Review: Solving Systems

x 2y+3x+y

12

Find the values of x and y that make the following triangles congruent.

Congruent Triangles (CPCTC)

Two triangles are congruent triangles if and only if the corresponding parts of those congruent triangles are congruent.

• Corresponding sides are congruent

• Corresponding angles are congruent

Congruence Statement

When naming two congruent triangles, order is very important.

Third Angle Theorem

If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.

Congruence Shortcuts

Side-Angle-Side (SAS) Congruence Postulate:

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Side-Side-Side Congruence Postulate

SSS Congruence Postulate:If the three sides of one triangle are congruent to

the three sides of another triangle, then the two triangles are congruent.

Congruence Shortcuts

Angle-Side-Angle (ASA) Congruence Postulate:

If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Congruence Shortcuts

Angle-Angle-Side (AAS) Congruence Theorem:If two angles and a non-included side of one

triangle are congruent to the corresponding two angles and the non-included side of another triangle, then the two triangles are congruent.

Base Angles Theorem:

If two sides of a triangle are congruent, then the angles opposite them are congruent.

Converse of the Base Angles Theorem:

If two angles of a triangle are congruent, then the sides opposite them are congruent.

Equilateral Triangle Theorem

A triangle is equilateral if and only if it is equiangular.

Practice

Practice

Practice

Congruence in Right Triangles

Vocabulary Right Triangles

Leg

Leg

HypotenuseA

CB

Hypotenuse-Leg Theorem

Hypotenuse-Leg (HL) Congruence Theorem:

If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and leg of another right triangle, then the two triangles are congruent.

• To use the HL Theorem, you must show that three conditions are met:

• There are two right triangles

• The triangles have congruent hypotenuses

• There is one pair of congruent legs

Using the HL Theorem

Using the HL Theorem

Statements Reasons

1.

2.

3.

4.

5.

1.

2.

3.

4.

5.

, ofbisector theis CEAD

of Defn.

s rt. are & EBACBD

EBCB

EACD

Thm. HL

Using the HL Theorem

Statements Reasons

1.

2.

3.

4.

1.

2.

3.

4.

srt are and RPQPRS

srt of Defn.

of Prop. Refl.

QRSP

RPQPRS

Which are congruent by HL?

3in3in

5in

5in

5in

3in

J

HG

FE

D

A

CB

Which are congruent by HL?

3in3in

5in

5in

5in

3in

J

HG

FE

D

A

CB

HFJ DEG

Prove the triangles are congruent

R

QP

S

4. SRP QPR

3. PR PR

2. QPR SRP

1. QPR and SRPare right, SP RQ

Given

All Right angles are congruent

Reflexive

HL Theorem

What else do you need to prove the triangles are congruent?

V T

R

XIs RT XT? orIs XV TV?

Prove the two triangles are congruent

T is the midpoint of RV

V

R

U

TS

3. RST TUV

1. T is the midpoint of RVS and U are right anglesRS TU

2. RT TV

1. Given

2. Definition of midpoint

3. HL Theorem