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Aim: Are there any shortcuts to prove triangles are congruent?

Aim: Are there any shortcuts to prove triangles are congruent?. Do Now:. In triangle ABC, the measure of angle B is twice the measure of angle A and an exterior angle at vertex C measures 120 o . Find the measure of angle A. S. T. W. V. U. Congruence.

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Aim: Are there any shortcuts to prove triangles are congruent?

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  1. Aim: Are there any shortcuts to prove triangles are congruent? Do Now: In triangle ABC, the measure of angle B is twice the measure of angle A and an exterior angle at vertex C measures 120o. Find the measure of angle A.

  2. S T W V U Congruence Is ABCDE the exact same size and shape as STUVW? A B E D 5 sides 5 angles C How would you prove that it is? Measure to compare. Measure what? If the 5 side pairs and 5 angle pairs measure the same, then the two polygons are exactly the same.

  3. S T W V           AB BC CD DE EA U A B C D E Corresponding Parts Corresponding Parts – pairs of segments or angles that are in similar positions in two or more polygons. A IF B CORRESPONDING PARTS E D ARE CONGUENT THEN THE POLYGONS ARE CONGRUENT C S T U V W ST TU UV VW WS

  4. Congruence Definitions & Postulates Two polygons are congruent if and only if 1. corresponding angles are . 2. corresponding sides are . Corresponding parts of congruent polygons are congruent. CPCPC True for all polygons, triangles our focus. Corresponding Parts of Congruent Triangles are Congruent. CPCTC

  5. Model Problem Hexagon ABCDEF  hexagon STUVWX. Find the value of the variables? AB and ST are corresponding sides x = 10 F & X are corresponding ’s x = 1200 ED and WV are corresponding sides 2y = 8 y = 4

  6. Corresponding Parts. Is ABC the exact same size and shape as GHI? How would you prove that it is? Measure corresponding sides and angles. What are the corresponding sides? angles?

  7. Side-Angle-Side I.SAS = SAS Two triangles are congruent if the two sides of one triangle and the included angle are equal in measure to the two sides and the included angle of the other triangle. S represents a side of the triangle and A represents an angle. A A’ B C B’ C’ If CA = C'A', A =A', BA = B'A', then DABC = DA'B'C' IfSAS  SAS, then the triangles are congruent

  8. Model Problem Each pair of triangles has a pair of congruent angles. What pairs of sides must be congruent to satisfy the SAS postulate?

  9. Model Problem Each pair of triangles is congruent by SAS. List the given congruent angles and sides for each pair of triangles.

  10. Aim: Are there any shortcuts to prove triangles are congruent? Do Now: Is the given information sufficient to prove congruent triangles? SAS = SAS Two triangles are congruent if the two sides of one triangle and the included angle are equal in measure to the two sides and the included angle of the other triangle.

  11. D C A B Side-Angle-Side Is the given information sufficient to prove congruent triangles?

  12. Side-Angle-Side Given that C is the midpoint of AD and AD bisects BE, prove that DABC  DCDA. B D C A E • C is the midpoint of AD means that CA  CD. (S  S) • BCA  DCE because vertical angles are congruent. (A  A) • AD bisects BE means that BE is cut in to congruent segments resulting in BC  CE. (S  S) The two triangles are congruent because of SAS  SAS

  13. Side-Angle-Side In ABC, AC  BC and CD bisects ACB. Explain how ACD  BCD C A B D

  14. Side-Angle-Side In ABC is isosceles. CD is a median. Explain why ADC  BDC. C A B D

  15. A’ B’ C’  A’ B’ C’ ABC  A’B’C’ Sketch 12 – Shortcut #1 A B C Copied 2 sides and included angle: AB  A’B’, BC  B’C’, B  B’ Measurements showed: Shortcut for proving congruence in triangles: SAS  SAS

  16. The Product Rule

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