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Geometry Notes TC – 1: Side - Angle - Side Congruent Polygons Review: Two polygons are congruent if Also, two polygons are congruent if (and only if) 1. 2. Ex: If ABC PQR, then a. All pairs or corresponding parts are congruent b. There is a rigid motion for which the image of ABC is PQR. P C R B A Q Problem: Saying two figures are congruent if one is the image of the other under a rigid motion is a good definition of congruence. But it is not always a convenient method to prove two figures are congruent. Ex: Are the triangles below congruent? If they are, then the transformation B C will map ABC onto DEF. F D A E C F But how can we be sure that the triangles actually map perfectly one onto the other? A D E B Proving Two Triangles Congruent If all three pairs of corresponding sides are congruent and all three pairs of corresponding angles are congruent, then two triangles must be congruent. Is it possible to prove two triangles congruent without proving all six pairs of corresponding parts congruent? If so, what is the least number of congruent pairs of corresponding parts we need? One pair of sides? One pair of angles? Two pairs of sides? Two pairs of angles? One pair of each sides, angles? Side-Angle-Side If two sides and the included angle of one triangle are all congruent to the corresponding sides and angle of a second triangle, then the two triangles are congruent. Given: ABC and A'B'C' AB A ' B ' , AC A ' C ' , and A A' B' B Note: A is called the included angle for sides AB and AC because it is the angle formed by those two sides (where those two sides meet). C' A' C Show via rigid motions that A'B'C' ABC. A B' C''' B C' B" A' C" C A Ex: Given: AB CD , AB || CD B A Prove: ABC CDA D C Geometry Notes TC – 2: Angle - Side - Angle Proving Two Triangles Congruent, Continued Angle-Side-Angle Theorem: If two angles and the included side of one triangle are all congruent to the corresponding angles and side of a second triangle, then the two triangles are congruent. Given: ABC and A'B'C' A A', A A'¸ and AB A ' B ' Note: AB is called the included side for A and B because it is the side joining the vertices of those two angles. B' B C' A' C Show via rigid motions that A'B'C' ABC. A B' C''' B C' B" A' C" C A Ex: Given: AC bisects BD at E, AB BD , CD BD Prove: C a. ABE CDE b. A C B A E D Ex: Given: CD bisects ACB, CD is an altitude of ABC, C Prove: AC BC A D B Geometry Notes TC – 3: Side - Side - Side Proving Two Triangles Congruent, Continued Side-Side-Side Theorem: If all three sides of one triangle are congruent to the corresponding sides of a second triangle, then the triangles are congruent. B' Given: ABC and A'B'C' AB A ' B ' , BC B ' C ' ¸ and CA C ' A ' Show via rigid motions that A'B'C' ABC. A' C' C' B C A B' A' C' C' B C''' C B" C" C' A Ex: Given: Isosceles triangle ABC with vertex angle C, CD is a median of the triangle. Prove: A B Note: Most proofs come with a diagram. If one doesn’t, try to draw a diagram that fits the givens. Geometry Notes TC – 4: Practice A Given: AB || CD , DPQB , BQ DP , B AP DB , CQ BD Q Prove: AP CQ P a. Tell what’s wrong with the following proof. Statement 1. AB || CD (S) Reason 1. Given 2. AP DB , CQ BD (A) 2. Given 3. BQ DP (S) 3. Given 4. AP CQ 4. SAS (1, 2, 3) b. Write a correct proof. Statement Reason D C Geometry Notes TC – 5: Isosceles Triangle Theorem Angle Bisectors (review) Definition: An angle bisector Postulate: Every angle has Isosceles Triangle Theorem We wish to prove: If two sides of a triangle are congruent, the angles opposite those sides are also congruent. Complete the Given and Prove below and draw a suitable diagram. Given: Prove: The plan is to bisect the vertex angle of the given triangle and then prove the two new triangles are congruent. Fill in missing statements and reasons (including the blanks in statement 2) to complete the proof below. Statement Reason 1. Given 1. 2. Let point be on base such that 2. is the angle bisector of vertex angle 3. 3. 4. 4. 5. 5. 6. 6. Ex: Draw a suitable diagram, then give an appropriate conclusion and reason: Given: In RAT, RA AT Conclusion: Reason: Ex: CAT, shown at right, has perimeter 52. A a. Find the value of x. 2y + 3z 2 x 5x z 5y + 30 C T 2x + 8 b. Find the values of y and z. The converse of the Isosceles Triangle Theorem is also true. (Proof is on a later HW). Write the converse: Ex: Determine which two sides of HUG are congruent. U 2x + 42 H 3x + 12 5x + 6 G Geometry Notes TC – 6: Review/Practice To prove two triangles congruent: To justify corresponding parts congruent: 1. 2. 3. 4. Coming soon to a theater near you. Don’t miss them! D 5. Ex: Given: DIN with DI DN , DI IM , DN NM , IAON , IO AN Prove: AM OM I A O M N Geometry Notes TC – 7: Angle - Angle - Side Proving Two Triangles Congruent, Continued Angle-Angle-Side Theorem: If two angles and a non-included side of one triangle are congruent to the corresponding angles and side of a second triangle, then the triangles are congruent. Given: ABC and DEF A D, C F, and AB DE Prove: ABC DEF. A B Ex: Give a reason why each pair of triangles is congruent. F C a. A B D E F C b. A F C B D E D E Ex: Given: Pentagon TOPAZ with diagonals TP and ZP , TZP is isosceles with vertex P, OTZ TZA, O OPA, A OPA T B Z A O Prove: OP AP P Geometry Notes TC – 8: Hypotenuse - Leg Proving Two Triangles Congruent, Continued Hypotenuse-Leg If the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the two triangles are congruent. Given: ABC with right angle C, DEF with right angle F, AB DE and BC EF E B Prove: ABC DEF C F E D Prove: AE || CF B A Ex: Given: AB DC , BFED , AB AE , CD CF , BF DE A C F D Geometry Notes TC – 10: Indirect Proofs C Ex: Theorem: If a triangle is isosceles, the altitude from the vertex angle bisects the base (i.e., the altitude from the vertex is also a median). Given: Isosceles triangle ABC with vertex C; CD is an altitude. Prove: CD bisects AB A D B This is called a direct proof. It starts with the givens and then uses deductive reasoning to lead to what is to be proved. Prove the inverse of the theorem above: C Given: Scalene ABC; CD is an altitude. Prove: CD does not bisect AB A D B Indirect proof: Start by assuming the negation (opposite) of what you want to prove. C Given: Triangle ABC is not isosceles; CD is an altitude. Prove: CD does not bisect AB Assume: A Ex: Given: Line AB and point P not on AB . D B .P Prove: There is only one line through P perpendicular to AB . l