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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 Triangle Angle Sum: The sum of the three angles of any triangle equals 180o. 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 90o angles, SECOND it creates congruent angles. Right Angles: ALL right angles are congruent. o *Creates 90 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 90o. *Complements of congruent angles are congruent Supplementary Angles: Two angles whose sum is 180o. *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 _________________. 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 QUIZ REVIEW Take Home QUIZ Day 5 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 Review Day 15 TEST DAY 1 By Angle: By Side: Classifying Triangles 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. Find the measures of the sides of ΔFGH. G 2y + 5 3y - 3 F H 5y – 19 Problem Set: DAY 2 More information on Triangles Problem Set: DAY 3 Corresponding Parts Problem Set: (w, x, y, and z) 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. Day 5 Two Column Proofs Write a two column proof. Given: L MN = PN LM = LP M Prove: ΔLMN = ΔLPN N P STATEMENT REASON Write a two column proof. B Given: BD ┴ AC BD bisects AC Prove: A ΔABD = ΔCBD C D STATEMENT REASON Problem Set: DAY 6 Two column proofs Write a two column proof. P Q Given: QS bisects <PQR <PSQ = < RSQ Prove: ΔPQS= ΔRQS S R STATEMENT REASON Write a two column proof. R S U Q STATEMENT Given: RQ = ST RQ ║ ST Prove: ΔRUQ = ΔTUS T REASON Write a two column proof: C B P Q A Given: PB ┴ AC, PD ┴ AE, AB = AD Prove: ΔABP = ΔADP D E STATEMENT Problem Set: REASON DAY 7 Two column proofs A B D C Given: BA = DC <BAC = <DCA Prove: AD = BC STATEMENT REASON Given: JM = NK L is the midpoint of JN and KM J K L Prove: <JMK = <NKM M STATEMENT REASON N A Given: BD bisects <B BD ┴ AC D Prove: <A = <C C STATEMENT B REASON A B D C Given: BA = DC <BAC = <DCA Prove: BC = DA STATEMENT REASON Problem Set: DAY 8 Two column proofs Q Given: Prove: ΔTPQ= ΔSPR <TQR = <SRQ R P ΔTQR= ΔSQR T STATEMENT Given: Prove: REASON HL = HM PM = KL PG = KJ GH = JH <G = <J STATEMENT S G J H P M REASON L K Given: Prove: <K = <M KP ┴ PR MR ┴ PR K <KPL = <MRL P STATEMENT M L REASON R Problem Set: B 1. Given: Prove: AB = AD AC bisects <BAD A C ΔABC= ΔADC D R 2. Given: Prove: AS ┴ RT A is the midpoint of RT A ΔRAS= ΔTAS T S A 3. Given: AR ┴ CB AR bisects <CAB Prove: ΔACR= ΔABR C B R E 4. Given: Prove: DCFA <E = <B ED = AB FD ┴ DE CA ┴ AB EF = CB D C F A B DAY 10 1. Isosceles/Equilateral Triangles In isosceles triangle MNP, <P is congruent to <M, side NM is 11 cm, and <N = 120o. Find the measures of <P and< M, and the length of PN. 1. Find x and y. 80 4y – 2 2y + 2 6x + 8 Write a two column proof. B A E STATEMENT Given: ΔABC is isosceles EB bisects <ABC Prove: ΔABE = ΔCBE C REASON Problem Set: 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. 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 26o N 80o W, 18.5o N 66o W, and 33o N 65o 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). Problem Set: 9. 10. 11.