Notes Page 1 3.1 Notes Wednesday, October 01, 2008 8:33 PM efinitions: 2. ongruent triangle: all pairs of corresponding parts are congruent. ongruent Polygons: all pairs of corresponding parts are congruent. Postulate: ny segment or angle is congruent to itself. (Reflexive Property)
Notes Page 2 3.2 notes Wednesday, October 01, 2008 8:43 PM our ways to prove triangles congruent SSS Postulate: If there exists a correspondence between the vertices of two triangles such that three sides of one triangle are congruent to the corresponding sides of the other, the two triangles are congruent. (SSS). 2. SS Postulate: If there exists a correspondence between the vertices of two triangles such that two sides and the included angle of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent. (SS) 3. S Postulate: If there exists a correspondence between the vertices of two triangles such that two angles and the included side of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent. (S)
Notes Page 3 4. S Postulate: If there exists a correspondence between the vertices of two triangles such that two angles and the adjacent side of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent. xample 1: Given : Pr ove : Given 2. 2. Reflexive property 3. 3. SS(1, 1, 2) xample 2: Given : 3 6 R KR PR KRO PRM Pr ove : KRM PRO 3 K 4 M O 5 6 P
Notes Page 4 xample 2: Given : 3 6 R KR PR KRO PRM Pr ove : KRM PRO 3 K 4 M O 5 6 P 3 6 KR PR KRO PRM Given 2. KM and PO are straight angles 2. ssumed from diagram 3. 3 is supp. to 4 3. ef. of supp. 's 5 is supp. to 6 4. 4 5 4. Supp. to 's 5. KRM PRO 5. Subtraction prop. 6. KRM PRO 6. S(4, 1, 5)
Notes Page 5 3.3 notes Monday, October 06, 2008 8:46 M PT: orresponding parts of congruent triangles are congruent. Introduction to circles: Point P is the center of the circle shown. y definition, every point of a circle is the same distance from the center. The center, however, is not an element of the circle; the circle consists only of the "rim". circle is named by its center: this circle is called circle P. Points,, and lie on circle P. P Theorem 19: ll radii of a circle are congruent. xample 1: Given : O T is comp. to MOT S is comp. to POS Pr ove : MO PO K T M O P R S O Given T is comp. to MOT S is comp. to POS 2. OT OS 2. Radii of a circle are 3. MOT POS 3. Vertical 's are
Notes Page 6 4. T S 4. omp. to 's MOT POS 5. S(3, 2, 4) 6. MO PO 6. PT
3.4 Notes Monday, October 06, 2008 9:17 M eyond PT Median: is a line segment drawn from any vertex of the triangle to the midpoint of the opposite side. ltitude: is a line segment drawn from any vertex of the triangle to the opposite side, extended if necessary, and perpendicular to that side. ase 1: cute triangle G ase 2: Right triangle G ase 3: obtuse triangle G J Notes Page 7
Notes Page 8 H G J Postulate: Two points determine a line, ray, or segment. Note: Many proofs involve lines, rays or segments that do not appear in the original figure. These additions to the diagram are called auxiliary lines. xample 1: Given : G is the midpoint of H H Pr ove : 1 2 1 H G 2 G is the midpoint of H Given H 2. HG G 2. y def. of midpoint 3. HG and G are straight 's 3. ssumed from diagram 4. G G 4. Reflexive property 5. GH G 5. SSS(1, 2, 4) 6. HG G 6. PT 7. HG is supp. to 1 7. ef. of supp. 's G is supp. to 2 8. 1 2 8. Supp. to 's
Notes Page 9 xample 2: Given : and are altitudes of Pr ove : and are altitudes of Given 2. and are right 's 2. y def. of altitude 3. 3. Right 's are 4. 4. Reflexive property 5. 5. S(3, 1, 4) 6. 6. PT 7. 7. Subtraction property
Notes Page 10 3.5 Notes Monday, October 06, 2008 10:16 M xample 1: Given : Pr ove : Given 2. 2. Subtraction prop. 3. 3. Reflexive prop. 4. 4. SS(1, 3, 2) 5. 5. PT Try this one!! xample 2: Given : NR NV N P and Q are midpoint s R V P Q PX QX X Pr ove : XS XT R S T V
Notes Page 11 NR NV P and Q are midpoint s R V PX QX Given 2. NX NX 2. Reflexive prop. 3. NP NQ and PR QV 3. ivision prop. 4. NPX NQX 4. SSS(1, 2, 3) 5. NPX NQX 5. PT 6. NPR and NQV are straight 's 6. ssumed from diagram 7. NPX is supp. to XPR 7. ef. of supp. 's NQX is supp. to XQV 8. XPR XQV 8. Supp. to 's 9. PRT QVS 9. S(1, 3, 8) 10. QS PT 10. PT 1 XS XT 1 Subtraction prop.
3.6 Notes Monday, October 06, 2008 12:29 PM efinitions: scalene triangle is a triangle in which no two sides are congruent. 2. n isosceles triangle is a triangle in which at least two sides are congruent. Segment and segment are called legs of the isosceles triangle. Segment is called the base of the isosceles triangle. and are called the base angles, and is called the vertex angle. 3. n equilateral triangle is a triangle in which all sides are congruent. Notes Page 12
Notes Page 13 4. 5. n equiangular triangle is a triangle in which all angles are congruent. n acute triangle is a triangle in which all angles are acute. 6. right triangle is a triangle in which one of the angles is a right angle. 7. n obtuse triangle is a triangle in which one of the angles is an obtuse angle. 110 raw and label a diagram. List, in terms of the diagram, what is
Notes Page 14 given and what is to be proved. Then write a two column proof. 2. In an isosceles triangle, if the vertex angle is bisected, then two congruent triangles are formed. In an isosceles triangle; if a segment is drawn from the vertex angle to the midpoint of the base, then congruent triangles are formed. Given: is isosceles with base 's and is bisected Prove: is isosceles base 's and is bisected Given 2. 2. y def. of bisector 3. 3. y def. isos. 4. 4. Reflexive prop.
Notes Page 15 5. SS(3, 2, 4) Given: is isosceles base 's and Point is a midpoint of segment Prove: is isosceles base 's and Point is a midpoint of segment Given 2. 2. y def. of midpoint 3. 3. y def. isos. 4. 4. Reflexive prop. 5. SSS(2, 3, 4)
Notes Page 16 3.7 Notes riday, October 10, 2008 12:17 PM Theorem 20: If two sides of a triangle are congruent, then the angles opposite the sides are congruent. Theorem 21: If two angles of a triangle are congruent, then the sides opposite the angles are congruent. Theorem: If two sides of a triangle are not congruent, then the angles opposite them are not congruent, and the larger angle is opposite the longer side. Theorem: If two angles of a triangle are not congruent, then the sides opposite them are not congruent, and the longer side is opposite the larger angle. Ways to prove that a triangle is isosceles. 2. If at least two sides of a triangle are congruent, then the triangle is isosceles. (by definition of isosceles triangle) If at least two angles of a triangle are congruent, then the triangle is isosceles. (base angles are congruent)
Notes Page 17 xample 1: given : 3 4 T X Y W Z Pr ove : WTZ is isosceles W 1 X 3 4 2 Y Z 3 4 X Y Given W Z 2. WT and ZT are straight 2. ssumed from the angles diagram 3. 3 is supp. to 1 4 is supp. to 2 3. y def. of supp. 's 4. 1 2 4. Supp. to 's 5. WX ZY 5. SS(1, 4, 1) 6. W Z 6. PT 7. WTZ is isos. 7. ase 's are
Notes Page 18 3.8 Notes Monday, October 13, 2008 7:19 M The HL Postulate: If there exists a correspondence between the vertices of two right triangles such that the hypotenuse and a leg of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent. xample 1: Given : O is an altitude of O OG O Pr ove : G G O is an altitude of O OG Given 2. O and OG are rt. 's 2. y def. of altitude 3. O OG 3. ll radii of a circle are 4. O O 4. Reflexive prop. 5. O OG 5. HL(2, 3, 4) 6. G 6. PT
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