4.1: Congruent Figures

Congruent Polygons:

Corresponding Angles:

Corresponding Sides:

Congruence Statement:

Naming Polygons:1) List the vertices by starting at one and working around (the corners of the figure)2) If it’s a triangle use a small triangle in front of the vertices, for anything else,

just use the word polygon

Example: CAT JSD. List each of the following.

1. three pairs of congruent sides

2. three pairs of congruent angles

Examples: Each pair of polygons is congruent. Find the measures of the numbered angles.

3. 4.

How can you show that two figures are congruent?

Example: Can you conclude that the triangles are congruent. If so, write a congruence statement.

5. FGH and JKH

Third Angle Theorem:

Example 6: Developing Proof Use the information given in the diagram. Give a reason that each statement is true.a. L Qb. LNM QNPc. M P

d. e. LNM QNP

Examples: Find the values of the variables.

7. 8.

Examples: ABCD FGHJ. Find the measures of the given angles or lengths of the given sides.

9. mB = 3y, mG = y + 50

Example 10: Given: is the angle bisector of ABC.

is the perpendicular bisector of . Prove: ADB CDB

4.2: Triangle Congruence by SSS and SAS

Parts of a Triangle 1) Vertex:

2) Included Side:

3) Included Angles:

Proving Triangles Congruent:If only a certain amount of information is given:

1) Apply the Side-Side-Side Postulate2) Apply the Side-Angle-Side Postulate

Side-Side-Side Postulate

B K

A C J L

Side-Angle-Side Postulate

B K

A C J L

Draw MGT. Use the triangle to answer the questions below.

1. What angle is included between and ?

2. Which sides include ?

3. What angle is included between and ?

Would you use SSS or SAS to prove the triangles congruent? If there is not enough information to prove the triangles congruent by SSS or SAS, write not enough information. Explain your answer.

4. 5. 6.

7. 8. 9.

10. Given: Prove: ABC EDC

15. Given: Prove: WXZ YZX

4.3: Triangle Congruence by ASA and AASRecall:Included Angle: Angle formed by a pair of sidesIncluded Side: Side connecting two angles

Angle-Side-Angle Postulate

A X

Reasons

Statements

What else would you need to know to prove the

triangles congruent by ASA?Given: AB≅ XY∧¿

B C Y Z

Angle-Angle-Side Theorem

A X Show on the pair of triangles one possibility of two

angles that could be congruent and the included side that would need to be congruent.

B C Y Z

Example: Are the two triangles congruent? If so, make a congruence statement.A B A

C D B C

D

Example: Given: Prove: A

B C

D

E F

Example: Given: Prove: A C

B

D E

4.4: Using Corresponding Parts of Congruent Triangles

CPCTC (Corresponding Parts of Congruent Triangles are Congruent)If you can prove that two triangles are congruent, then you can prove any pair of corresponding parts is congruent.

Your goal is to prove that two triangles are congruent!

What have we used to prove that triangles are congruent?

Example: State why the two triangles are congruent. Then list all other pairs of corresponding parts.

Example:

Given: H JProve:

L K

Example:

Given: A is the midpoint of M RProve:

A

S T

D

Example: Given: Prove: 2 4 C 1 E 3 A

B

4.5: Isosceles, Equilateral, and Right Triangles

Isosceles:

Legs:

Base:

Base Angles:

Vertex Angle:

Example: Label the legs, the base angles and the vertex in the triangle. Use symbols to show which parts are congruent.

Isosceles Triangle Theorem: If two sides of a triangles are congruent, then

Converse of the Isosceles Theorem: If two angles of a triangle are congruent, then

Given: 1 2Prove:

Theorem 4-5: If a line bisects the vertex angle of an isosceles triangle, then the line is also perpendicular bisector of the base.

Diagram:Corollary: If a triangle is equilateral, then it’s equiangular.Corollary: If a triangle is equiangular, then it’s equilateral.

Examples: Complete each statement. Explain why it is true.

1.

2. BDE

3. CBE BCE

X

ZB

Y

Example: Solve for x and y.1) 2) 3) y 50 50 y x

12 2x + 8 50 y x 40

4) 5)

4.6: Congruence in Right Triangles

Right Triangle:

Hypotenuse:

Leg:

Diagram: Label the Hypotenuse and Leg in the Right Triangle Below

Hypotenuse – Leg Thm. (HL): If the hypotenuse and leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent.

Conditions For Using HL:1) The two triangles must be .

2) The hypotenuses must be .

3) There is one pair of congruent .

Example: Complete the proof.

Given: V and W are right angles ; Prove: WVZ VWX

Statements Reasons

1) 1) Given

2) 2) Given

3) 3) Reflexive Property of Congruence

4) 4) HL Theorem

Examples: For what values of x and y are the triangles congruent by HL?1) 2)

Examples: What additional information would prove each pair of triangles congruent by the Hypotenuse-Leg Theorem?

3) 4)

Practice with Proofs A

B

Given: AB DC , D CProve:

Statements Reasons

Q

K

Q

A

B

K A

B

Statements Reasons

Statements Reasons

Statements Reasons