Geometry proofs

Congruence and similarity written out as a proof — two per sheet, because a proof is not a question you answer in a line.

4 sheets · 8 questions · one answer key · 5 pages

Recommended for grades 9–10

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Congruence proofs Sheet 1 of 4

GeometryCongruence proofs

Name: Date:

Read each proof and choose the missing reason.

Given E is the midpoint of BD; AE ≅ CE

Prove △AEB ≅ △CED

Statements	Reasons
1. BE ≅ DE	?
2. ∠AEB ≅ ∠CED	Vertical Angles
3. AE ≅ CE	Given
4. △AEB ≅ △CED	SAS

Reason for step 1: GivenReflexive PropertyVertical AnglesAlternate Interior AnglesDefinition of MidpointSSSSASASAAASHLDefinition of Perpendicular

Given RS ∥ UT; RS ≅ UT

Prove △RSV ≅ △TUV

Statements	Reasons
1. ∠SRV ≅ ∠UTV	?
2. ∠RSV ≅ ∠TUV	Alternate Interior Angles
3. RS ≅ UT	Given
4. △RSV ≅ △TUV	ASA

Reason for step 1: GivenReflexive PropertyVertical AnglesAlternate Interior AnglesDefinition of MidpointSSSSASASAAASHLDefinition of Perpendicular

GeometrySimilarity proofs

Name: Date:

Read each proof and choose the missing reason.

Given PQ ∥ SR

Prove △PQT ~ △RST

Statements	Reasons
1. ∠PTQ ≅ ∠RTS	Vertical Angles
2. ∠QPT ≅ ∠SRT	?
3. △PQT ~ △RST	AA ~

Reason for step 2: GivenReflexive PropertyVertical AnglesCorresponding AnglesAlternate Interior AnglesAA ~SAS ~SSS ~

Given UV ∥ ST

Prove △RUV ~ △RST

Statements	Reasons
1. ∠R ≅ ∠R	?
2. ∠RUV ≅ ∠RST	Corresponding Angles
3. △RUV ~ △RST	AA ~

Reason for step 1: GivenReflexive PropertyVertical AnglesCorresponding AnglesAlternate Interior AnglesAA ~SAS ~SSS ~

Write a triangle proofWrite a congruence proof

Name: Date:

Write a full two-column proof for each. Reasons may be used more than once.

Given GH ≅ JH; ∠GHK ≅ ∠JHK

Prove △GHK ≅ △JHK

Statements	Reasons
1.
2.
3.
4.

Given AB ∥ DC; AB ≅ DC

Prove △ABE ≅ △CDE

Statements	Reasons
1.
2.
3.
4.

Write a triangle proofWrite a similarity proof

Name: Date:

Write a full two-column proof that the triangles are similar.

Given ∠BAC is a right angle; AD ⊥ BC

Prove △DAC ~ △ABC

Statements	Reasons
1.
2.
3.

Given ∠PRS ≅ ∠Q

Prove △PRS ~ △PQR

Statements	Reasons
1.
2.
3.

Power Pack · Geometry proofsAnswer key

Congruence proofs

1. △AEB ≅ △CED (E is the midpoint of BD, AE ≅ CE), step 1: Definition of Midpoint2. △RSV ≅ △TUV (RS ∥ UT, RS ≅ UT), step 1: Alternate Interior Angles

Similarity proofs

1. △PQT ~ △RST (PQ ∥ SR), step 2: Alternate Interior Angles2. △RUV ~ △RST (UV ∥ ST), step 1: Reflexive Property

Write a congruence proof

1. △GHK ≅ △JHK — 1. GH ≅ JH — Given; 2. ∠GHK ≅ ∠JHK — Given; 3. HK ≅ HK — Reflexive Property; 4. △GHK ≅ △JHK — SAS2. △ABE ≅ △CDE — 1. ∠BAE ≅ ∠DCE — Alternate Interior Angles; 2. ∠ABE ≅ ∠CDE — Alternate Interior Angles; 3. AB ≅ DC — Given; 4. △ABE ≅ △CDE — ASA

Write a similarity proof

1. △DAC ~ △ABC — 1. ∠C ≅ ∠C — Reflexive Property; 2. ∠ADC ≅ ∠BAC — Right angles are congruent; 3. △DAC ~ △ABC — AA ~2. △PRS ~ △PQR — 1. ∠P ≅ ∠P — Reflexive Property; 2. ∠PRS ≅ ∠Q — Given; 3. △PRS ~ △PQR — AA ~

Geometry proofs

What is in the pack

The geometry track

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