1: Mark the Given Step 2: Mark reflexive sides Step 3: Choose a Method (SSS /SAS/ASA) Step 4: List the Parts in the order of the method Step 5: Fill in the reasons Statements Step 6: Is there more? X W ASA Reasons Given Reflexive Postulate Y Given ASA Postulate Z Lesson 4 -3: SSS, SAS, ASA 43