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Transcript
Honors Geometry Chapter 4 Section Target 4.1 1 Triangle Sum Conjecture 4.2 1 Properties of Isosceles Triangles Pg 208 1-10,18-21,24,25 4.3 1 Triangle Inequalities Pg 218 1-17,19-21 4.4 2 Triangle Shortcuts Pg 224 1-10,12-19 4.5 2 More Triangle Shortcuts Pg 229 1-19,22 4.4/4.5 2/3 Triangle Proofs 4.4 & 4.5 WS 4.6 3 Corresponding Parts of Congruent Triangles Pg 233 1-9,12-15,18,22,23 4.7 3 Flow Chart Thinking Pg 239 1-5,7-14 3 Congruence in Right Triangles Congruence in Right Triangles WS 3 Proving Special Triangle Conjectures Pg245 1-7,12-14 4.8 Section Name Notes Date Homework Pg 203 2-9 Chapter 4 Learning Targets 1. The student will be able to know and apply side and angle properties of triangles. 2. The student will be able to recognize congruent triangles. 3. The student will be able to construct proofs for congruent triangles and their corresponding parts. 4.2 Properties of Isosceles Triangle Honors Geometry (C-17) Triangle Sum Conjecture – The sum of the measures of the angles of in every triangle is ______________. Ex 1: Ex 2: Isosceles Triangle - An isosceles triangle has at least two congruent sides. (C-18) Isosceles Triangle Conjecture - If a triangle is isosceles, then base angles are _______ . Ex 3: Ex 4: (C-19) Converse of the Isosceles Triangle Conjecture - If a triangle has two congruent angles, then ________________ . Ex 5: Ex 6: 4.3 Triangle Inequalities (C-20) Honors Geometry Triangle Inequality Conjecture - The sum of the lengths of any two sides of a triangle is _____________ the length of the third side. Ex 1: Are the following sides of a triangle possible? a. 5 in, 7 in, 9in b. 10 cm, 10 cm, 20 cm c. 12 in, 24in, 2 ft Ex 2: Two sides of a triangle are given. What is the range for the third side of the triangle? a. 4 m & 7m b. 10 mm & 14 mm c. 20 yd & 15 yd (C-21) Side-Angle Inequality Conjecture - In a triangle, if one side is longer than another side, then the angle opposite the longer side is __________________ . Put the unknown measures in order from least to greatest. Ex 3: Ex 5: Ex 4: If you extend one side of a triangle from one vertex, then you have constructed an exterior angle at that vertex. (C-22) Triangle Exterior Angle Conjecture - The measure of an exterior angle of a triangle is __________________________ . Ex 6: Find the unknown angle Ex 7: Find the unknown variables x x = ______ 7y y = ______ z = ______ z 31y 4.4 Triangle Shortcuts Honors Geometry What does it mean to have congruent triangles? Is it necessary to prove all angles and all sides congruent in order to conclude the triangles are congruent? Today we will take a look at some possible shortcuts to prove triangles congruent. C-23 If the three sides of one triangle are congruent to the three sides of another triangle, then _____________________. C-24 If two sides and the angle between them in one triangle are congruent to two sides and the angle between them in another triangle, then _____________________. C-25 If two sides and an angle that is not between the two sides in one triangle are congruent to two sides and an angle that is not between the two sides in another triangle, then _____________________. Determine if the following triangles are congruent . If they are, name the congruent triangles and then name why they are congruent. D 1) 2) A 3) H G C B I X W E 4) Z Y 5) R E D J 6) L O F S T U H M G N 4.5 More Triangle Shortcuts Honors Geometry C-25 If the two angles and the side between them in one triangle are congruent to _____________________. C-26 If two angles and a side that is not between them in one triangle are congruent to the corresponding _____________________. C-27 If the three angles of one triangle are congruent to _____________________. Determine if the following triangles are congruent . If they are, name the congruent triangles and then name why they are congruent. 1) X 2) R 3) S O L T Y N M 4) D C Q Z 5) P W FGHIJK is regular X is the midpoint of KH F 6) DCEA is an isosceles Trapezoid C D K B G A B J X E A H I E 4.4 and 4.5 Proofs 4.6 Corresponding Parts of Congruent Triangles Honors Geometry CPCTC – If two triangles are congruent, then _____________________________________ ______________________________________________________________________ . Determine whether the triangles are congruent, then state which conjecture (SSS, SAS, ASA, SAA) proves them congruent. 1) m1 = m2 m3 = m4 2) ABDC mA = mC 3) O is the midpoint of RQ O is the midpoint of PS R A Q 1 2 1 4 R B 3 T 4 O 3 2 S Is QR = QS? Why? S C D Is AD = BC? Why? Given: mA = mB AM = BM Is mP = mS Why? B C 2 1 M Prove: AD = BC A P D Q E Given: AR = ER EC = AC Prove: mE = mA R C A A Given: C is the midpoint of BD AB DE Prove: AB = DE B C D E 4.7 Flow Chart Thinking Honors Geometry Prove: AD = BC B C Given: mA = mB AM = BM 2 1 M D A E Given: AR = ER EC = AC Prove: mE = mA R C A A Given: C is the midpoint of BD AB DE Prove: AB = DE B C D E Congruence in Right Triangles Honors Geometry Hypotenuse-Leg (HL) Conjecture – 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. Ex 2: Using the HL Conjecture 1) 1) 2) 2) 3) 3) 4) 4) 5) 5) 6) 6) V R Ex 3: Using the HL Conjecture Given: RV RU and UT RU ; S is midpoint of TV; RV = TU S Prove: mV mT 1) 1) 2) 2) 3) 3) 4) 4) 5) 5) 6) 6) 7) 7) T U Additional Ex: For what values x and y are the triangles congruent by HL? x+y 5x + 1 2x + y +3 2y - 4 What additional information you need to prove the triangles congruent with HL? 1. 2. 4.8 Proving Special Triangle Conjectures Honors Geometry (C-27) Vertex Angle Bisector Conjecture - In an isosceles triangle, the bisector of the vertex angle is also _______________________ and _______________________________ . Given: ABC is isosceles AC BC CD is the bisector of C Prove: ADC BDC Given: ABC is isosceles AC BC CD is the bisector of C Prove: CD is an altitude (C-28)Equilateral/Equiangular Triangle Conjecture - Every equilateral triangle is __________________, and, conversely, every equiangular triangle is _________________________ . This conjecture is a biconditional conjecture: Both the statement and its converse are true. A triangle is equilateral if and only if it is equiangular.