Name: ________________________________ 14-8 Congruency ReviewGeometry PD. _____ Date: _______________

Concept Key Ideas/Tips

Beginning a Proof

*mark your picture *annotate question (givens)*use the tools from your tool box *1st write your givens*last step = what you're trying to prove *make a plan*use a checklist for your shortcuts! *# your steps!*All of your givens and markings should be a step in your proof!

Triangle Congrue

ncy Shortcut

s

*WE CANNOT USE AAA or SSA!!! *

Proving Parts are

Congruent

*To prove triangles are congruent, you need to use a shortcut.*To prove parts (sides and angles) of triangles are congruent, we: 1st: prove triangles are congruent 2nd: use CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

Proving Definitio

ns

Addition Postulate

With Substituti

on

Subtraction

PostulateWith

Substitution

Transitive Property

AB = 5; AB ≅ DE ; DE ≅ GH

So, AB = GH(by the TRANSITIVE PROPERTY)

Using Suppleme

nts

<1 < 4≅So, <2 <3≅(since 2 is the supplement of <1and <3 is the supplement of < 4)

Indirect Proof

*Use when proving something is not true or ≇ or to show something is not true.Steps:1. Assume the opposite of the prove statement2. Prove normally3. Contradict something in the given.

**Proof Pieces are available online in our Regents Review Folder!

IF AB = EFCD = CD (Reflexive Property)Then, AB + CD = EF + CD (Addition Postulate)AD = ED (Substitution Postulate)

=

IF AC = BD,BC = BC (Reflexive Property)Then, AC- BC = BD - BC (Subtraction Postulate)AB = CD (Substitution Postulate)

1) Figures are congruent when: A shortcut method is shown All corresponding sides and all corresponding angles are congruent

2) If two images are congruent, then its corresponding parts are congruent!

Try it!

1. Given: <A ≅<C, <1 ≅<2, and B is the midpoint of AC, what short cut proves ∆ ABD≅ ∆CBE?

2. Given , which statement is not always true?

1)2)3) area of area of 4) perimeter of perimeter of

3. Given: AD bisects <BAC and AD⊥BC what short cut proves ∆ ADB≅ ∆ ADC?

4. GivenAB⊥BC ,AD ⊥DC , AB≅ AD , what short cut proves ∆ ABC ≅∆ ADC?

`

5. Given: CD bisects AB, AB bisects CDProve: ∡ ACE≅∡BDE

6. Given AB∨¿DC , AB≅ CD,what short cut proves ∆ ABD≅ ∆CDB?

7. Given trapezoid ABCD where ∆ DAB≅ ∆CBA, which of the following statements is true based on the given?

o BD ≅CDo DA ≅ DCo <BCE ≅<DABo <ACB≅<BDA

8. A pair of congruent triangles are shown at the right, as marked. Which statement must be true?

o AB ≅CDo AD ≅ ABo <DAC ≅<ACBo <DCB≅<DAB

9. BD is a perpendicular bisector of AC, which CAN NOT always be proven:

o AD ≅CDo DC ≅ BDo <ADB ≅<CDBo ∆ ABD≅ ∆CBD

10. Given: AB⊥BC , AD⊥DC , AB≅ ADProve: ∆ ABC ≅∆ ADC

11. Shown in the diagram, ∆ FAC ≅ ∆GEC . Which statement CAN NOT Be proven using CPCTC?

o AF ≅≥¿o AC ≅ ECo <FCA ≅<GCEo <CAF≅<CDG

12. Line segment EA is the perpendicular bisector of , and and are drawn.

Which conclusion can not be proven?1) bisects angle ZET.2) Triangle EZT is equilateral.3) is a median of triangle EZT.4) Angle Z is congruent to angle T.

13. Given: Quadrilateral ABCD is a parallelogram with diagonals and intersecting at E

Prove:

Continue that thought….Describe a single rigid motion that maps onto .

14.

15. In the diagram of trapezoid ABCD below, diagonals and intersect at E and .

Which statement is true based on the given information?1) 3)2) 4)

16. Two right triangles must be congruent if1) All of its angles are congruent

2) the lengths of the hypotenuses are equal

3) the corresponding legs are congruent

4) the areas are equal

Given:

Prove:

17. Given:XY ≅ ZY and YW bisects <XYZ

Prove: <YWZ is a right angle.

18. In the diagram of ∆ LAC∧∆DNC below, LA≅ DN ,CA ≅ CN ,∧DAC⊥ LCN

What short cut method would prove ∆LAC ≅ ∆DNC ?

What is one important thing you would have to state in this specific proof?

19. Given: CB ≅ DC and AB ≅ DA

Prove: ACis an angle bisector

Statements Reasons1. CB ≅ DC and AB ≅ DA 1. Given

2. 2.

3. 3.

4. 4. CPCTC

5. 5. Definition of an angle bisector

20. Given: ∆ABC is isosceles with legs AB and ACAH is the altitude to side BC

Prove: BH ≅CH

Statements Reasons

1. ∆ABC is isosceles with legs AB and ACAH is the

altitude to side BC1. Given

2. AH⊥BC 2. Definition of an altitude3. ∠ AHB and ∠ AHC are right angles 3.

4. ∆ AHB∧∆ AHC are¿ triangles 4. Triangles with right angles are right triangles

5. AB≅AC 5.

6. AH≅AH 6. Reflexive Property

7. ∆ AHB≅ ∆ AHC 7.

8. BH ≅CH 8.

DID YOU CHECK YOUR WORK?

DID YOU MARK YOUR DIAGRAM?