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Math 1312 Section 3.1 Congruent Triangles To have a correspondence between two triangles, you must “match up” the angles and sides of one triangle with the angles and sides of the other triangle. Each corresponding angle and side must have the same measure. B L 23 90 J A 23 K 90 C The correspondence between the above two triangles can be stated as ABC JKL. The order in which the letters are written matters since it shows which angles and sides of one triangle match up with the angles and sides of the other triangle: If ABC JKL, the corresponding angles are: ABC JKL AND the corresponding segments are: ABC JKL ABC J K L ABC JKL AB JK BC KL AC JL The correspondence may be written in more than one way: CAB LJK is the same as ABC JKL. CONGRUENT TRIANGLES Each triangle has six parts: three sides and three angles. If the six parts of one triangle are congruent to the corresponding six parts of another triangle, then the triangles are congruent triangles. Two triangles are congruent if: 1. All pairs of corresponding angles are congruent. 2. All pairs of corresponding sides are congruent. Below, ABC and DEF are congruent, this means that the corresponding parts of each triangle are the same measures. The congruence is written ABC DEF. B E A D B E C F A AB DE BC EF CA FD F D C Definition of Congruent Triangles (CPCTC) - two triangles are congruent if and only if their corresponding parts are congruent. (CPCTC - corresponding parts of congruent triangles are congruent) Example 1: If ABC DEF, name the corresponding congruent angles and sides: E B C A F D Congruent Angles Congruent Sides - SSS Postulate - (side-side-side) if the three sides of one triangle are congruent to the three sides of a second triangle, then the triangles are congruent. E B Since all three sides in ABC are congruent to all three sides in DEF, then ABD DEF A C D F Example 2: Here is an example of SSS. If the lines AC = DC and AB = BD. Show that ABC DBC using SSS. C A B D SAS Postulate - if two sides and the “included” angle of one triangle are congruent to two sides and the “included” angle of another triangle, then the triangles are congruent. In the above definition, the “included” angle is the angle that is formed by the intersection of two sides of a triangle. For example, below B and E are the “included” angles. E B Since AB DE , and B E, and BC EF , then ABC DEF C A Example 3: Show that D F ABE DBC using SAS. C A B D E B is the midpoint of both AD and EC ASA Postulate - if two angles and the “included” side of one triangle are congruent to two angles and the “included” side of another triangle, the triangles are congruent. In the above definition, the “included” side is the segment that connects two angles. For example, AB and DE are the “included” sides . E B Since A D, and AB DE , and B E, then ABC DEF A C D F Example 4: Given the following information: P PN MQ : MPN NPQ PNM PNQ by ASA 1 2 M Q N AAS Theorem - if two angles and a “non-included” side of one triangle are congruent to the corresponding two angles and side of a second triangle, the two triangles are congruent. B E Since AB DE , and B E, and C F, then ABC DEF A C D F Example 5: Given the following information: A A D BC bisects both ACD and ABD B How can we show congruence using AAS? C D Note: You can not use AAA and SSA because they are not valid for proving triangles are congruent. A triangle can have equal angles can have the same shape but the triangles are not necessarily congruent. SUMMARY OF METHODS: METHOD QUALIFICATIONS Def. of All six parts of one triangle must be congruent with all six parts of the other triangle. SSS The three sides of one triangle must be congruent to the three sides of the other triangle. SAS Two sides and the included angle of one triangle must be congruent to two sides and the included angle of the other triangle. ASA Two angles and the included side of one triangle must be congruent to two angles and the included side of the other triangle. AAS Two angles and a non-included side of one triangle must be congruent to the corresponding two angles and non-included side of the other triangle. PICTURE Use for Popper 5 questions 1 and 2: a||b a b 2y 2 56° 7x Popper 5 question 1: Find the value for y. A. 56 B. 8 C. 29 D 58 E None of these Popper 5 question 2: Find the value for x. A. 56 B. 8 C. 29 D 58 E None of these Use for Popper 5 questions 3 and 4. ∠1 3 ∠2 12 1 29 2 16 Popper 5 questions 3: Find the value of x. A. 3 B. 5 C. 45 D. 9 E. None of these Popper 5 question 4: Find the value of angle 1. A. 135° B. 65° C. 45° D. 15° E. None of these Popper 5 question 5: If angle 1 = angle 2, angle 2 = angle 3, then angle 1 = angle 3. What property is being used? A. Distributive B. Reflexive C. Substitution D. Transitive Solution to Popper 3: Popper 3 question 1: ∆ is a right triangle with angle C is the right angle. If angle B = 2x +40 and angle A = x +20. Fine the measure of angle B. A. 30° B. 60° Popper 3 question 2: C. 40° D. 10° E. None of these Fine the mA = __________ for the following: 2x x (x-20) A A. 50° B. 100° C. 80° D. 30° E. None of These Popper 3 question 3: Given: m Find: ∠3 A. 45° B. 135° 1= x and m 3 = 3x C. 60° D None of these Popper 3 question 4 and question 5: B 80 Use the figure to the right. A Popper 3 question 4: Find measure for ∠ . A. 100° B. 131° C. 51° D None of these Popper 3 question 5: Find measure for ∠ A. 100° B. 131° C. 51° . D None of these x (3x – 22) C D Solutions for Popper 4: Popper 4 question 1: Given Nonagon, find the number of diagonals. A. 54 B. 9 C. 27 D. Need more information E. None of these Popper 4 questions 2: Find the sum of the interior angles of the eleven sided regular polygon. A. 1980° B. 1620° C. 1440° D. Need more information E. None of these Popper 4 questions 3: Find the measure of each interior angles of the given twelve sided regular polygon. A. 30° B. 150° C. 120° D. Need more information E. None of these Popper 4 question 4: Find the measure of each exterior angle of a regular octagon. A. 45° B. 135° C. 30° D. Need more information E. None of these. Popper 4 question 5: Each interior angle of a regular polygon is120 . Find the number of sides. A. 10 B. 5 C. 6 D. 7 E. None of these