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Congruent Triangles. Helpful websites. http://www.regentsprep.org/Regents/mathb/1c/preprooftriangles.htm http://www.mathwarehouse.com/geometry/congruent_triangles/ http://www.cliffsnotes.com/WileyCDA/CliffsReviewTopic/Congruent-Triangles.topicArticleId-18851,articleId-18788.html.
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Helpful websites • http://www.regentsprep.org/Regents/mathb/1c/preprooftriangles.htm • http://www.mathwarehouse.com/geometry/congruent_triangles/ • http://www.cliffsnotes.com/WileyCDA/CliffsReviewTopic/Congruent-Triangles.topicArticleId-18851,articleId-18788.html
4-1 Congruent Figures Objective: To recognize congruent figures and their corresponding parts
Vocabulary/ Key Concept • Congruent polygons- two polygons are congruent if their corresponding sides and angles are congruent • If two angles in a triangle are congruent to two angles in another triangle, then the third angles are congruent.
Each pair of polygons are congruent. Find the measure of each numbered angle • M1 = 110 • m 2 = 120 • M3 = 90 • m 4 = 135
WXYZ JKLM. List 4 pairs of congruent sides and angles. • WX JK • XY KL • YZ LM • ZW MJ • W J • K X • Y L • Z M
Determine if the polygons are congruent. Explain. 1.G I , GJH IJH, 2.GHJ IHJ 3. JI GJ, GH IH, 4. JH JH GHJ and IHJ are congruent because all corresponding angle and sides are congruent 1.Given –congruent markings 2.If two angles in a triangle are congruent to two angles in another triangle, then the third angles are congruent. 3.Given-congruent markings 4.Reflexive
Determine if the polygons are congruent. Explain. 1.Q T 2.QSR TSV 3.R V No, the triangles do no have corresponding sides that are congruent 1.Given – congruent measures 2.Vertical angles 3. If two angles in a triangle are congruent to two angles in another triangle, then the third angles are congruent.
PROOF Statement 1. LM QP LN QN MN PN L Q 2. LNM QNP 3. P M 4. LMN QPN Reasons • GIVEN • VEERITCAL ANGLES • If two angles in a triangle are congruent to two angles in another triangle, then the third angles are congruent • 4. Def. of congruent polygons
4-2 Congruent Triangles by SSS and SAS Objective: To prove two triangles are congruent using SSS and SAS Postulates
Key Concepts • SSS – Side-side-side corresponding congruence. • SAS – Side-Angle-Side corresponding Congruence. ANGLE MUST BE IN BETWEEN THE TWO SIDE
Determine if you can use SSS or SAS to prove two triangles are congruent. Write the congruence statement. AB CB --CONGRUENCE MARKING BD BD -- REFLEXIVE ABD CBD–CONGRUENCE MARKING ABD CBD by SAS
Determine if you can use SSS or SAS to prove two triangles are congruent. Write the congruence statement. What do you know? NP = QP -- CONGRUENT MARKS NR = QR -- CONGRUENT MARKS RP = RP -- REFLEXIVE PROPERTY OF PRN PRQ by SSS
Determine if you can use SSS or SAS to prove two triangles are congruent. Write the congruence statement. EM PQ EN PR CONGRUENCE MARKING N R Not possible
Proof Statements • AB DC, BAC DCA • AC CA • ABC CDA Reasons • Given • Reflexive property of • SAS A B D C
Try: Write a proof. Statement • EF GF, HF DF • DFE HFG • DFE HFG Reasons • Given • Vertical angles • SAS
4-3 Congruent Triangles by AAS and ASA Objective: To prove two triangles are congruent using AAS and ASA Postulates
Key Concepts • ASA – Two angles and an included side. • AAS – Two angles and a non-included side. SIDE IS IN BETWEEN THE ANGLES
Determine if you can use ASA or AAS to prove two triangles are congruent. Write the congruence statement. TUV VST by ASA not possible BCEDCF by AAS
Write a Proof Statement • KL ML, K M • JLK PLM • JKL PML Reasons • Given • Vertical angles • ASA
What did you learn today? • What are some simple steps to writing proofs? • What are the four ways to prove triangles are congruent?