[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-448586-105":59,"doc-detail-448586-en":131},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":124,"head_meta":126,"extra_data":128,"updated_unix":130},105,"en","mr-schlansky-geometry-proofs-regents-review","Mr. Schlansky Geometry - Proofs Regents Review","","Study notes for Geometry Regents proof strategies focused on triangle congruence and similarity. The document lists core congruence tools such as CPCTC, using three pairs of sides/angles, and common starting mini-proofs involving altitude, median, and angle/segment bisectors. It also covers when parallel lines imply congruent corresponding, alternate interior, or alternate exterior angles, plus vertical angles, reflexive property, perpendicular lines, and perpendicular bisectors. Additional sections address Euclidean proofs using parallelogram and circle theorems, and similarity via AA and proportional side reasoning.",{"@graph":69,"@context":123},[70,84,106],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":20,"@type":76,"position":81},"https://docshare.wps.com/document/exam/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/mr-schlansky-geometry-proofs-regents-review/448586/",{"url":83,"name":65,"@type":85,"image":86,"author":91,"headline":65,"publisher":94,"fileFormat":97,"inLanguage":63,"description":67,"dateModified":98,"datePublished":99,"encodingFormat":97,"isAccessibleForFree":100,"interactionStatistic":101},"DigitalDocument",{"url":87,"@type":88,"width":89,"height":90},"https://docshare.wps.com/thumbnails/mr-schlansky-geometry-proofs-regents-review/448586.png","ImageObject",300,407,{"name":92,"@type":93},"Đào","Person",{"url":74,"name":95,"@type":96},"DocShare","Organization","application/pdf","2026-10-22","2026-09-30",true,{"@type":102,"interactionType":103,"userInteractionCount":105},"InteractionCounter",{"@type":104},"ViewAction",12,{"@type":107,"mainEntity":108},"FAQPage",[109,115,119],{"name":110,"@type":111,"acceptedAnswer":112},"What is CPCTC and when should it be used in congruence proofs?","Question",{"text":113,"@type":114},"CPCTC means corresponding parts of congruent triangles are congruent. Use it after you have proven the triangles congruent so you can transfer side/angle equalities to the parts you need.","Answer",{"name":116,"@type":111,"acceptedAnswer":117},"How do you prove triangles are congruent in these notes?",{"text":118,"@type":114},"The notes instruct proving three pairs of sides/angles are congruent, then using CPCTC to conclude additional congruences within the triangles.",{"name":120,"@type":111,"acceptedAnswer":121},"What methods are used to prove triangles are similar?",{"text":122,"@type":114},"The notes state using AA similarity by proving two angles congruent, then using proportional corresponding sides (CSSTIP) and, when needed, the equal cross products method for proportion-style questions.","https://schema.org",{"og:url":83,"og:type":125,"og:title":65,"og:site_name":95,"og:description":67},"article",{"robots":127,"canonical":83},"index,follow",{"doc_id":129,"site_id":62},448586,1790993256,{"code":4,"msg":5,"data":132},{"doc_id":129,"user_id":133,"nickname":92,"user_avatar":134,"doc_module":4,"category_id":19,"category_name":20,"doc_title":65,"doc_description":67,"doc_content":135,"file_id":136,"file_url":137,"file_type":138,"file_size":139,"view_count":105,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":46,"language":140,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":67,"update_tm":144,"read_time":31},1374402968488,"https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0","Name   Date    \nMr. Schlansky Geometry  \nProofs Regents Review  \nEuclidean Proofs (Basic)  \nIf it is not specified, prove triangles are congruent  \nTo prove triangles are congruent, prove 3 pairs of sides/angles are congruent  \nTo prove segments or angles, use CPCTC  \n*If you get stuck, make something up and keep on going!  \n1) Do a mini proof with your givens Altitude creates congruent right angles Median creates congruent segments Line bisector creates congruent segments Midpoint creates congruent segments Angle bisector creates congruent angles  \nPerpendicular lines create congruent right angles  \nWhen given parallel lines:  \nCorresponding angles are congruent OR Alternate interior angles are congruent OR Alternate exterior angles are congruent  \n2) Use additional tools:  \nVertical Angles are congruent (Look for an X)  \nReflexive Property (A side/angle is congruent to itself)  \n1. Given: BD bisects 􀂑 ADC  \nAD 􀀣  DC  \nProve: AB 􀀣 BC  \nD  \nA C  \n2. Given: LN bisects 􀂑 KLM  \n􀂑 LKM 􀀣  􀂑 LMK Prove: NM 􀀣 NK  \nL  \nM N K  \n3. Given: HN 􀁁 KA , KN 􀀣 AN Prove: 􀂑HAN 􀀣 􀂑HKN  \nH  \nK  \nA N  \n4. Given:  NO and HA bisect each other Prove: NA 􀀣 HO  \nN  \n.  \nA  \nH  \nO  \n5. Given: QR is the perpendicular bisector of NP Prove: 􀂑NQR 􀀣 􀂑PQR  \nN  \nR  \nP  \n6. Given: IE || RN , TR 􀁁 RN , HS 􀁁 IE , EH 􀀣 AT Prove: SH 􀀣 RT  \nEuclidean Similar Triangle Proofs To prove triangles are SIMILAR, prove AA 􀀣 AA If asked to prove a proportion/multiplication:  \n1) Prove triangles are similar  \n2) Corresponding Sides of Similar Triangle are In Proportion (CSSTIP)  \n3) Cross Products are Equal Work Backwards!  \n1. Given AB || DC Prove: DABE ~ DCDE  \n2. Given: BC 􀁁 AC DE 􀁁 AB  \nProve: DABC ~ DADE  \n3. Given: 􀂑HCE 􀀣 􀂑LIE  \nCE EI  \nProve: ~~   ~~ 􀀠 ~~    ~~CH IL  \nI  \nH  \nC  \n4. Given: UO 􀁁 RG , UR 􀁁 EG  \nUS EU  \nProve: ~~   ~~ 􀀠 ~~  ~~SO OG  \nU  \nR O G  \n5. Given: CA bisects 􀂑 CAD , 􀂑ABC 􀀣 􀂑ADE  \nProve: BC 􀁸 AE 􀀠 DE 􀁸 AC  \n6. Given: AB 􀀣 AC , AF 􀁁 BC , PD 􀁁 AB  \nProve: FC 􀁸 PB 􀀠 DB 􀁸 AC  \n7. In  shown below, points T and O are on  and , respectively. Segment OT is drawn  \nso that  .  \nProve: SC 􀁸 OU 􀀠 OT 􀁸 SU  \nEuclidean Proofs with Parallelogram and Circle Theorems  \n\n| Parallelogram Theorems | Circle Theorems |\n| --- | --- |\n| A parallelogram/rectangle/rhombus/square has: Two pairs of opposite sides congruent\u003Cbr>Two pairs of opposite sides parallel\u003Cbr>Diagonals that bisect each other\u003Cbr>Opposite angles congruent | All radii/diameters of a circle are congruent Angles inscribed to the same arc are congruent An angle inscribed to a semicircle is a right angle\u003Cbr>A tangent and a radius/diameter form a right angles\u003Cbr>Congruent arcs have congruent chords have congruent central angles\u003Cbr>Parallel Lines intercept congruent arcs Tangents drawn from the same point are congruent |\n| A rectangle/square has:\u003Cbr>A right angle\u003Cbr>Congruent diagonals |  |\n| A rhombus/square has:\u003Cbr>All sides congruent\u003Cbr>Perpendicular diagonals\u003Cbr>Diagonals that bisect the angles |  |\n\n1. Given: ABCD is a parallelogram  \nProve: DAED 􀀣 DCEB  \nA  B   \nD C  \n2. Given: ABCD is a rectangle, M is the midpoint of AC  \nProve: DM 􀀣 BM  \nD B  \nA M C  \n3. Given: ABCD is a rhombus, AE 􀀣 CE  \nProve: 􀂑ADE 􀀣 􀂑 CDE  \nB  \nE  \n4. Given: SPIN is a square Prove: DSNI 􀀣 DSPI  \nN I  \n5. Given: ABCD  is a square, FA 􀀣 AE  \nProve: BF 􀀣 DE  \nB C  \nF A DE  \n6. Given: Parallelogram ABCD, , and diagonal  Prove:   \n7. In parallelogram ABCD, and  are perpendicular  \nto diagonal  at points F and E.  \nProve:   \n8. Given: Chords AD and BC of circle O intersect at E, AB 􀀣 CD Prove: BC 􀀣 AD  \n9. Given: Circle O with diameters MOT and AOH . Prove: MA 􀀣 HT  \n10. In circle Y, tangent LE is drawn to diameter TYL  \n~~ ~~ ~~ ~~ TE TL  \nand YR 􀁁 TE . Prove that ~~   ~~ 􀀠 ~~   ~~ .  \nTY TR  \n11. In the diagram below, secant and tangent are drawn from external point A to circle O.  \nProve the theorem: If a secant and a tangent are drawn to a circle from an external point, the product of the lengths of the secant segment and its external s","cbCaiu0RJatvY5EM","https://ap.wps.com/l/cbCaiu0RJatvY5EM","pdf",1199679,"English","# Congruent Triangle Proofs (Regents Review)\n## Core Rules (CPCTC, three pairs)\n## Mini-Proof Templates (altitude, median, bisectors)\n## Parallel Lines and Extra Tools\n## Worked Congruence Problems\n# Similar Triangle Proofs\n## AA Similarity and Proportion Reasoning\n## Cross Products Equal\n## Worked Similarity Problems\n# Parallelogram and Circle Theorems\n## Parallelogram Theorems\n## Circle Theorems\n## Worked Theorems and Proofs\n# Triangle Proofs with Additional Tools\n## Vertical Angles, Reflexive Property, Isosceles Triangles\n## Addition/Subtraction Property\n## Worked Problems","[{\"question\":\"What is CPCTC and when should it be used in congruence proofs?\",\"answer\":\"CPCTC means corresponding parts of congruent triangles are congruent. Use it after you have proven the triangles congruent so you can transfer side/angle equalities to the parts you need.\"},{\"question\":\"How do you prove triangles are congruent in these notes?\",\"answer\":\"The notes instruct proving three pairs of sides/angles are congruent, then using CPCTC to conclude additional congruences within the triangles.\"},{\"question\":\"What methods are used to prove triangles are similar?\",\"answer\":\"The notes state using AA similarity by proving two angles congruent, then using proportional corresponding sides (CSSTIP) and, when needed, the equal cross products method for proportion-style questions.\"}]","Mr. Schlansky Geometry - Proofs Regents Review | PDF",1790727938]