Geometry Chapter 5.1 and 5.2 Triangles - Home - Jal … Chapter 5.5 SAS –Side-Angle-Side This leads us to say that two sides and the included angle must be congruent for the triangles

GeometryChapter 5.1 and 5.2 Triangles

Objectives:• Students will classify triangles.• Students will define and use the Angle Sum Theorem.

A triangle is formed when three non-collinear points are connected by segments.

R

T

S side

sidevertex

Each pair of segments forms an angle with a vertex that is a vertex of the triangle.

side

Triangles are named by the letters of their vertices. The above triangle is named triangle RST, written Δ 𝑅𝑆𝑇.



Recall that angles are classified as either acute, obtuse, or right.

All triangles have two acute angles. A triangle can be classified by its third angle as either acute, obtuse, or right.

Triangles can also be classified by their sides.

A scalene triangle has no sides congruent.

An isosceles triangle has two sides congruent.

An equilateral triangle has all sides congruent.

Are equilateral triangles isosceles triangles? YES!

Are isosceles triangles equilateral triangles? NO!



The angle formed by the two congruent sides is called the vertex angle.

legleg

The side opposite the vertex angle is called the base.

The congruent sides are called legs.

The angles formed by the base and one of the congruent sides are called the base angles.



What can you tell me about the sum of the angles of a triangle?

Equals 180°



Angle Sum Theorem: The sum of the measures of the angles of a triangle is 180°.

Theorem 5-2: The acute angles of a right angle triangle are complementary.

An equilateral triangle is a triangle with all angles congruent.

Theorem 5-3: The measure of each angle of an equilateral triangle is 60°.

Bookwork: page 191 problems 8-17; page 196 problems 8-20.

GeometryChapter 5.3 Geometry in Motion

Objectives:• Students will identify translations, reflections, and rotations.

TRANSLATION, sometimes called slides REFLECTION, image flipped over a line

ROTATION, image is turned around a point

GeometryChapter 5.3 Geometry in Motion

Objectives:• Students will identify translations, reflections, and rotations.

A

B

X

Y

ZC

∆𝐴𝐵𝐶 → ∆𝑋𝑌𝑍

𝐴 → 𝑋;𝐵 → 𝑌; 𝐶 → 𝑍𝐴𝐵 → 𝑋𝑌;𝐵𝐶 → 𝑌𝑍; 𝐶𝐴 → 𝑍𝑋

GeometryChapter 5.3 and 5.4 Assignments

Chapter 5.3: page 201 problems 9-24.

Chapter 5.4: page 203 problems 11-25.

Chapter 5 Review: page 220 problems 1-21.

GeometryChapter 5.4 Congruent Triangles

Objectives:• Students will identify corresponding parts of congruent triangles.

A

B

X

Y

ZC

If a triangle can be translated, rotated, or reflected onto another triangle so that all the vertices correspond, the triangles are congruent triangles.



The parts of congruent triangles that match are called corresponding parts.

A

B

C F

E

D∆𝐴𝐵𝐶 ≅ ∆𝐷𝐸𝐹

Congruent Angles

∠𝐴 ≅ ∠𝐷

∠𝐵 ≅ ∠𝐸

∠𝐶 ≅ ∠𝐹

Congruent Sides

𝐴𝐵 ≅ 𝐷𝐸

𝐵𝐶 ≅ 𝐸𝐹

𝐶𝐴 ≅ 𝐹𝐷



Definition of Congruent Triangles (CPCTC): If the corresponding parts of two triangles are congruent, then the triangles are congruent. If two triangles are congruent, then the corresponding parts are congruent.

A

B

C F

E

D

CPCTC: Corresponding Parts of Congruent Triangles are Congruent.

Geometry

Objectives:• Students will identify Congruent triangles by SSS and SAS.

If we draw a triangle… Then attempt to draw a congruent triangle; however side 2 is longer…

Do we have congruent triangles?

No! Why?

CPCTC

Chapter 5.5 SSS – Side-Side-Side

GeometryChapter 5.5 SSS – Side-Side-Side

This leads us to say that the three sides of a two triangles must be congruent for the triangles to be congruent.

The SSS Postulate: If three sides of one triangle are congruent to three corresponding sides of a second triangle, then the triangles are congruent.


Geometry

If we draw a triangle… Then attempt to draw a congruent triangle; however the included angle between side one and side two is bigger…

Do we have congruent triangles?

No! Why?

CPCTC

Chapter 5.5 SSS – Side-Angle-Side


GeometryChapter 5.5 SAS – Side-Angle-Side

This leads us to say that two sides and the included angle must be congruent for the triangles to be congruent.

The SAS Postulate: If two sides and the included angle of one triangle are congruent to the corresponding sides and the included angle of a second triangle, then the triangles are congruent.

Notice that Side-Angle-Side states that the angle must be between the sides, not angle-side-side.

There is no ASS in geometry.


Bookwork: page 213, problems 8-22

GeometryChapter 5.6 ASA – Angle-Side-Angle

Objectives:• Students will identify congruent triangles by ASA and AAS.

If we draw a triangle… Then attempt to draw a congruent triangle by keeping side one congruent and the two angles by that side congruent…

Then sides 2 and 3 will intersect at a point such that they are congruent.

GeometryChapter 5.6 ASA – Angle-Side-Angle

This leads us to say that two angles and the included side must be congruent for the triangles to be congruent.

The ASA Postulate: If two angles and the included side of one triangle are congruent to the corresponding angles and the included side of a second triangle, then the triangles are congruent.


GeometryChapter 5.6 AAS – Angle-Angle-Side

Objectives:• Students will identify congruent triangles by ASA and AAS.

From what we know, can we prove these triangles congruent?

A

B

C

X

Y

Z

SSS? No! SAS? No! ASA? No! Wait a minute!

If ∠𝐴 ≅ ∠𝑋 𝑎𝑛𝑑 ∠𝐵 ≅ ∠𝑌, is ∠𝐶 ≅ ∠𝑍? Yes, the sum of a triangle’s interior angles =180

So these triangles are congruent by ASA, And, by extension, AAS

GeometryChapter 5.6 AAS – Angle-Angle-Side

This leads us to say that two angles and the non-included side must be congruent for the triangles to be congruent.

The AAS Theorem: If two angles and a non-included side of one triangle are congruent to the corresponding angles and non-included side of a second triangle, then the triangles are congruent.


Bookwork: page 218, problems 11-24

Geometry Chapter 5.1 and 5.2 Triangles - Home - Jal … Chapter 5.5 SAS –Side-Angle-Side This leads us to say that two sides and the included angle must be congruent for the triangles

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