4.1 Triangles and Angles

Triangle: a figure formed by three segments joining three noncollinear points. It can be classified by its sides and by its angles.

Classification by Sides

EquilateralIsoscelesScalene

3 congruent sidesat least 2 congruent sidesno congruent sides

Classification by Angles

Acute Equiangular Right Obtuse

3 acute angles 3 congruent angles 1 right angle 1 obtuse angle

Vertex: each of three points joining the sides of a triangle.

A, B, and C are vertices of triangle ABC. A

B C

Adjacent sides: two sides sharing a common vertex. ABand BC share vertex B.

Right and Isosceles Triangles

Right Triangle Isosceles Triangle

hypotenuse

leg leg leg

leg

base

Triangle Sum Theorem:

The sum of the measures of the interior angles of a triangle is 180*.

m<A + m<B + m<C = 180* A

B C

Exterior Angle Theorem:

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.

m<1 = m<A + m<B A

B C

Corollary to the Triangle Sum Theorem:

The acute angles of a right triangle are complementary.

m<A + m<B = 90* A

C B

4.2 Congruence and Triangles

Congruent: two geometric figures that have exactly the same size and shape.When two figures are congruent, there is a correspondence between their angles and sides such that corresponding sides are congruent and corresponding angles are congruent.

ABC = PQR A P

B C R Q

Corresponding AnglesCorresponding Sides

<A = <PAB = PQ

<B = <QBC = QR

<C = <RCA = RP

Third Angle Theorem:

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

If <A = <P and <B = <Q, then <C = <R

Reflexive Property of Congruent Triangles:

Every triangle is congruent to itself.

Symmetric Property of Congruent Triangles:

If ABC = DEF, then DEF = ABC.

Transitive Property of Congruent Triangles:

If ABC = DEF and DEF = JKL, then ABC = JKL

4.3 Proving Triangles are Congruent: SSS and SAS

SSS Congruence Postulate

If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.

If SideAB = DE C F

SideBC = EF

SideCA = FD

Then ABC = DEF

A B E D

SAS Congruence Postulate

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

If Side AB = DEC F

Angle <B = <E

Side BC = EF

Then ABC = DEF A B E D

Choosing Which Congruence Postulate to Use

Name the included angle between the given pair of sides.

AB and BC

CE and DC

AC and BC

4.4 Proving Triangles are Congruent: ASA and AAS

ASA Congruence Postulate

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

If Angle <C = <F C F

SideBC = EF

Angle <B = <E

Then ABC = DEF

A B E D

AAS Congruence Postulate

If two angles and the nonincluded side of one triangle are congruent to two angles and the nonincluded side of a second triangle, then the two triangles are congruent.

If Angle <C = <FC F

Angle <A = <D

Side AB = DE

Then ABC = DEF

4.5 Using Congruent Triangles

EQ: What does the acronym CPCTC represent?

Knowing that all pairs of corresponding parts of congruent triangles are congruent can help one prove congruent parts of triangles.

Given: PS = RS, PQ = RQ Q

Prove: <PQS = <RQS

StatementsReasons

1. PS = RS1. Given P R

2. PQ = RQ2. Given

3. QS = QS3. Reflexive S

4. PQS = RQS4. SSS

5. <PQS = <RQS5. CPCTC

Once one proves two triangles are congruent, then any pair of congruent parts are congruent by CPCTC: Corresponding Parts of Congruent Triangles are Congruent.These may be corresponding sides or angles.

4.6 Isosceles, Equilateral, and Right Triangles

Base Angle Theorem: if two sides of a triangle are

congruent, then the angles opposite them are congruent.

If AB = AC, then <B = <C

Base Angle Converse: if two angles of a triangle are

Congruent, then the sides opposite them are congruent.

If <B = <C, then AB = AC

Corollary: if a triangle is equilateral, then it is equiangular.

Corollary: if a triangle is equiangular, then it is equilateral.

HL: Hypotenuse-Leg: if the hypotenuse and a leg

of one right triangle are congruent to the hypotenuse

and a leg of a second right triangle, then the two

triangles are congruent.

In the two right triangles

if leg BC = EF

and hypotenuse AC = DF

then ABC = DEF by HL

Triangle may be proved congruent by any of five ways

SSSSASASAAASHL

4.7 Triangles and Coordinate Proof

Place a 2 unit by 6 unit rectangle ina coordinate plane.

A right triangle has legs of 5 and 12 units. Place the triangle in a coordinate plane and then find the length of the hypotenuse.

In the diagram ∆MLO = ∆KLO. Find the coordinate of L

Write a plan to prove that SQ bisects <PSR.

Can the two triangles be proved congruent? If so, which postulate or theorem can be used?

1. 2. 3.

4. 5. 6.

7. 8. 9.

10. 11. 12.

  1. Write the SSS Congruence Postulate in your own words.
  1. Write the SAS Congruence Postulate in your own words.
  1. Write the ASA Congruence Postulate in your own words.
  1. Write the AAS Congruence Theorem in your own words.

Find the values of the missing angles or sides.

17.18.

19.20.

21.