Chapter 4: Congruent Triangles

Lesson 1: Classifying Triangles

Classifying Triangle by Angles Acute Triangle: all of the angles are acute

Obtuse Triangle: one angle is obtuse, the other two are acute

Right Triangle: one angle is right, the other two are acute

Equiangular Triangle: all the angles are 60 degrees

Classifying Triangles by Sides

Scalene Triangle: all sides are different measures

Isosceles Triangle: at least two sides have the same measure

Equilateral Triangle: all sides have the same measure

3

5

7

* vertex angle= formed by the two congruent sides of an isosceles triangle

* base= the side of an isosceles triangle not congruent to the others

If point Y is the midpoint of VX, and WY = 3.0 units, classify ΔVWY as equilateral, isosceles, or scalene. Explain your reasoning.

ALGEBRA Find the measure of the sides of isosceles triangle KLM with base KL.

__

ALGEBRA Find x and the measure of each side of equilateral triangle ABC if AB = 6x – 8, BC = 7 + x, and AC = 13 – x.

Find the measure of each side of Triangle JKL and classify the triangle based on its sides. J(-3, 2) K(2, 1) L(-2, -3)

Find y___ ___

Chapter 4: Congruent Triangles

Lesson 2: Angles of Triangles

The sum of the measures of the angles of a triangle is always 180 degrees.

The acute angles of a right triangle are complementary

There can be at most one right or one obtuse angle in a triangle

Third Angle Theorem If two angles of one

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

A

BC

X

YZ

If A X, and B Y, then

C Z.

Interior and Exterior Angles of Triangles

Exterior angle: formed by one side of a triangle and the extension of another side

The interior angles farthest from the exterior angle are its remote interior angles. (remote interior angles are not adjacent to the exterior angle)

1

2

3 4

Exterior angle

Remote interior angles An exterior angle is equal to the

sum of its remote interior angles.

ex: 1 + 2 = 4

Anticipation Guide: read each statement. State whether the sentence is true or false. If the statement is false- rewrite it with the correct term in place of the underlined word

The acute angles of a right triangle are supplementary The sum of the measures of the angles of any triangle

is 100 A triangle can have at most one right angle or acute

angle If two angles of one triangle are congruent to two

angles of another triangle, then the third angle of the triangles are congruent

The measure of an exterior angle of a triangle is equal to the difference of the measures of the two remote interior angles

If the measures of two angles of a triangle are 62 and 93, then the measure of the third angle is 35

An exterior angle of a triangle forms a linear pair with an interior angle of the triangle

SOFTBALL The diagram shows the path of the softball in a drill developed by four players. Find the measure of each numbered angle.

Find the measure of each numbered angle.

GARDENING Find the measure of FLW in the fenced flower garden shown.

The piece of quilt fabric is in the shape of a right triangle. Find the measure of ACD.

Find the measure of each numbered angle.

Find m3.

Chapter 4: Congruent Triangles

Lesson 6: Isosceles Triangles

Isosceles Triangles

Vertex Angle

Base angles

leg

.

leg

- If two sides of a triangle are congruent, the two angles opposite of them are also congruent

-If two angles of a triangle are congruent, then two sides opposite of them are also congruent

- If a triangle is equilateral, it is also equiangular

A. Find mR.

B. Find PR

A. Find mT.

ALGEBRA Find the value of each variable

Chapter 4: Congruent Triangles

Lesson 3: Congruent Triangles

Definition of Congruent Triangles

Congruent triangles are triangles with exactly the same size and shape

CPCTC: Corresponding Parts of Congruent Triangles are Congruent

Two triangles are congruent if and only if their corresponding parts are congruent

Corresponding Parts

Corresponding parts have the same congruence markings

AB HI AC HJ BC IJ A H B I C J

A

B C

H

I J

Congruence Transformations Slide or Translation: the triangle is in

the same position farther down, up, or across the page

Turn or Rotation: the triangle is spun around a point (usually one of the angles)

Flip or reflection: the triangle is shown in a mirror image across a line of symmetry

Write a congruence statement for the triangles.

Name the corresponding congruent angles for the congruent triangles.

In the diagram, ΔITP ΔNGO. Find the values of x and y.

In the diagram, ΔFHJ ΔHFG. Find the values of x and y.

Find the missing information in the following proof.

Prove: ΔQNP ΔOPN

Proof:

3. Q O, NPQ PNO 3. Given

5. Definition of Congruent Polygons5. ΔQNP ΔOPN

4. _________________4. QNP ONP ?

2. 2. Reflexive Property ofCongruence

1. 1. Given

Write a two-column proof.

Prove: ΔLMN ΔPON

Chapter 4: Congruent Triangles

Lesson 4 and 5: Proving Congruence- SSS, SAS, ASA, AAS, and HL

SSS

Side-Side-Side If all three sets of corresponding sides are

congruent, the triangles are congruent

ABC MNO

A M

ONCB

SAS Side-Angle-Side

If two corresponding sides and the included angles of two triangles are congruent, then the triangles are congruent

XYZ FGH

X

Y Z

F

G H

* The included angle is the angle between the congruent sides

ASA Angle-Side-Angle

If two sets of corresponding angles and the included sides are congruent, then the triangles are congruent

JKL RST

J

L K T

R

S

* The included side is the side between the two congruent angles

AAS Angle-Angle-Side

If two sets of corresponding angles and one of the corresponding non-included sides are congruent, then the triangles are congruent

EFG TUV

E

G F

T

V U

HL Hypotenuse-Leg

If the hypotenuse and one set of corresponding legs of two right triangles are congruent, then the triangles are congruent

CDH RAM

AD H

RC

M

Determine if the triangles are congruent. If they are, write the congruence statement.

Given: AC ABD is the midpoint of BC.

Prove: ΔADC ΔADB

___ ___

Determine whether ΔABC ΔDEF for A(–5, 5), B(0, 3), C(–4, 1), D(6, –3), E(1, –1), and F(5, 1).

Determine if the triangles are congruent. If they are, write the congruence statement.

Determine which postulate can be used to prove that the triangles are congruent. If it is not possible to prove congruence, choose not possible.

Write a two column proof.