GCR Proof Power

ABC DEF |

AB ≅ DE

BC ≅ EF

[Add AB + BC, DE + EF]

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Entire segments:

A ___ ≅ D ___

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Segment ____________

Property

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#6 Subtract Segments (−)

MNP QST |

MP ≅ QT (whole)

NP ≅ ST (part)

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Remaining part:

M ___ ≅ Q ___

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Segment ____________

Property

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#7 Common Angle (+)

B A D E C |

∠ABD ≅ ∠EBC

Shared angle: ∠DBE

[Add ∠DBE to both]

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Larger combined angles:

∠ A B ___ ≅ ∠ C B ___

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Angle ____________

Property (common angle)

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#8 Common Angle (−)

O X Y Z W |

∠XOZ ≅ ∠YOW

Shared angle: ∠YOZ

[Subtract ∠YOZ from both]

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Outer angles:

∠ X O ___ ≅ ∠ Z O ___

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Angle ____________

Property (common angle)

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GCR Practice 4 • Guided Edition Page 2 of 5

GCR Practice: Problems 9 – 12 (Guided Edition) Geometry Name: _______________________

AMB CND |

AB ≅ CD

M is midpoint of AB

N is midpoint of CD

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Matching halves:

A ___ ≅ C ___

(or MB ≅ ND)

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____________ Property

(Halves of ≅ segments)

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#10 Chain (A = B = C)

JK LM PQ |

JK ≅ LM

LM ≅ PQ

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1st & 3rd segments:

JK ≅ ___ ___

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____________ Property

of Congruence

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#11 Doubles (× 2)

XWYZ KJML |

∠WXY ≅ ∠JKM (halves)

Ray XY bisects ∠WXZ

Ray KM bisects ∠JKL

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Entire double angles:

∠ W X ___ ≅ ∠ J K ___

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____________ Property

(Doubles of ≅ ∠s are ≅)

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#12 Common Segment (+)

WXYZ |

WX ≅ YZ

Shared segment: XY

[Add XY to both ends]

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Overlapping spans:

W ___ ≅ X ___

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Segment ____________

Property (common segment)

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GCR Practice 4 • Guided Edition Page 3 of 5

GCR Practice: Problems 13 – 16 (Guided Edition) Geometry Name: _______________________

PQRS |

PR ≅ QS

Shared segment: QR

[Subtract QR from both]

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End segments:

P ___ ≅ R ___

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Segment ____________

Property (common segment)

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#14 Add Angles (+)

VFED 12 XYZW 34 |

∠1 ≅ ∠3

∠2 ≅ ∠4

[∠1 + ∠2 = ∠FVD]

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Total angles:

∠ F V ___ ≅ ∠ Y X ___

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Angle ____________

Property

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#15 Doubles (× 2)

EMF GNH |

EM ≅ GN (halves)

M is midpoint of EF

N is midpoint of GH

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Entire double segments:

E ___ ≅ G ___

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____________ Property

(Doubles of ≅ segments)

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#16 Subtract Angles (−)

ACBD 12 ZXWY 34 |

∠CAD ≅ ∠XZY (whole)

∠1 ≅ ∠3 (part)

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Remaining angle:

∠2 ≅ ∠ ___

(∠BAD ≅ ∠WZY)

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Angle ____________

Property

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GCR Practice 4 • Guided Edition Page 4 of 5

GCR Practice: Problems 17 – 20 (Guided Edition) Geometry Name: _______________________

XUTSR |

∠UXT ≅ ∠TXS

∠TXS ≅ ∠SXR

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1st & 3rd angles:

∠ U X T ≅ ∠ ___ ___ ___

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____________ Property

of Congruence

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#18 Add Segments (+)

ABC KLM |

AB ≅ KL

BC ≅ LM

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Total segment lengths:

A ___ ≅ K ___

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Segment ____________

Property

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#19 Split Angle (∠)

QPSR |

Ray QS bisects ∠PQR

[Bisect means cut into 2 ≅ angles]

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Two congruent parts:

∠ P Q ___ ≅ ∠ ___ Q R

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Definition of

Angle ____________

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#20 Split Segment (Mid)

AMB |

M is the midpoint of AB

[Midpoint splits segment into 2 ≅ halves]

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Two congruent halves:

A ___ ≅ ___ B

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Definition of

____________

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GCR Practice 4 • Guided Edition Page 5 of 5

If a common angle (∠DBE) is added to ≅ angles, sums are ≅.

| | #8 OXYZW |

∠XOZ ≅ ∠YOW

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∠XOY ≅ ∠ZOW

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Angle Subtraction Property

If a common angle (∠YOZ) is subtracted from ≅ angles, differences are ≅.

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GCR Practice 4 • Answer Key & Solutions Page 2 of 5

GCR Practice: Problems 9 – 12 (Solutions) Teacher Reference Key

M is midpoint of AB

N is midpoint of CD

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AM ≅ CN

(or MB ≅ ND)

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Division Property

Halves of ≅ segments are ≅ (Like divisions of ≅ segments).

| | #10 JK LM PQ |

JK ≅ LM

LM ≅ PQ

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JK ≅ PQ

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Transitive Property

If two segments are ≅ to the same segment, they are ≅ to each other.

| | #11 XWYZ KJML |

∠WXY ≅ ∠JKM

Ray XY bisects ∠WXZ

Ray KM bisects ∠JKL

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∠WXZ ≅ ∠JKL

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Multiplication Property

Doubles of ≅ angles are ≅ (Like multiples of ≅ angles).

| | #12 WXYZ |

WX ≅ YZ

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WY ≅ XZ

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Segment Addition Property

If a common segment (XY) is added to ≅ segments, sums are ≅.

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GCR Practice 4 • Answer Key & Solutions Page 3 of 5

GCR Practice: Problems 13 – 16 (Solutions) Teacher Reference Key

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PQ ≅ RS

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Segment Subtraction Property

If a common segment (QR) is subtracted from ≅ segments, differences are ≅.

| | #14 VFED 12 XYZW 34 |

∠1 ≅ ∠3

∠2 ≅ ∠4

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∠FVD ≅ ∠YXW

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Angle Addition Property

If ≅ angles are added to ≅ angles, sums are ≅ (∠1+∠2=∠FVD).

| | #15 EMF GNH |

EM ≅ GN

M is midpoint of EF

N is midpoint of GH

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EF ≅ GH

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Multiplication Property

Doubles of ≅ segments are ≅ (Like multiples of ≅ segments).

| | #16 ACBD 12 ZXWY 34 |

∠CAD ≅ ∠XZY

∠1 ≅ ∠3

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∠2 ≅ ∠4

(∠BAD ≅ ∠WZY)

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Angle Subtraction Property

If ≅ angles are subtracted from ≅ angles, differences are ≅.

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GCR Practice 4 • Answer Key & Solutions Page 4 of 5

GCR Practice: Problems 17 – 20 (Solutions) Teacher Reference Key

∠TXS ≅ ∠SXR

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∠UXT ≅ ∠SXR

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Transitive Property

If two angles are ≅ to the same angle (∠TXS), they are ≅ to each other.

| | #18 ABC KLM |

AB ≅ KL

BC ≅ LM

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AC ≅ KM

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Segment Addition Property

If ≅ segments are added to ≅ segments, sums are ≅.

| | #19 QPSR |

Ray QS bisects ∠PQR

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∠PQS ≅ ∠SQR

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Definition of Angle Bisector

An angle bisector divides an angle into two congruent adjacent angles.

| | #20 AMB |

M is the midpoint of AB

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AM ≅ MB

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Definition of Midpoint

The midpoint divides a segment into two congruent segments.

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GCR Practice 4 • Answer Key & Solutions Page 5 of 5