Sec 2.6 Geometry – Triangle Proofs Name
Sec 2.6 Geometry – Triangle ProofsName:COMMON POTENTIAL REASONS FOR PROOFSDefinition of Congruence: Having the exact same size and shape and there by having the exact same measures.Definition of Midpoint: The point that divides a segment into two congruent segments.Definition of Angle Bisector: The ray that divides an angle into two congruent angles.Definition of Perpendicular Lines: Lines that intersect to form right angles or 90 Definition of Supplementary Angles: Any two angles that have a sum of 180 Definition of a Straight Line: An undefined term in geometry, a line is a straight path that has no thickness andextends forever. It also forms a straight angle which measures 180 Reflexive Property of Equality: any measure is equal to itself (a a)Reflexive Property of Congruence: any figure is congruent to itself (𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝐴𝐴 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 𝐴𝐴)Addition Property of Equality: if a b, then a c b cSubtraction Property of Equality: if a b, then a – c b – cMultiplication Property of Equality: if a b, then ac bcaDivision Property of Equality: if a b, then cbcTransitive Property: if a b & b c then a c OR if a b & b c then a c.Segment Addition Postulate: If point B is between Point A and C then AB BC ACAngle Addition Postulate: If point S is in the interior of PQR, then m PQS m SQR m PQRSide – Side – Side Postulate (SSS) : If three sides of one triangle are congruent to three sides of another triangle,then the triangles are congruent.Side – Angle – Side Postulate (SAS): If two sides and the included angle of one triangle are congruent to two sidesand the included angle of another triangle, then the triangles are congruent.Angle – Side – Angle Postulate (ASA): If two angles and the included side of one triangle are congruent to twoangles and the included side of another triangle, then the triangles are congruent.Angle – Angle – Side Postulate (AAS) : If two angles and the non-included side of one triangle are congruent totwo angles and the non-included side of another triangle, then the triangles are congruentHypotenuse – Leg Postulate (HL): If a hypotenuse and a leg of one right triangle are congruent to a hypotenuseand a leg of another right triangle, then the triangles are congruentRight Angle Theorem (R.A.T.): All right angles are congruent.Vertical Angle Theorem (V.A.T.): Vertical angles are congruent.Triangle Sum Theorem: The three angles of a triangle sum to 180 Linear Pair Theorem: If two angles form a linear pair then they are adjacent and are supplementary.Third Angle Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the thirdpair of angles are congruent.Alternate Interior Angle Theorem (and converse): Alternate interior angles are congruent if and only if thetransversal that passes through two lines that are parallel.Alternate Exterior Angle Theorem (and converse): Alternate exterior angles are congruent if and only if thetransversal that passes through two lines that are parallel.Corresponding Angle Theorem (and converse) : Corresponding angles are congruent if and only if thetransversal that passes through two lines that are parallel.Same-Side Interior Angles Theorem (and converse) : Same Side Interior Angles are supplementary if and only ifthe transversal that passes through two lines that are parallel.Pythagorean Theorem (and converse): A triangle is right triangle if and only if the given the length of the legs aand b and hypotenuse c have the relationship a2 b2 c2Isosceles Triangle Theorem (and converse): A triangle is isosceles if and only if its base angles are congruent.Triangle Mid-segment Theorem: A mid-segment of a triangle is parallel to a side of the triangle, and its length ishalf the length of that side.CPCTC: Corresponding Parts of Congruent Triangles are Congruent by definition of congruence.
1. Tell which of the following triangle provide enough information to show that they must be congruent.If they are congruent, state which theorem suggests they are congruent (SAS, ASA, SSS, AAS, HL)and write a congruence statement.Circle one of the following:SSSSASASA AASHLNot EnoughInformationSSSSASASA AASHLNot EnoughInformationSSSSASASA AASHLNot EnoughInformationSSSSASASA AASCongruence Statementif necessary:HLASA AASHLCircle one of the following:SSSSASASA AASHLSASNot EnoughInformationASA AASHLSASNot EnoughInformationNot EnoughInformationASA AASHLNot EnoughInformationCongruence Statementif necessary:M. WinkingSASASA AASHLNot EnoughInformationHLNot EnoughInformationHLNot EnoughInformationCircle one of the following:SSSSASASA AASCircle one of the following:SSSSASASA AASCongruence Statementif necessary:Circle one of the following:SSSSSSCongruence Statementif necessary:Circle one of the following:SSSCircle one of the following:Congruence Statementif necessary:Congruence Statementif necessary:Congruence Statementif necessary:Circle one of the following:SASCongruence Statementif necessary:Congruence Statementif necessary:Circle one of the following:SSSNot EnoughInformationCongruence Statementif necessary:Congruence Statementif necessary:Circle one of the following:Circle one of the following:Circle one of the following:SSSSASASA AASCongruence Statementif necessary:Unit 2-6page 49HLNot EnoughInformation
2. Prove which of the following triangles congruent if possible by filling in the missing blanks: and ⃖ ⃗a. Given 𝑪𝑪𝑪𝑪 𝑨𝑨𝑨𝑨𝑪𝑪𝑪𝑪 ⃖ ⃗𝑨𝑨𝑨𝑨StatementsReasons 𝐴𝐴𝐴𝐴 1. 𝐶𝐶𝐶𝐶⃖ ⃗ 𝐴𝐴𝐴𝐴⃖ ⃗2. 𝐶𝐶𝐶𝐶3. 𝐶𝐶𝐶𝐶𝐶𝐶 𝐴𝐴𝐴𝐴𝐴𝐴4. 𝐵𝐵𝐵𝐵 𝐵𝐵𝐵𝐵5. 𝐵𝐵𝐵𝐵𝐵𝐵 𝐷𝐷𝐷𝐷𝐷𝐷 𝒀𝒀𝒀𝒀 and Point A is theb. Given 𝑷𝑷𝑷𝑷midpoint of 𝑷𝑷𝑷𝑷Statements ⃖ ⃗ ⃖ ⃗c. Given 𝑽𝑽𝑽𝑽𝑹𝑹𝑹𝑹 and ven3.Definition of Midpoint4.Reflexive property ofcongruence5.By steps 1,3,4 and SSSStatements1.Given2.Given3.4.5. 𝑃𝑃𝑃𝑃𝑃𝑃 𝐸𝐸𝐸𝐸𝐸𝐸M. WinkingReasonsUnit 2-6page 50
Prove the Isosceles Triangle Theorem and the rest of the suggested proofs.d. Given 𝐖𝐖𝐖𝐖𝐖𝐖 is isosceles and point R isthe midpoint of 𝐖𝐖𝐖𝐖Statements1. 𝑊𝑊𝑊𝑊𝑊𝑊 isisoscelesReasons2. 𝑊𝑊𝑊𝑊 𝐾𝐾𝐾𝐾3. R is the midpoint of 𝑊𝑊𝑊𝑊4. 𝑊𝑊𝑊𝑊 𝐾𝐾𝐾𝐾5. 𝑂𝑂𝑂𝑂 𝑂𝑂𝑂𝑂6. 𝑊𝑊𝑊𝑊𝑊𝑊 𝐾𝐾𝐾𝐾𝐾𝐾7. 𝑂𝑂𝑂𝑂𝑂𝑂 𝑂𝑂𝑂𝑂𝑂𝑂 ande. Given point I is the midpoint of 𝑿𝑿𝑿𝑿 point I is the midpoint of 𝑨𝑨𝑨𝑨Statements1. I is the midpoint of 𝑋𝑋𝑋𝑋2.ReasonsDefinition of Midpoint3. I is the midpointof 𝐴𝐴𝐴𝐴4.5. 𝐴𝐴𝐴𝐴𝐴𝐴 𝑂𝑂𝑂𝑂𝑂𝑂6. 𝐴𝐴𝐴𝐴𝐴𝐴 𝑂𝑂𝑂𝑂𝑂𝑂f.Given 𝐌𝐌𝐌𝐌𝐌𝐌 and 𝐇𝐇𝐇𝐇𝐇𝐇 are right angles and 𝑴𝑴𝑴𝑴 𝑻𝑻𝑻𝑻Statements1. 𝑀𝑀𝑀𝑀𝑀𝑀 & 𝐻𝐻𝐻𝐻𝐻𝐻are right angles2. 𝑀𝑀𝑀𝑀 𝑇𝑇𝑇𝑇3. 𝑀𝑀𝑀𝑀 𝑀𝑀𝑀𝑀4. 𝑀𝑀𝑀𝑀𝑀𝑀 𝐻𝐻𝐻𝐻𝐻𝐻M. WinkingUnit 2-6page 51Reasons
Prove the suggested proofs by filling in the missing blanks.⃖ ⃗ 𝑷𝑷𝑷𝑷⃖ ⃗ and 𝑮𝑮𝑮𝑮⃖ ⃗ ⃖ ⃗g. Given . ⃖ ⃗𝐺𝐺𝐺𝐺 ⃖ ⃗𝑃𝑃𝑃𝑃2.3. ⃖ ⃗𝐺𝐺𝐺𝐺 ⃖ ⃗𝐶𝐶𝐶𝐶4.5.6. 𝐺𝐺𝐺𝐺𝐺𝐺 𝑃𝑃𝑃𝑃𝑃𝑃 𝑯𝑯𝑯𝑯 , 𝐓𝐓𝐓𝐓𝐓𝐓 𝐌𝐌𝐌𝐌𝐌𝐌 andh. Given 𝑹𝑹𝑹𝑹⃖ ⃗𝑴𝑴𝑴𝑴𝑻𝑻𝑻𝑻 ⃖ ⃗StatementsReasons 𝑂𝑂𝑂𝑂1. 𝐻𝐻𝐻𝐻2. 𝑹𝑹𝑹𝑹 𝑯𝑯𝑯𝑯3.4.𝑹𝑹𝑹𝑹 𝑂𝑂𝑂𝑂 𝑯𝑯𝑯𝑯 𝐸𝐸𝐸𝐸𝑹𝑹𝑹𝑹 𝑂𝑂𝑂𝑂 𝑅𝑅𝑅𝑅 and𝑯𝑯𝑯𝑯 𝐸𝐸𝐸𝐸 𝐻𝐻𝐻𝐻5. 𝑅𝑅𝑅𝑅 𝐻𝐻𝐻𝐻 𝑂𝑂𝑂𝑂6. 𝑅𝑅𝑅𝑅⃖ ⃗ ⃖ ⃗7. 𝑇𝑇𝑇𝑇𝑀𝑀𝑀𝑀8. 𝑀𝑀𝑀𝑀𝑀𝑀 𝑇𝑇𝑇𝑇𝑇𝑇9. 𝑇𝑇𝑇𝑇𝑇𝑇 𝑀𝑀𝑀𝑀𝑀𝑀10. 𝑇𝑇𝑇𝑇𝑇𝑇 𝑀𝑀𝑀𝑀𝑀𝑀M. WinkingUnit 2-6page 52Substitution Property
Prove the suggested proofs by filling in the missing blanks.i.Given 𝐇𝐇𝐇𝐇𝐇𝐇 𝐓𝐓𝐓𝐓𝐓𝐓 , 𝐓𝐓𝐓𝐓𝐓𝐓 𝐇𝐇𝐇𝐇𝐇𝐇,and Point A is the midpoint of 𝑬𝑬𝑬𝑬ii.StatementsReasons1. 𝑇𝑇𝑇𝑇𝑇𝑇 𝐻𝐻𝐻𝐻𝐻𝐻2. 𝐻𝐻𝐻𝐻𝐻𝐻 is anisosceles triangle3. 𝐻𝐻𝐻𝐻 𝑇𝑇𝑇𝑇4. A is the midpointof 𝐸𝐸𝐸𝐸 5. 𝐸𝐸𝐸𝐸𝐴𝐴𝐴𝐴6. 𝑇𝑇𝑇𝑇𝑇𝑇 𝐻𝐻𝐻𝐻𝐻𝐻7. 𝑇𝑇𝑇𝑇𝑇𝑇 𝐻𝐻𝐻𝐻𝐻𝐻j. ⃗ bisects 𝐍𝐍𝐍𝐍𝐍𝐍.Given that 𝑨𝑨𝑨𝑨 𝑨𝑨𝑨𝑨 are radii of theAlso, 𝑨𝑨𝑨𝑨 and same circle with center A.Statements ⃗ bisects 𝑁𝑁𝑁𝑁𝑁𝑁1. 𝐴𝐴𝐴𝐴2.Definition of Angle Bisector3.Radii of the same circle arecongruent.4.Reflexive property ofcongruence5. 𝐴𝐴𝐴𝐴𝐴𝐴 𝐴𝐴𝐴𝐴𝐴𝐴M. WinkingReasonsUnit 2-6page 53
Prove the suggested proofs by filling in the missing blanks.i.Given: 𝟏𝟏 𝟗𝟗 𝑨𝑨𝑨𝑨 𝑮𝑮𝑮𝑮 𝚫𝚫𝚫𝚫𝚫𝚫𝚫𝚫 forms an isosceles trianglewith base 𝑪𝑪𝑪𝑪Prove: 𝐴𝐴𝐴𝐴𝐴𝐴 𝐺𝐺𝐺𝐺𝐺𝐺StatementsReasons1. 1 92. 𝐴𝐴𝐴𝐴 𝐺𝐺𝐺𝐺3. 2 44. 𝐶𝐶𝐶𝐶𝐶𝐶 is anisosceles triangle5.Isosceles Triangle Theorem6. 6 77. 2 78.Prove the suggested proofs by filling in the missing blanks.k. Given: The circle has a center at point C 𝚫𝚫𝚫𝚫𝚫𝚫𝚫𝚫 forms an isoscelestriangle with base 𝑨𝑨𝑨𝑨Prove: 𝐴𝐴𝐴𝐴𝐴𝐴 𝐷𝐷𝐷𝐷𝐷𝐷Statements1. 𝐴𝐴𝐴𝐴𝐴𝐴 is an isoscelestriangle w/ base 𝐴𝐴𝐴𝐴Reasons 2. 𝐴𝐴𝐴𝐴𝐷𝐷𝐷𝐷3. The circle is centered atpoint C 4. 𝐴𝐴𝐴𝐴𝐷𝐷𝐷𝐷Reflexive Property ofCongruence5.6. 𝐴𝐴𝐴𝐴𝐴𝐴 𝐷𝐷𝐷𝐷𝐷𝐷M. WinkingUnit 2-6page 54
Sec 2.6 Geometry – Triangle Proofs Name: COMMON POTENTIAL REASONS FOR PROOFS . Definition of Congruence: Having the exact same size and shape and there by having the exact same measures. Definition of Midpoint: The point that divides a segment into two congruent segments. Definition of Angle Bisector: The ray that div
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acute triangle equiangular triangle right triangle obtuse triangle equilateral triangle isosceles triangle scalene triangle Vocabulary. Holt McDougal Geometry 4-2 Classifying Triangles Recall that a triangle ( ) is a polygon with three sides. Triangles can be classified in
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Name _ 1 Geometry 1 Chapter 4 – Triangle Congruence Terms, Postulates and Theorems 4.1 Scalene triangle - A triangle with all three sides having different lengths. Equilateral triangle - All sides of a triangle are congruent. Isosceles triangle - A tri
The triangle has three acute angles that are not all equal. It is an acute triangle. 62/87,21 One angle of the triangle measures 90, so it is a right angle. Since the triangle has a right angle, it is a right triangle. 62/87,21 Since all the angles are congruent, it is a equiangular triangle. 62/87,21 The triangle
So, ADÆis a median of the triangle. The three medians of a triangle are concurrent. The point of concurrency is called the The centroid, labeled Pin the diagrams below, is always inside the triangle. acute triangle right triangle obtuse triangle The medians of a triangle have a special concurrency property, as described in Theorem 5.7.File Size: 695KBPage Count: 7
Glencoe/McGraw-Hill 184 Glencoe Geometry Classify Triangles by Sides You can classify a triangle by the measures of its sides. Equal numbers of hash marks indicate congruent sides. If all three sides of a triangle are congruent, then the triangle is an equilateral triangle. If at least two sides of a triangle are congruent, then the triangle is an isosceles triangle.