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Triangle proof key to correction, Exercises of Geometry

Cavite State University (CSU)Geometry

Answer key answer key answer key fo triangle proof key for students

Typology: Exercises

2020/2021

Uploaded on 06/03/2021

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Proving Triangles Congruent
SSS - SideSideSide
If 3 sides in one triangle are
congruent to 3 sides of a
second triangle, then the
triangles are congruent.
SAS - SideAngleSide
If two sides and the included
angle of one triangle are
congruent to the corresponding
parts of another triangle, the
triangles are congruent.
ASA - AngleSideAngle
If two angles and included side
of one triangle are congruent to
the corresponding parts of
another triangle, the triangles
are congruent.
AAS - AngleAngleSide
If two angles and the non-
included side of one triangle are
congruent to the corresponding
parts of another triangle, the
triangles are congruent.
HL - HL HypLeg
If the hypotenuse and leg of one
right triangle are congruent to
the corresponding parts of
another right triangle, the right
triangles are congruent
(CPCTC) Corresponding
parts of congruent triangles are
congruent.
Isosceles Triangles Theorems
• If two sides of a triangle are congruent, the angles opposite these sides are congruent.
• If two angles of a triangle are congruent, the sides opposite these angles 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 is perpendicular to the base.
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Proving Triangles Congruent

SSS - SideSideSide If 3 sides in one triangle are congruent to 3 sides of a second triangle, then the triangles are congruent. SAS - SideAngleSide If two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. ASA - AngleSideAngle If two angles and included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. AAS - AngleAngleSide If two angles and the non- included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent. HL - HL HypLeg If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, the right triangles are congruent

(CPCTC) Corresponding

parts of congruent triangles are congruent.

Isosceles Triangles Theorems

  • If two sides of a triangle are congruent, the angles opposite these sides are congruent.
  • If two angles of a triangle are congruent, the sides opposite these angles 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 is perpendicular to the base.

Given: ΔABC CD bisects AB CD ⊥ AB Prove: ΔACD ≅ ΔBCD Statement

  1. ΔABC CD bisects AB CD ⊥ AB
  2. AD ≅ DB Side
  3. ∠CDA and ∠CDB are right ∠
  4. ∠CDA ≅ ∠CDB Angle
  5. CD ≅ CD Side
  6. ΔACD ≅ ΔBCD Reasons
  7. Given
  8. A bisector cuts a segment into 2 ≅ parts.
  9. ⊥ lines form right ∠.
  10. All rt ∠ are ≅.
  11. Reflexive post.
  12. SAS ≅ SAS

Given: ABC and DBE bisect each other. Prove: ΔABD ≅ ΔCBD

Statement

  1. ABC and DBE bisect each other.
  2. AB ≅ BC Side BD ≅ BE Side
  3. ABD and BEC are vertical ∠
  4. ∠ABD ≅ ∠BEC Angle
  5. ΔABD ≅ ΔCBD Reasons
    1. Given
    2. A bisector cuts a segment into 2 ≅ parts.
    3. Intersecting lines form vertical ∠.
    4. Vertical ∠ are ≅.
    5. SAS ≅ SAS

Given: AEB & CED intersect at E E is the midpoint AEB AC ⊥ AE & BD ⊥ BE Prove: ΔAEC ≅ ΔBED Statement

  1. AEB & CED intersect at E E is the midpoint AEB AC ⊥ AE & BD ⊥ BE
  2. AEC and BED are vertical
  3. ∠AEC ≅ ∠BED Angle
  4. AE ≅ EB Side
  5. ∠A & ∠B are rt. ∠
  6. ∠A ≅ ∠B Angle
  7. ΔAEC ≅ ΔBED Reasons
  8. Given
  9. Intersecting lines form vertical ∠.
  10. Vertical ∠ are ≅.
  11. A midpoint cut a segment into 2 ≅ parts
  12. ⊥ lines form right ∠.
  13. All rt ∠ are ≅.
  14. ASA ≅ ASA

Given: AEB bisects CED AC ⊥ CED & BD ⊥ CED Prove: ΔEAC ≅ ΔEBD

Statement

  1. AEB bisects CED AC ⊥ CED & BD ⊥ CED
  2. CE ≅ ED Side
  3. ACE & EDB are rt ∠
  4. ∠ACE ≅ ∠EDB Angle
  5. ∠AEC & ∠DEB are vertical ∠
  6. ∠AEC ≅ ∠DEB Angle
  7. ΔEAC ≅ ΔEBD Reasons
    1. Given
    2. A bisector cuts an angle into 2 ≅parts.
    3. ⊥ Lines form rt ∠.
    4. All rt ∠ are ≅
    5. Intersect lines form vertical ∠
    6. Vertical ∠ are ≅
    7. ASA ≅ ASA

Given: ΔABC is equilateral D midpoint of AB Prove: ΔACD ≅ ΔBCD Statement

  1. ΔABC is equilateral D midpoint of AB
  2. AC ≅ BC Side
  3. AD ≅ DB Side
  4. CD ≅ CD Side
  5. ΔACD ≅ ΔBCD Reasons
  6. Given
  7. All sides of an equilateral Δ are ≅
  8. A midpoint cuts a segment into 2 ≅parts.
  9. Reflexive Post
  10. SSS ≅ SSS

Given: m∠A = 50, m∠B = 45, AB = 10cm, m∠D = 50 m∠E = 45 and DE = 10cm Prove: ΔABC ≅ ΔDEF

Statement

  1. m∠A = 50, m∠B = 45, AB = 10cm, m∠D = 50 m∠E = 45 and DE = 10cm
  2. ∠A = ∠D Angle and ∠B = ∠E Angle AB = DE Side
  3. ΔABC ≅ ΔDEF Reasons
    1. Given
    2. Transitive Prop
    3. ASA ≅ ASA

Given: DE ≅ DG EF ≅ GF Prove: ΔDEF ≅ ΔDFG Statement

  1. DE ≅ DG Side EF ≅ GF Side
  2. DF ≅ DF Side
  3. ΔDEF ≅ ΔDFG Reasons
  4. Given
  5. Reflexive Post
  6. SSS ≅ SSS

Given: KM bisects ∠LKJ LK ≅ JK Prove: ΔJKM ≅ ΔLKM

Statement

  1. KM bisects LKJ LK ≅ JK Side
  2. ∠ 1 ≅ ∠ 2 Angle
  3. KM ≅ KM Side
  4. ΔJKM ≅ ΔLKM Reasons
    1. Given
    2. An ∠ bisectors cuts the ∠ into 2 ≅ parts
    3. Reflexive Post
    4. SAS ≅ SAS

Given:. PR ≅ QR ∠P ≅ ∠Q RS is a median Prove: ΔPSR ≅ ΔQSR Statement

  1. PR ≅ QR Side ∠P ≅ ∠Q Angle RS is a median Side
  2. PS ≅ SQ
  3. ΔPSR ≅ ΔQSR Reasons
  4. Given
  5. A median cuts the side into 2 ≅ parts
  6. SAS ≅ SAS

Given: EG is ∠ bisector EG is an altitude Prove: ΔDEG ≅ ΔGEF

Statement

  1. EG is ∠ bisector EG is an altitude
  2. ∠ 3 ≅ ∠ 4 Angle
  3. EG ⊥ DF
  4. ∠1 & ∠2 are rt ∠
  5. ∠ 1 ≅ ∠ 2 Angle
  6. GE ≅ GE Side
  7. ΔDEG ≅ ΔGEF Reasons
    1. Given
    2. An ∠ bisector cuts an ∠ into 2 ≅ parts.
    3. An altitude form ⊥ lines.
    4. ⊥ lines form right angles.
    5. All right angles are ≅
    6. Reflexive Post
    7. ASA ≅ ASA

Given:. AB ≅ CD ∠CAB ≅ ∠ACD Prove: AD ≅ CB Statement

  1. AB ≅ CD Side ∠CAB ≅ ∠ACD Angle
  2. AC ≅ AC Side
  3. ΔACD ≅ ΔABC
  4. AD ≅ CB Reasons
  5. Given
  6. Reflexive Post
  7. SAS SAS
  8. Corresponding parts of ≅ Δ are ≅.

Given: AEB & CED bisect each Other Prove: ∠C ≅ ∠D

Statement

  1. AEB & CED bisect each other
  2. CE ≅ ED Side & AE ≅ EB Side
  3. ∠1 and ∠2 are vertical
  4. ∠ 1 ≅ ∠ 2 Angle
  5. ΔAEC ≅ ΔDEB
  6. ∠C ≅ ∠D Reasons
    1. Given
    2. A bisector cuts segments into 2 ≅ parts.
    3. Intersect lines form vertical ∠
    4. Vertical ∠ are ≅
    5. SAS ≅ SAS
    6. Corresponding parts of ≅ Δ are

Given: ∠KLM & ∠NML are rt ∠ KL ≅ NM Prove: ∠K ≅ ∠N Statement

  1. ∠KLM & ∠NML are rt ∠ KL ≅ NM Side
  2. ∠KLM ≅ ∠NML Angle
  3. LM ≅ LM Side
  4. ΔKLM ≅ ΔLNM
  5. ∠K ≅ ∠N Reasons
  6. Given
  7. All rt ∠ are ≅
  8. Reflexive Post.
  9. SAS ≅ SAS
  10. Corresponding parts of ≅ Δ are ≅.

Given: AB ≅ BC ≅ CD PA ≅ PD & PB ≅ PC Prove: a) ∠APB ≅ ∠DPC b) ∠APC ≅ ∠DPB

Statement

  1. AB ≅ BC ≅ CD Side PA ≅ PD Side & PB ≅ PC Side
  2. ΔABP ≅ ΔCDP
  3. ∠APB ≅ ∠DPC
  4. ∠BPC ≅ ∠BPC
  5. ∠APB + ∠BPC ≅ ∠DPC + ∠BPC or ∠APC ≅ ∠DPB Reasons
    1. Given
    2. SSS ≅ SSS
    3. Corresponding parts of ≅ Δ are ≅.
    4. Reflexive Post.
    5. Addition Prop.

Given: AB ≅ CB & AD ≅ CD Prove: ∠BAD ≅ ∠BCD Statement

  1. AB ≅ CB & AD ≅ CD
  2. ∠ 1 ≅ ∠ 2 ∠ 3 ≅ ∠ 4
  3. ∠1 + ∠ 3 ≅ ∠2 + ∠ 4 or ∠BAD ≅ ∠BCD Reasons
  4. Given
  5. If 2 sides are ≅ then the ∠ opposite are ≅.
  6. Addition Post.

Given: **Δ** ABC ≅ **Δ** DEF M is midpoint of AB N is midpoint DE Prove: ΔAMC ≅ ΔDNF

Statement

  1. Δ ABC ≅ Δ DEF
  2. M is midpoint of AB N is midpoint DE
  3. ∠D ≅ ∠A Angle and DF ≅ AC Side
  4. AM ≅ AB and DN ≅ DE
  5. AM ≅ DN Side
  6. ΔAMC ≅ ΔDNF Reasons
    1. Given
    2. Given
    3. Corresponding parts of ≅ Δ are ≅
    4. A midpoint cuts a segment in
      1. of ≅ sides are ≅
    5. SAS ≅ SAS

Given: ΔABC ≅ ΔDEF CG bisects ∠ACB FH bisects ∠DFE Prove: CG ≅ FH Statement

  1. ΔABC ≅ ΔDEF CG bisects ∠ACB FH bisects ∠DFE Reasons

Given: **Δ** AME ≅ **Δ** BMF DE ≅ CF Prove: AD ≅ BC

Statement

  1. Δ AME ≅ Δ BMF DE ≅ CF
  2. EM ≅ MF AM ≅ MB Side ∠ 1 ≅ ∠ 2 Angle
  3. DE + EM ≅ CF + MF Reasons
    1. Given
    2. Corresponding parts of ≅ Δ are ≅
    3. Addition Post.

Given: BC ≅ BA BD bisects ∠CBA Prove: DB bisects ∠CDA Statement

  1. BC ≅ BA Side BD bisects CBA
  2. ∠ 1 ≅ ∠ 2 Angle
  3. BD ≅ BD Side
  4. ΔABD ≅ ΔBCD
  5. ∠ 3 ≅ ∠ 4
  6. DB bisects ∠CDA Reasons
  7. Given
  8. A bisector cuts an angle into 2 ≅ parts.
  9. Reflexive Post.
  10. SAS ≅ SAS
  11. Corresponding parts of ≅ Δ are ≅
  12. A angle bisector cuts an angle into 2 ≅ parts.

Given: AE ≅ FB DA ≅ CB ∠A and ∠B are Rt. ∠ Prove: ΔADF ≅ ΔCBE DF ≅ CE Statement

  1. AE ≅ FB DA ≅ CB Side ∠A and ∠B are Rt. ∠
  2. EF ≅ EF
  3. AE + EF ≅ FB + EF or AF ≅ EB Side
  4. ∠A ≅ ∠B Angle
  5. ΔADF ≅ ΔCBE
  6. DF ≅ CE Reasons
  7. Given
  8. Reflexive Post
  9. Addition Property
  10. All rt. ∠ are ≅.
  11. SAS ≅ SAS
  12. Corresponding parts of ≅ Δ are ≅

Given: SPR ≅ SQT PR ≅ QT Prove: ΔSRQ ≅ ΔSTP ∠R ≅ ∠T

Statement

  1. SPR ≅ SQT Side PR ≅ QT
  2. ∠S ≅ ∠S Angle
  3. SPR – PR ≅ SQT – QT or SR ≅ ST Side Reasons
    1. Given
    2. Reflexive Post
    3. Subtraction Property

Given: TM ≅ TN M is midpoint TR N is midpoint TS Prove: RN ≅ SM Statement

  1. TM ≅ TN Side M is midpoint TR N is midpoint TS
  2. ∠T ≅ ∠T Angle
  3. RM is ½ of TR NS is ½ of TS
  4. RM ≅ NS
  5. TM + RM ≅ TN + NS or RT ≅ TS Side
  6. ΔRTN ≅ ΔMTS
  7. RN ≅ SM Reasons
  8. Given
  9. Reflexive Post.
  10. A midpoint cuts a segment in 1.
  11. ½ of ≅ parts are ≅.
  12. Addition Property
  13. SAS ≅ SAS
  14. Corresponding parts of ≅ Δ are ≅.

Given: AD ≅ CE & DB ≅ EB Prove: ∠ADC ≅ ∠CEA Statement

  1. AD ≅ CE & DB ≅ EB Side
  2. ∠B ≅ ∠B Angle
  3. AD + DB ≅ CE + EB or AB ≅ BC Side
  4. ΔABE ≅ ΔBCD
  5. ∠ 1 ≅ ∠ 2
  6. ∠1 & ∠3 are supplementary ∠2 & ∠4 are supplementary
  7. ∠ 3 ≅ ∠4 or ∠ADC ≅ ∠CEA Reasons
  8. Given
  9. Reflexive Post
  10. Addition Post.
  11. SAS ≅ SAS
  12. Corresponding parts of ≅ Δ are ≅.
  13. A st. line forms supplementary ∠.
  14. Supplements of ≅ ∠ are ≅.