Math-Essentials. Lesson 6-2. Triangle Congruence

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1 Math-Essentials Lesson 6-2 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 Examples A B C ΔABC ΔACB ΔBAC ΔBCA ΔCAB ΔCBA X Y ΔXYZ Z

3 Your Turn Give all six names of the triangles 1) Q ΔPQR ΔPRQ ΔRPQ ΔRQP ΔQPR 2) P R ΔQRP D E F ΔDEF ΔDFE ΔEDF ΔEFD ΔFDE ΔFED

Correspondence In triangles, there are two types of correspondence that we consider: Corresponding angles Corresponding sides Corresponding Angles of

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

4 Correspondence In triangles, there are two types of correspondence that we consider: Corresponding angles Corresponding sides 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). 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). A corresponds to D since they are opposite the longest side of their triangles 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 Congruence Angles two angles are congruent if they have the same measure (degrees) A B if m A = m B Segments (sides) two line segments are congruent if they have the same length AB CD if AB = CD 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 polygon. In short we say corresponding parts of congruent triangles are congruent or CPCTC

7 Congruence Statements When naming a triangle, the order of the vertices is not important But when stating congruence, the order is important The vertices of must be put in order so that the corresponding parts in the names of the triangles match the corresponding parts in the triangles themselves For Example, ΔABC ΔZYX because A corresponds to Z B corresponds to Y C corresponds to X AB corresponds to ZY BC corresponds to YX CA corresponds to XZ

8 Your Turn Express the congruence of the following pairs of triangles 1) 2) ΔDEF ΔPRQ ΔRST ΔGFH

9 Congruence Conditions Why are ΔRST and ΔZYX congruent? (That is, how do we prove it?) The corresponding parts are congruent (CPCTC) This is just the definition of congruence Are there other ways we can know triangles are congruent?...

Background Vocab 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.

10 Background Vocab 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 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

11 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 3) DF is the included side of which two angles? D and F 4) What is the included side of D and E DE E F

More Congruence Conditions Side-Angle-Side (SAS) Congruence Axiom: if an angle in one triangle is congruent to an angle in another triangle and if the sides of the angle

12 More Congruence Conditions Side-Angle-Side (SAS) Congruence Axiom: if an angle in one triangle is congruent to an angle in another triangle and if the sides of the angle in the first triangle are congruent to the sides of the angle in the other triangle, then the two triangles are congruent. XZ QR ZXY RQF XY QF Therefore, ΔXYZ ΔQFR by SAS

13 More Congruence Conditions Angle-Side-Angle (ASA) Congruence Theorem: 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

14 More Congruence Conditions Side-Side-Side (SSS) Congruence Theorem: if all corresponding sides of a triangle are congruent, then the triangles are congruent AB DE BC EF CA FD Therefore, ΔABC ΔDEF by SSS

15 More Congruence Conditions Angle-Angle-Side (AAS) Congruency Theorem: If two corresponding angles are congruent between two triangles and a 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

16 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) 3) 4)

17 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)

18 Angle-Angle-Angle (AAA) Condition Let s look at an example of AAA A A ABC ADE (why?) ACB AED (why?) Does this mean that ΔABC ΔADE? How do you know?

19 Angle-Side-Side (ASS) Condition Let s look at an example of ASS A D AB DE BC EF Does this mean that ΔABC ΔDEF? How do you know?

20 Congruence Conditions Conditions that work (axioms) SSS ASA SAS AAS Conditions that do not work AAA ASS

Math-2. Lesson 5-2. Triangle Congruence

Math-2. Lesson 5-2. Triangle Congruence Math-2 Lesson 5-2 Triangle Congruence 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