Geometry
Chapter 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
side vertex
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.
Geometry
Chapter 5.1 and 5.2 Triangles
Objectives:
• Students will classify triangles.
• Students will define and use the Angle Sum Theorem.
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!
Geometry
Chapter 5.1 and 5.2 Triangles
Objectives:
• Students will classify triangles.
• Students will define and use the Angle Sum Theorem. The angle formed by the two congruent sides is called the
vertex angle.
leg leg
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.
Geometry
Chapter 5.1 and 5.2 Triangles
Objectives:
• Students will classify triangles.
• Students will define and use the Angle Sum Theorem.
What can you tell me about the sum of the angles of a triangle? Equals 180°
Geometry
Chapter 5.1 and 5.2 Triangles
Objectives:
• Students will classify triangles.
• Students will define and use the Angle Sum Theorem.
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°.
Geometry
Chapter 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
Geometry
Chapter 5.3 Geometry in Motion
Objectives:
• Students will identify translations, reflections, and rotations. A B X Y Z C
∆𝐴𝐵𝐶 → ∆𝑋𝑌𝑍
𝐴 → 𝑋; 𝐵 → 𝑌; 𝐶 → 𝑍
𝐴𝐵 → 𝑋𝑌; 𝐵𝐶 → 𝑌𝑍; 𝐶𝐴 → 𝑍𝑋
Geometry
Chapter 5.3 and 5.4 Assignments
Chapter 5.3: page 201 problems 9-24.
Chapter 5.4: page 203 problems 11-25.
Geometry
Chapter 5.4 Congruent Triangles
Objectives:
• Students will identify corresponding parts of congruent triangles. A B X Y Z C
If a triangle can be translated, rotated, or reflected onto another triangle so that all the vertices correspond, the triangles are congruent triangles.
Geometry
Chapter 5.4 Congruent Triangles
Objectives:
• Students will identify corresponding parts of 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
Geometry
Chapter 5.4 Congruent Triangles
Objectives:
• Students will identify corresponding parts of congruent triangles.
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
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
Geometry
Chapter 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.
Objectives:
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
Objectives:
Geometry
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 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.
Objectives:
• Students will identify Congruent triangles by SSS and SAS.
Geometry
Chapter 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.
Geometry
Chapter 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.
Objectives:
Geometry
Chapter 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
Geometry
Chapter 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.
Objectives:
• Students will identify Congruent triangles by SSS and SAS.
