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1 Postulate - Side-Side-Side (SSS) Congruence: If three sides of one triangle are congruent to three sides of a second triangle, then the triangles are congruent. Picture: Postulate - Side-Angle-Side (SAS) Congruence: If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent. Picture: Postulate Angle-Side-Angle (ASA) Congruence: If two angles and the included side of one triangle are congruent to two angles and the included sided of another triangle, then the triangles are congruent. Picture: Postulate Angle-Angle-Side (AAS) Congruence: If two angles and the non-included side of one triangle are congruent to two angles and the non-included sided of another triangle, then the triangles are congruent. Picture: Postulate Hypotenuse-Leg (HL) Congruence: If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent. Picture: Corresponding Parts of Congruent Triangles Are Congruent!! *Must be used AFTER you have two congruent triangles

2 Triangle Angle Sum: The sum of the three angles of any triangle equals 180 o. Exterior Angle: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles. Angle Bisector: An angle bisector divides an angle into two congruent parts. *Creates congruent angles **Halves of equal angles are equal Segment Bisector: A segment bisector creates a midpoint. *Creates a midpoint and congruent segments Midpoint: A midpoint divides a segment into two congruent parts. *Creates congruent segments Vertical Angles: Vertical angles are congruent. *Creates congruent angles Reflexive Property: Anything equals itself. *Creates a shared piece step Perpendicular Lines: ( ) Perpendicular lines form right angles. *Creates 2 steps in a proof: FIRST it creates 90 o angles, SECOND it creates congruent angles. Right Angles: ALL right angles are congruent. *Creates 90 o congruent angles Parallel Lines: ( ) Parallel lines never intersect. *Creates either congruent angles (corresponding, alternate interior, alternate exterior) or supplementary angles (consecutive interior) Complementary Angles: Two angles whose sum is 90 o. *Complements of congruent angles are congruent Supplementary Angles: Two angles whose sum is 180 o. *Supplements of congruent angles are congruent Distance Formula: *Proves type of triangle: Scalene, Isosceles, Equilateral Isosceles Triangle: 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. Corollaries for the Equilateral Triangle: A triangle is equilateral if and only if it is equiangular. *An equilateral triangle has angles that each measure.

3 Geometry CONGRUENT TRIANGLES Day 1 Classifying Triangles/Finding the Missing Side Day 2 Angles of Triangles/Congruent Triangles Day 3 Corresponding Parts/Postulates/Definitions Day 4 Day 5 QUIZ REVIEW Take Home QUIZ Proving Triangles - SSS, SAS Day 6 Proving Triangles ASA, AAS, HL two column proofs Day 7 CPCTC Day 8 Overlapping Triangles Day 11 OPEN NOTE QUIZ Day 12 Isosceles and Equilateral Triangles Day 13 Congruence Transformations/Triangles and Coordinate Proofs Day 14 Day 15 Review TEST

4 DAY 1 Classifying Triangles By Angle: By Side:

5 1. Given: Isosceles Triangle ABC AC = CB AB = 9x - 1 AC = 4x + 1 BC = 5x 0.5 Find the measures of the sides of ΔABC. 2. Given: ΔFGH is equilateral. G Find the measures of the sides of ΔFGH. 2y + 5 3y - 3 F 5y 19 H

6 Problem Set:

7 DAY 2 More information on Triangles

8 Problem Set:

9 DAY 3 Corresponding Parts

10 Problem Set: (w, x, y, and z)

11 DAY 4 QUIZ REVIEW Use a protractor and ruler to classify the triangle as right, acute, obtuse and scalene, isosceles, equilateral. If XYZ = ABC, side XY corresponds with side. Determine whether MNO = QRS and classify each triangle using sides and angles. M(0, -3) N(1, 4) O(3, 1) Q(4, -1) R(6, 1) S(9, -1) In triangle ABC, m<a = 48, and m<c = 24. What type of triangle is triangle ABC? The angles of a triangle are in the ratio of 1:3:5. Find the measure of the largest angle of the triangle. In triangle DEF, m<d = 37 and m<f = 56. Find the measure of an exterior angle at E.

12 Day 5 Two Column Proofs Write a two column proof. Given: MN = PN L N M Prove: LM = LP ΔLMN = ΔLPN P STATEMENT REASON Write a two column proof. Given: BD AC B BD bisects AC Prove: ΔABD = ΔCBD A D C STATEMENT REASON

13 Problem Set:

14 DAY 6 Two column proofs Write a two column proof. Given: QS bisects <PQR P <PSQ = < RSQ S Q Prove: ΔPQS= ΔRQS R STATEMENT REASON Write a two column proof. Given: RQ = ST R S RQ ST U Prove: ΔRUQ = ΔTUS Q T STATEMENT REASON

15 Write a two column proof: C B P Q A Given: PB AC, PD AE, AB = AD Prove: ΔABP = ΔADP D E STATEMENT REASON Problem Set:

16 DAY 7 Two column proofs Given: BA = DC <BAC = <DCA Prove: AD = BC STATEMENT A D REASON B C Given: JM = NK L is the midpoint of JN and KM J K Prove: <JMK = <NKM L M N STATEMENT REASON

17 A Given: BD bisects <B D B BD AC Prove: <A = <C STATEMENT REASON C Given: BA = DC <BAC = <DCA Prove: BC = DA STATEMENT A D REASON B C

18 Problem Set:

19 DAY 8 Given: Two column proofs ΔTPQ= ΔSPR <TQR = <SRQ Q P R Prove: ΔTQR= ΔSQR T S STATEMENT REASON Given: HL = HM PM = KL G J PG = KJ H GH = JH Prove: <G = <J P M L K STATEMENT REASON

20 Given: <K = <M K M KP PR L MR PR Prove: <KPL = <MRL P R STATEMENT REASON

21 Problem Set: B 1. Given: AB = AD AC bisects <BAD A C Prove: ΔABC= ΔADC D 2. Given: AS RT A is the midpoint of RT A S R Prove: ΔRAS= ΔTAS T

22 3. Given: AR CB AR bisects <CAB A Prove: ΔACR= ΔABR C R B E D 4. Given: DCFA <E = <B C ED = AB FD DE CA AB F Prove: EF = CB A B

23 DAY 10 Isosceles/Equilateral Triangles 1. In isosceles triangle MNP, <P is congruent to <M, side NM is 11 cm, and <N = 120 o. Find the measures of <P and< M, and the length of PN Find x and y. 4y 2 2y + 2 6x + 8 Write a two column proof. Given: ΔABC is isosceles B EB bisects <ABC Prove: ΔABE = ΔCBE A E C STATEMENT REASON

24 Problem Set:

25 DAY 11 CONGRUENCE TRANSFORMATIONS Reflection: Translation: Rotation: 1. Triangle XZY with vertices X(2, -8), Z(6, -7) and Y(4, -2) is a transformation of ΔABC with vertices of A(2, 8), B(6, 7), and C(4, 2). Graph the original figure and its image. Identify the transformation and verify that it is a congruence transformation.

26 2. The Bermuda Triangle is a region formed by Miami (Florida), San Jose (Puerto Rico), and Bermuda. The approximate coordinates of each location, respectively, are 26 o N 80 o W, 18.5 o N 66 o W, and 33 o N 65 o W. Write a coordinate proof to prove that the Bermuda Triangle is scalene. 3. Write a coordinate proof to prove that ΔABC is an isosceles triangle if the vertices are A(0,0), B(a,b), and C(2a, 0).

27 Problem Set:

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