4-1 Classifying Triangles A triangle is a three-sided polygon.

A triangle is made up of sides, angles, and vertices.

A triangle can be classified by sides and by angles.

When a triangle is classified by its sides, it can be scalene, isosceles or equilateral.

Label the following triangles as scalene, isosceles, or equilateral.

___________________ All 3 sides are different lengths.

___________________ At least two of the sides are the

same length.

___________________ ALL 3 sides are the same length

When a triangle is classified by its angles, it can be acute, obtuse, right or equiangular.

Label the following triangles as acute, obtuse, right, or equiangular.

All of the angles are equal.

___________________ All the angles are less

than 90.

___________________One of the angles is

exactly 90.

___________________ One of the angles is

greater than 90.

Your Turn!Match the following triangles to their descriptions._______1. a. Right scalene

_______2. b. Acute isosceles

_______3. c. Obtuse isosceles

_______4. d. Right isosceles

_______5. e. Equiangular equilateral

Your Turn, Again!Classify the following triangles by sides AND by angles. You will have two answers for each!!

___________________ , ___________________ ___________________ , ___________________

___________________ , ___________________ ___________________ , ___________________

Special TrianglesRight TrianglesRight triangles have two legs and a hypotenuse. The hypotenuse is the always the side that is across from the right angle. One special thing about a right triangle is that the two angles other than the right angle are always acute and complementary.

Isosceles TrianglesIsosceles triangles have two legs and a base. The base is the bottom of the triangle. Isosceles triangles also have two base angles and a vertex angle. The base angles are on either side of the base and are congruent. The vertex angle is in between the two congruent sides.

Example 1Triangle ABC is an equilateral triangle. Find x and the measure of each side if AB = 4x - 3 and BC = 3x + 4.

4-2 Angles of Triangles The Angle Sum Theorem states that the sum of the measures of the interior angles of a

triangle is always 180o.

Example 1Find the measure of <T.

Our Turn!Solve for each numbered angle.

The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior angles.

Example 2Find m<1.

Our Turn!Solve for each numbered angle.

Your Turn!Solve for each numbered angle.

4-3 Congruent TrianglesRecall from Chapter 1 that the word congruent means equal, or the same. Similarly, if two triangles are congruent they will have exactly the same three sides and exactly the same three angles.

Your Turn!

Which triangle is not congruent to the other three?

CPCTC states thatC ______________________

P _____________ of

C _________________

T ________________ are

C ________________

Two triangles are congruent ONLY if their corresponding parts are congruent. This means the corresponding angles must be congruent and the corresponding sides must be congruent as well.

Example 1Take a look at the example below. Triangles RST and XYZ are congruent.

Name all the congruent angles. Name all the congruent sides.

Congruency statements will tell you what angles and sides are equal in triangles.

Example 2Write a congruency statement for the triangles above.

Your Turn!Using the congruency statement below, name the congruent angles and sides.

Name all the congruent angles. Name all the congruent sides.

Turns, flips, and slides are called congruence transformations. They do not change the congruence of the triangle.

Your Turn!Circle the right description for the transformations below.

Slide

Turn

Flip

4-4 and 4-5 Proving Triangles CongruentTwo triangles are congruent if they have exactly the same three sides AND exactly the same three angles. But we don't have to know all three sides and all three angles ...usually three out of the six is enough.

Ways to Determine Triangle CongruenceThere are five ways to find if two triangles are congruent: SSS, SAS, ASA, AAS and HL.

1. SSS (_______,_______,_______)

SSS stands for "side, side, side" and means that we have two triangles with all three sides equal.For example:

is congruent to:

If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.

2. SAS (_______,_______,_______)

SAS stands for "side, angle, side" and means that we have two triangles where we know two sides and the included angle are equal.For example:

is congruent to:

If two sides and the included angle of one triangle are equal to the corresponding sides and angle of another triangle, the triangles are congruent. 3. ASA (_______,_______,_______)

ASA stands for "angle, side, angle" and means that we have two triangles where we know two angles and the included side are equal.For example:

is congruent to:

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

4. AAS (_______,_______,_______)

AAS stands for "angle, angle, side" and means that we have two triangles where we know two angles and the non-included side are equal.

For example:

is congruent to:

If two angles and the non-included side of one triangle are equal to the corresponding angles and side of another triangle, the triangles are congruent. 5. HL (_____________________, _______)This one applies only to right triangles!

or

HL stands for "Hypotenuse, Leg" (the longest side of the triangle is called the "hypotenuse", the other two sides are called "legs")For example:

is congruent to:

If the hypotenuse and one leg of one right triangle are equal to the corresponding hypotenuse and leg of another right triangle, the two triangles are congruent.

Caution ! Don't Use "AAA" OR the Donkey Theorem!

AAA means we are given all three angles of a triangle, but no sides.

This is not enough information to decide if two triangles are congruent!Because the triangles can have the same angles but be different sizes:

is not congruent to:

Without knowing at least one side, we can't be sure if two triangles are congruent.

The Donkey Theorem is pretty self-explanatory. DON’T use it!Your Turn!Draw two triangles that are congruent by each of the following congruence theorems.

SSSIf three sides of one triangle are equal to three

sides of another triangle, the triangles are congruent.

SASIf two sides and the included angle of one triangle are equal to the corresponding sides and angle of

another triangle, the triangles are congruent.

ASAIf two angles and the included side of one triangle are equal to the corresponding angles and side of

another triangle, the triangles are congruent.

AASIf two angles and the non-included side of one

triangle are equal to the corresponding angles and side of another triangle, the triangles are

congruent.

HLIf the hypotenuse and one leg of one right triangle are equal to the corresponding hypotenuse and leg

of another right triangle, the two triangles are congruent.