Chapter 4 Congruent Triangles Classifying Triangles l Triangle

Chapter 4 Congruent Triangles

Classifying Triangles l Triangle – a three sided polygon – – Sides are segments Vertices are points Has three angles Two ways to classify: l Angles – l All triangles have at least two acute angles, but the third angle is the classifying angle Number of congruent sides they have – Equal number of hash marks are drawn on corresponding sides to indicate that sides are congruent

Classifying Triangles by Angles l Acute Triangle – – A triangle where all of the angles are acute All angle measures are less than 90 degrees l An acute angle with all angles congruent is an equiangular triangle

Classifying Triangles by Angles l Obtuse Triangle – – A triangle where one angle that is obtuse One angle measure is greater than 90 degrees

Classifying Triangles by Angles l Right Triangle – – A triangle where one angle is right One angle measure is equal to 90 degrees

Classifying Triangles by Sides l Scalene Triangle – A triangle where no two sides are congruent

Classifying Triangles by Sides l Isosceles Triangle – A triangle with at least two sides are congruent

Classifying Triangles by Sides l Equilateral Triangle – A triangle where all the sides are congruent l An equilateral triangle is a special kind of isosceles triangle

Example One l Identify the indicated type of triangles in the figure: – – Isosceles Triangles Scalene Triangles Acute Triangles Right Triangles

You Do It l Identify the indicated triangles in the figure if UV = VX = UX. – – Isosceles Triangles Scalene Triangles Obtuse Triangles Acute Triangles

Example Two l Find x and the measure of each side of equilateral triangle RST if RS = x + 9, ST = 2 x, and RT = 3 x – 9

You Do It l Find d and the measure of each side of equilateral triangle KLM if KL = d + 2, LM = 12 – d, and KM = 4 d – 13.

Let’s Try another one l Find x and the measure of each side of equilateral triangle HKT if HK = x + 7, HT = 4 x – 8, and KT = 2 x + 2

Using Distance Formula? l Find the measures of the sides of ΔDEC. Classify the triangle by sides. – – – D(3, 9) E(-5, 3) C(2, 2)

You Do It l Find the measures of the sides of ΔRST. Classify the triangle by sides. – – – R(-1, 3) S(4, 4) T(8, -1)

Angle Sum Theorem l The sum of the measures of a triangle is 180. – Example: l Angle l W + Angle X + Angle Y = 180 If we know the measures of two angles of a triangle, then we can find the third.

Example l Find the missing angle measures.

You Do It l Find the missing angle measures.

Third Angle Theorem l If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent.

Exterior Angles l l l Each angle of a triangle has an exterior angle. Exterior Angle: formed by one side of a triangle and the extension of another side Remote Interior Angles: the interior angles of the triangle not adjacent to a given exterior angle

Exterior Angle Theorem l The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. – Example: l Angle YZP = Angle X + Angle Y

Example l Find the measure of each numbered angle in the figure

You Do It l Find the measure of each numbered angle in the figure

Corollary l Statements that can be used to prove theorems – Corollary 1: l The acute angle of a right triangle are complementary – Corollary 2: l There can be at most one right or obtuse angle in a triangle

Classwork/Homework l Pages 180 – 181 – – – l 14 – 18 (even) 22 – 28 (even) 32 – 34 (even) Pages 189 – 190 – – 12 – 26 (even) 32 and 34

Congruent Triangles l l Triangles that are the same size and shape All three angles and three sides must be congruent Corresponding parts of the triangles must be congruent Definition: Two triangles are congruent if and only if their corresponding parts are congruent. Also known as CPCTC.

Properties l l Just like segments and angles, congruence of triangles is reflexive, symmetric, and transitive. If you slide, flip, or turn a triangle, the size and shape do not change. – This is called the congruence transformations

Side-Side Congruence l l If the sides of one triangle are congruent to the sides of a second triangle, then the triangles are congruent. Abbreviation: SSS

Example l Determine whether ΔRTZ is congruent ΔJKL for R(2, 5), Z(1, 1), T(5, 2), L(-3, 0), K(-7, 1), and J(-4, 4). Explain.

You Do It l Determine whether ΔWDV is congruent to ΔMLP if W(-7, -4), D(-5, -1), V(-1, -2), M(4, 7), L(1, -5), and P(2, -1). Explain.

Side-Angle-Side Congruence l l If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. Abbreviation: SAS

Example l Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove that they are congruent, write not possible.

Angle-Side-Angle Congruence l l If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent Abbreviation: ASA

Angle-Side Congruence l l If two angles and a nonincluded side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent. Abbreviation: AAS

Methods to Prove Triangle Congruence Definition of Congruent Triangles All corresponding parts of one triangle are congruent to the corresponding parts of the other triangle. SSS The three sides of one triangle must be congruent to the three sides of the other triangle SAS Two sides and the included angle of one triangle must be congruent to two sides and the included angle of the other triangle. ASA Two angles and the included side of one triangle must be congruent to two angles and the included side of the other triangle AAS Two angles and a nonincluded side of one triangle must be congruent to two angles and side of the other triangle

Example l Suppose the redwood supports of a chapel, TU and TV, measure 3 feet, TY = 1. 6 feet, and the measurement of angle U and the measurement of angle V are 31. Determine whether ΔTYU is congruent to ΔTYV. Justify your answer.

You Do It l When Ms. Gomez puts her hands on her hips, she forms two triangles with her upper body and arms. Suppose her are lengths AB and DE measure 9 inches, and AC and EF measure 11 inches. Also suppose that you are given BC = DF. Determine whether ΔABC is congruent to ΔEDF.

Classwork/Homework l Worksheet – – Please complete both sides for tomorrow Remember you have a quiz tomorrow on sections 4. 1 – 4. 5

Quiz Time l l l You have 20 – 25 minutes to take your quiz. If you have any questions, please ask Good luck and take your time

Congruence in Right Triangles l l Just like regular triangles, right triangles have congruencies. They are: – – Leg-Leg Congruence Hypotenuse-Angle Congruent Leg-Angle Congruence Hypotenuse-Leg Congruence

Leg-Leg Congruence l l If the legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent. Abbreviation: LL

Hypotenuse-Angle Congruence l l If the hypotenuse and acute angle of one right triangle are congruent to the hypotenuse and corresponding acute angle of another right triangle, then the triangles are congruent. Abbreviation: HA

Leg-Angle Congruence l l If one leg and an acute angle of one right triangle are congruent to the corresponding leg and acute angle of another right triangle, then the triangles are congruent. Abbreviation: LA

Hypotenuse-Leg Congruence l l If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent. Abbreviation: HL

Isosceles Triangles Theorem l If two sides of a triangle are congruent, then the angles opposite those sides are congruent

Example l If GH = HK, HJ = JK, and the measurement of angle GJK is 100, what is the measure of angle HGK?

You Do It l If DE = CD, BC = AC, and the measurement of angle CDE is 120, what is the measure of angle BAC?

What is the converse of the Isosceles Triangle Theorem? l l Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite those sides are congruent. Converse: If two angles of a triangle are congruent, then the sides opposite those angles are congruent. – Abbreviation: Conv. Of Isos. Δ Th.

Example l Name two congruent angles l Name two congruent segments

You Do It l Name two congruent angles

Properties of Equilateral Triangles l l A triangle is equilateral if and only if it is equiangular. Each angle of an equilateral triangle measures 60º

Example l ΔEFG is equilateral, and EH bisects angle E. – Find the measurement of angle 1 and angle 2 – Find x

You Do It l Using the same figure as above, draw EJ so that EJ bisects angle 2 and J lies on FG. – Find the measurement of HEJ and the measurement of EJH – Find the measurement of EJG

Coordinate Proof l l Uses figures in the coordinate plane and algebra to prove geometric concepts. Steps: – – Use the origin as a vertex or center of the figure Place at least on side of a polygon on an axis Keep the figure within the first quadrant if possible Use coordinates that make computations as simple as possible

Example l Position and label isosceles triangle JKL on a coordinate plane so that base JK is a units long

You Do It