Math-2A. Lesson 8-3 Triangle Congruence
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1 Math-2A Lesson 8-3 Triangle Congruence
2 Naming Triangles Triangles are named using a small triangle symbol and the three vertices of the triangles. The order of the vertices does not matter for NAMING a triangle A Examples B C ΔABC ΔACB ΔBAC ΔBCA ΔCAB ΔCBA
3 Corresponding Angles of Triangles: an angle in one triangle that has the same position (relative to its sides) as an angle in another triangle (relative to its sides). A corresponds to D since they are opposite the longest side of their triangles
4 Corresponding Sides of Triangles: a side in one triangle that has the same position (relative to its angles) as a side in another triangle (relative to its angles). BC corresponds to EF since they are opposite the largest angle of their triangles
5 Your Turn: 1) What angle does A correspond to? 2) What angle does X correspond to? 3) What side does XY correspond to? 4) What side does AC correspond to? Z C CB ZX
6 Vocabulary Included side: If two angles in a triangle are given, the included side is the side that is between the two angles or side that both of the angles have in common. RS is the included side of R and S What is the included Side for S and T? ST is the included side of S and T
7 Vocabulary Included angle: If two sides of a triangle are given, the included angle is the angle formed by those two sides. T is the included angle of RT and TS What is the included angle of SR and RT? R is the included angle of SR and RT
8 Congruence Angles two angles are congruent if they have the same measure (degrees) IF A B then m A = m B (converse is true also) Segments (sides) two line segments are congruent if they have the same length If AB CD then AB = CD (converse is true also)
9 Congruence Triangles two triangles are congruent If each angle in one triangle is congruent to its corresponding angle in the other triangle AND if each side in one triangle is congruent to its corresponding side in the other triangle.
10 Congruence Statements When stating congruence, the order is important The vertices of must be put in order so that the vertices in one triangle correspond to the vertexes in the other triangle. If Corresponding parts are congruent, then the triangles are congruent. For example: ΔABC ΔZYX because A Z B Y C X AB ZY BC YX CA XZ corresponding parts of congruent triangles are congruent or CPCTC
11 Your Turn Each pair of triangles is congruent: Write a congruence statement for each pair of triangle. ΔDEF ΔPRQ ΔRST ΔGFH
12 Triangle Congruence Why are ΔRST and ΔZYX congruent? (That is, how do we prove it?) All corresponding parts are congruent (CPCTC) This is just the definition of congruence Do we need all 6 pairs of angles and sides to be congruent to prove the triangles are congruent?...
13 D Your Turn 1) D is the included angle of which two sides? DF and DE 2) What is the included angle of sides DF and EF? F E F 3) DF is the included side of which two angles? D and F 4) What is the included side of D and E DE
14 We can prove Triangle Congruence using congruence of only three pairs of corresponding parts. Side-Side-Side (SSS) Congruency Axiom: if all three pairs of corresponding sides of a triangle are congruent, then the triangles are congruent AB DE BC EF CA FD Therefore, ΔABC ΔDEF by SSS
15 Angle-Side-Angle (ASA) Congruency Axiom: if two angles and their included side are congruent, then the two triangles are congruent. ABC DEG BC EG BCA EGD Therefore, ΔABC ΔDEG by ASA
16 Side-Angle-Side (SAS) Congruency Axiom: if two pairs of corresponding sides and the pair of included angles are congruent, then the triangles are congruent. XZ QR ZXY RQF XY QF Therefore, ΔXYZ ΔQFR by SAS
17 Angle-Angle-Side (AAS) Congruency Axiom: If two pairs of corresponding angles are congruent and one pair of corresponding sides are congruent (which are NOT the included side), then the two triangles are congruent. ZXY EFD XYZ FDE XZ FE Therefore, ΔXYZ ΔFDE by AAS
18 Your Turn Determine which congruence condition proves the congruence for each the following pairs of triangles. Write a congruence statement for each of the following pairs of triangles. 1) 2)
19 Your Turn Determine which congruence condition proves the congruence for each the following pairs of triangles. Write a congruence statement for each of the following pairs of triangles. 3) 4)
20 Your Turn You may have notice a pattern of needing 3 parts of a triangle. What other 3-part groups are we missing? AAA ASS (usually referred to by the more appropriate SSA)
21 Angle-Angle-Angle (AAA) Condition AAA controls shape only, not size
22 Angle-Side-Side (ASS) Condition Let s look at an example of ASS A D AB DE BC EF
23 Symbols that show congruence (without giving measures) AB FG
24 Symbols that show congruence (without giving measures) AB FG AC EF Write a congruence statement that identifies the additional information needed to prove these two triangles are congruent by ASA. A F Write a triangle congruence statement: ΔABC ΔFGE
25 Are the triangles congruent? If so, write a congruence statement for the two triangles, and identify why are they congruent? ΔABC ΔFGE by AAS
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