Overlapping

TrianglesLesson 3.6Mr. Migo Mendoza

Getting Ready:

Each division in the given triangle is • 1 unit long. Hence, the side of the largest triangle is 4-unit long.

Something to think about…

• How many 1 unit triangles are there? 2-units triangles? 3-unit triangles? 4-unit triangles?

Developing Skills:

• Sometimes a geometric figure contains two or more triangles

that overlap each other. In such case, we need to untangle the figure to identify the triangles

involved.

Developing Skills:

• Sometimes two triangles that we want to prove congruent have common parts with the two other triangles that can easily be proven congruent.

Developing Skills:

In this case we may be able •

to use corresponding parts of these triangles to prove that the original triangles

are congruent.

Pre-Example 3.6.1:

• Name the triangles in the figure.

Pre-Example 3.6.2:

• Name the pairs of triangles that can be proven congruent in the given figures.

Example 3.6.1

Theorems on

Isosceles

TrianglesLesson 3.7

Mr. Migo Mendoza

Theorem 3.7.1 The Isosceles Triangle

Theorem

• If two sides of a triangle are congruent, then the angles

opposite them are also congruent.

Example 3.7.1

Corollary 3.7.1.1

• An equilateral triangle is also equiangular.

Corollary 3.7.1.2

• An equilateral triangle has three 60 degrees angles.

Theorem 3.7.2 The Converse of the

Isosceles Triangle Theorem

• If two angles of a triangle are congruent, then the

sides opposite them are also congruent.

Example 3.7.2

Corollary 3.7.1.3

• An equiangular triangle is also equilateral.

Let’s Practice:

Direction:• Using theorems and

corollaries about Isosceles, Equilateral and Equiangular triangles find the values of x.

Let’s Practice:

Let’s Practice:

Example 3.7.4:

Example 3.7.5

Assignment:Direction:

•Complete the proof of the following.

Measuring

Angles in a

TriangleLesson 3.8

Mr. Migo Mendoza

Theorem 3.8.1 The Sum of the Measures

of the Angles of a Triangle Theorem

• The Sum of the degree measures of the angles of a

triangle is 180.

Example 3.8.1

Theorem 3.8.2 The Third

Angles Theorem

• If two angles of one triangle are congruent to two angles

of another, then the third angles are congruent.

Theorem 3.8.2

• If one angle of a triangle is right or obtuse, then

the other two angles are acute.

Theorem 3.8.3

The acute angles of a right •

triangle are complementary.

Exterior and

Interior Angles of

a Triangle

Lesson 3.9Mr. Migo Mendoza

Getting Ready:

• If you extend one side of a triangle from one

vertex, then you have constructed an exterior angle at that vertex.

Getting Ready:

Definition 3.9.1 Exterior Angle

It is an angle which •

forms a linear pair with an angle of the

triangle.

Definition 3.9.2 Remote

Interior Angles

• These are two angles in the triangle that do not have

the same vertex as the exterior angle.

Definition 3.9.3 Adjacent

Interior Angle

•For each exterior angle of a triangle, there

corresponds an adjacent interior angle and a pair of remote

interior angles.

Theorem 3.9.1 The Exterior

Angle Theorem (EAT)

• The measure of an exterior angle of a triangle is equal to the sum of the measures of its two remote interior

angles

Theorem 3.9.1 The Exterior

Angle Theorem (EAT)

Corollary 3.9.1.1

• The measure of an exterior angle of a triangle is

greater than the measure of either of its remote interior

angle.

Assignment:Direction:

• Using the definitions, theorems and corollary in Lesson 3.9, find the value

of x for the following given.