congruent polygons and congruent parts
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Congruent Polygons and Congruent Parts Two polygons are congruent if and only if there is a one-to-one correspondence between their vertices such that corresponding angles are congruent and corresponding sides are congruent Corresponding parts of congruent polygons are congruent.
Congruent Triangles The following also holds true for Triangles: Reflexive Property ΔABC ≅ ΔABC Symmetric Property If ΔABC ≅ ΔDEF then ΔABC ≅ ΔDEF Transitive Property If ΔABC ≅ ΔDEF and ΔDEF ≅ ΔRST then ΔABC ≅ ΔRST
Proving Triangles Congruent
Side Angle Side Two triangles are congruent if two sides and the included angle of one triangle are congruent, respectively, to two sides and the included angle of the other.
SAS ≅ SAS
Angle Side Angle Two triangles are congruent if two angles and the included side of one triangle are congruent, respectively, to two angles and the included side of the other.
ASA ≅ ASA
Side Side Side Two triangles are congruent if three sides of one triangle are congruent respectively, to three sides of the other.
SSS ≅ SSS
Congruence Based on Triangles An Altitude of a triangle is a line segment drawn from any vertex of the triangle perpendicular to and ending in the line contains the opposite side. A Median of a triangle is a line segment that joins any vertex of the triangle to the midpoint of the opposite side.
An Angle Bisector of a triangle is a line segment that bisects any angle of the triangle and terminates in the side opposite that angles. Using congruent triangles to prove line segments congruent and angles congruent. (Corresponding parts of congruent triangles are congruent)(CPCTC)
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Isosceles Triangles
If two sides of a triangle are congruent, the angles opposite these sides are congruent. The median from the vertex angle of an isosceles triangle bisects the vertex angle. The median from the vertex angle of an isosceles triangle is perpendicular to the base.
Equilateral Triangle Every equilateral triangle is equiangular.
Perpendicular Bisector The Perpendicular Bisector of a line segment is any line or subset of a line that is perpendicular to the line at its midpoint. If two points are each equidistant from the endpoints of a line segment, then the points determine the perpendicular bisector of the line segment If a point is equidistant from the endpoints of a line segment, then it is on the perpendicular bisector of the line segment. If a point is on the perpendicular bisector of a line segment, then it is equidistant from the endpoints of the line segment. A point is on the perpendicular bisector of a line segment if and only if it is equidistant from the endpoints of a line segment. The perpendicular bisectors of the sides of a triangle are congruent.
Methods of Proving Lines or Line Segments Perpendicular 1. The two lines form right angles at their point of intersection.
2. The two lines form congruent adjacent angles at their point of intersection.
3. Each of two points on one line is equidistant from the endpoints of a segment of the other.
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#1
Given: ΔABC CD bisects AB CD ⊥ AB Prove: ΔACD ≅ ΔBCD
Statement 1. ΔABC CD bisects AB CD ⊥ AB
Reasons 1. Given
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#2
Given: ABC and DBE bisect each
other.
Prove: ΔABD ≅ ΔCBD
Statement
Reasons
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#3
Given: AB = CD and BC = AD ∠DAB, ∠ABC, ∠BCD and ∠CDA
are rt ∠ Prove: ΔABC ≅ ΔADC
Statement
Reasons
#4
Given: ∠PQR ≅ ∠RQS PQ ≅ QS Prove: ΔPQR ≅ ΔRQS
Statement
Reasons
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#5
Given: AEB & CED intersect at E E is the midpoint AEB AC ⊥ AE & BD ⊥ BE Prove: ΔAEC ≅ ΔBED
Statement
Reasons
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#6
Given: AEB bisects CED AC ⊥ CED & BD ⊥ CED Prove: ΔEAC ≅ ΔEBD
Statement
Reasons
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#7
Given: ΔABC is equilateral D midpoint of AB Prove: ΔACD ≅ ΔBCD
Statement
Reasons
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#8
Given: m∠A = 50, m∠B = 45, AB = 10cm, m∠D = 50 m∠E = 45 and DE = 10cm Prove: ΔABC ≅ ΔDEF
Statement
Reasons
#9
Given: GEH bisects DEF m∠D = m∠F Prove: ΔGFE ≅ ΔDEH
Statement
Reasons
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#10
Given: PQ bisects RS at M ∠R ≅ ∠S Prove: ΔRMQ ≅ ΔSMP
Statement
Reasons
#11
Given: DE ≅ DG EF ≅ GF Prove: ΔDEF ≅ ΔDFG
Statement
Reasons
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#12
Given: KM bisects ∠LKJ LK ≅ JK Prove: ΔJKM ≅ ΔLKM
Statement
Reasons
#13
Given: . PR ≅ QR ∠P ≅ ∠Q RS is a median Prove: ΔPSR ≅ ΔQSR
Statement
Reasons
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#14
Given: EG is ∠ bisector EG is an altitude Prove: ΔDEG ≅ ΔGEF
Statement
Reasons
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#15
Given: ∠A and ∠D are a rt ∠ AE ≅ DF AB ≅ CD Prove: EC ≅ FB
Statement
Reasons
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#16
Given: AC ≅ BC D midpoint of AB Prove: ∠A ≅ ∠B
Statement
Reasons
#17
Given: . AB ≅ CD ∠CAB ≅ ∠ACD Prove: AD ≅ BC
Statement
Reasons
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#18
Given: AEB & CED bisect each
Other Prove: ∠C ≅ ∠D
Statement
Reasons
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#19
Given: ∠KLM & ∠NML are rt ∠ KL ≅ NM Prove: ∠K ≅ ∠N
Statement
Reasons
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#20
Given: AB ≅ BC ≅ CD PA ≅ PD & PB ≅ PC Prove: a) ∠APB ≅ ∠DPC b) ∠APC ≅ ∠DPB
Statement
Reasons
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#21
Given: PM is Altitude PM is median Prove: a) ΔLNP is isosceles b) PM is ∠ bisector
Statement
Reasons
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#22
Given: AC ≅ BC Prove: ∠CAD ≅ ∠CBE
Statement
Reasons
#23
Given: AB ≅ BC & AD ≅ CD Prove: ∠BAD ≅ ∠BCD
Statement
Reasons
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#24
Given: ΔABC ≅ ΔDEF M is midpoint of AB N is midpoint DE Prove: ΔAMC ≅ ΔDNF
Statement
Reasons
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#25
Given: ΔABC ≅ ΔDEF CG bisects ∠ACB FH bisects ∠DFE Prove: CG ≅ FH
Statement
Reasons
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#26
Given: ΔAME ≅ ΔBMF DE ≅ CF Prove: AD ≅ BC
Statement
Reasons
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#27
Given: AEC & DEB bisect each other Prove: E is midpoint of FEG
Statement
Reasons
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#28
Given: BC ≅ BA BD bisects ∠CBA Prove: BD bisects ∠CDA
Statement
Reasons
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#29
Given: AE ≅ FB AD ≅ BC ∠A and ∠B are Rt. ∠ Prove: ΔADF ≅ ΔCBE DF ≅ CE
Statement
Reasons
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#30
Given: SPR ≅ SQT PR ≅ QT Prove: ΔSRQ ≅ ΔSTP ∠R ≅ ∠T
Statement
Reasons
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#31
Given: AD ≅ BC AD ⊥ AB & BC ⊥ AB Prove: ΔDAB ≅ ΔCBA
AC ≅ BD
Statement
Reasons
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#32
Given: ∠BAE ≅ ∠CBF ∠BCE ≅ ∠CDF AB ≅ CD Prove: AE ≅ BF ∠E ≅ ∠F
Statement
Reasons
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#33
Given: TM ≅ TN M is midpoint TR N is midpoint TS Prove: RN ≅ SM
Statement
Reasons
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#34
Given: AD ≅ CE & BD ≅ EB Prove: ∠ADC ≅ ∠CEA
Statement
Reasons
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#35
Given: AE ≅ BF & AB ≅ CD ∠ABF is the suppl. of ∠A Prove: ΔAEC ≅ ΔBFD
Statement
Reasons
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#36
Given: AB ≅ BC BD bisects ∠ABC Prove: AE ≅ CE
Statement
Reasons
#37
Given: PB ≅ PC Prove: ∠ABP ≅ ∠DCP
Statement
Reasons
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#38
Given: AC and BD are ⊥ bisectors of each other. Prove: AB ≅ BC ≅ CD ≅ AD
Statement
Reasons
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#39
Given: AEFB, ∠1 ≅ ∠2 CE ≅ DF , AE ≅ BF Prove: ΔAFD ≅ ΔBEC
Statement
Reasons
#40
Given: SX ≅ SY , XR ≅ YT Prove: ΔRSY ≅ ΔTSX
Statement
Reasons
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#41
Given: AD ≅ BC AD ⊥ AB , BC ⊥ AB Prove: ΔDAB ≅ ΔCBA
Statement
Reasons
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#42
Given: AF ≅ CE ∠1 ≅ ∠2, ∠3 ≅ ∠4 Prove: ΔABE ≅ ΔCDF
Statement
Reasons
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#43
Given: AB ⊥ BF , CD ⊥ BF ∠1 ≅ ∠2, BD ≅ FE Prove: ΔABE ≅ ΔCDF
Statement
Reasons
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#44
Given: ∠BAC ≅ ∠BCA CD bisects ∠BCA AE bisects ∠BAC Prove: ΔADC ≅ ΔCEA
Statement
Reasons
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#45
Given: TR ≅ TS , MR ≅ NS Prove: ΔRTN ≅ ΔSTM
Statement
Reasons
#46
Given: CEA ≅ CDB, ΔABC AD and BE intersect at P ∠PAB ≅ ∠PBA Prove: PE ≅ PD
Statement 1.
Reasons
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#47
Given: AB ≅ AD and BC ≅ CD Prove: ∠1 ≅ ∠2
Statement
Reasons
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#48
Given: BD is both median and
altitude to AC Prove: BA ≅ BC
Statement
Reasons
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#49
Given: ∠CDE ≅ ∠CED and AD ≅ EB Prove: ∠ACD ≅ ∠BCE
Statement
Reasons
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#50
Given: Isosceles triangle CAT CT ≅ AT and ST bisects ∠CTA Prove: ∠SCA ≅ ∠SAC
Statement
Reasons
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#51
Given: ∠1 ≅ ∠2 BD ⊥ AC Prove: ΔABD ≅ ΔCBD
Statement
Reasons
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#52
Given: ∠P ≅ ∠S R is midpoint of PS Given: ΔPQR ≅ ΔSTR
Statement
Reasons
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#53
Given: FG ⊥ DE G is midpoint of DE Given: ΔDFG ≅ ΔEFG
Statement
Reasons
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#54
Given: AC ≅ BC D is midpoint of AB Prove: ΔACD ≅ ΔBCD
Statement
Reasons
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#55
Given: PT bisects QS PQ ⊥ QS and TS ⊥ QS Prove: ΔPQR ≅ ΔRST
Statement
Reasons
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#56
Given: AB ≅ DE and FE ≅ BC FE ⊥ AD and BC ⊥ AD Prove: ΔAEF ≅ ΔCBD
Statement
Reasons
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#57
Given: SM is ⊥ bisector of LP RM ≅ MQ ∠a ≅ ∠b Prove: ΔRLM ≅ ΔQPM
Statement
Reasons
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#59
Given: AC ≅ BC CD ⊥ AB Prove: ΔACD ≅ ΔBCD
Statement
Reasons
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#60
Given: FQ bisects AS ∠A ≅ ∠S Prove: ΔFAT ≅ ΔQST
Statement
Reasons
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#61
Given: ∠A ≅ ∠D and ∠BCA ≅ ∠FED AE ≅ CD ∠AEF ≅ ∠BCD Prove: ΔABC ≅ ΔDFE
Statement
Reasons
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#62
Given: SU ≅ QR , PS ≅ RT ∠TSU ≅ ∠QRP Prove: ΔPQR ≅ ΔSTU ∠Q ≅ ∠U
Statement
Reasons
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#63
Given: ∠M ≅ ∠D ME ≅ HD ∠THE ≅ ∠SEM Prove: ΔMTH ≅ ΔDSE
Statement
Reasons
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#64
Given; SQ bisects ∠PSR ∠P ≅ ∠R Prove: ΔPQS ≅ ΔQSR
Statement
Reasons
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#65
Given: PQ ⊥ SQ and TS ⊥ QS R midpoint of QS Prove: ∠P ≅ ∠T
Statement
Reasons
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#66
Given: BC ≅ FB , BT ≅ BV DV ≅ TS , CD ≅ FS Prove: ∠D ≅ ∠S
Statement
Reasons
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#67
Given: PQ ≅ DE and PB ≅ AE QA ⊥ PE and BD ⊥ PE Prove: ∠D ≅ ∠Q
Statement
Reasons
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#68
Given: TS ≅ TR ∠P ≅ ∠Q Prove: PS ≅ QR
Statement
Reasons
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#69
Given: HY and EV bisect each other Prove: HE ≅ VY
Statement
Reasons
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#70
Given: ∠E ≅ ∠D and ∠A ≅ ∠C B is the midpoint of AC Prove: AE ≅ DC
Statement
Reasons
#71
Given: E is midpoint of AB AD ⊥ AB and BC ⊥ AB ∠1 ≅ ∠2 Prove: AD ≅ CB
Statement
Reasons
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