1. ANGLES OF TRIANGLES
HEXAHEDRON: This is your standard cube, but "Hex" comes from the Greek meaning "six."  About the only fascinating thing about this solid is that when you rotate it to where a point, apex, or vertex is perpendicular to your nose, you should notice that the outer edges form a "hexagon."  And, this solid was associated with the "Earth" for some reason. Now, when this solid is constructed as a frame, without welding or solder, it is very unstable.  The "Earth" is a very unstable thing because of its volcanoes and earthquakes.
TETRAHEDRON:
The word "Tetra" is a Greek word and it has something to do with the number "four." Now, with a tetrahedron you have an equilateral triangle as its base. Then you place an equilateral triangle at each edge of the base then they all come up to a point, apex, or vertex if you will. So, with "tetra" meaning "four", you have four surfaces. The tetrahedron was also associated somehow with "Fire" in Plato's day.

2. THM 4.1-ANGLE SUM THEOREM The Sum of the measure of the interior Angles of a Triangle is ALWAYS 180°
HEXAHEDRON: This is your standard cube, but "Hex" comes from the Greek meaning "six."  About the only fascinating thing about this solid is that when you rotate it to where a point, apex, or vertex is perpendicular to your nose, you should notice that the outer edges form a "hexagon."  And, this solid was associated with the "Earth" for some reason. Now, when this solid is constructed as a frame, without welding or solder, it is very unstable.  The "Earth" is a very unstable thing because of its volcanoes and earthquakes.
TETRAHEDRON:
The word "Tetra" is a Greek word and it has something to do with the number "four." Now, with a tetrahedron you have an equilateral triangle as its base. Then you place an equilateral triangle at each edge of the base then they all come up to a point, apex, or vertex if you will. So, with "tetra" meaning "four", you have four surfaces. The tetrahedron was also associated somehow with "Fire" in Plato's day.

3. THM 4.1-ANGLE SUM THEOREM Example Problem: Find x
55°
(3x+5)° x°
55°
55°
30°
95°
HEXAHEDRON: This is your standard cube, but "Hex" comes from the Greek meaning "six."  About the only fascinating thing about this solid is that when you rotate it to where a point, apex, or vertex is perpendicular to your nose, you should notice that the outer edges form a "hexagon."  And, this solid was associated with the "Earth" for some reason. Now, when this solid is constructed as a frame, without welding or solder, it is very unstable.  The "Earth" is a very unstable thing because of its volcanoes and earthquakes.
TETRAHEDRON:
The word "Tetra" is a Greek word and it has something to do with the number "four." Now, with a tetrahedron you have an equilateral triangle as its base. Then you place an equilateral triangle at each edge of the base then they all come up to a point, apex, or vertex if you will. So, with "tetra" meaning "four", you have four surfaces. The tetrahedron was also associated somehow with "Fire" in Plato's day.
Step 3
Use Algebra to Solve for x
60+4x=180
4x=120
x=30
Step 6
We now know the values of all the angles
Check to make sure they sum 180°
Step 4
Now that we know x=30° we can find the measure of all the interior angles.
Step 5
Solving for Shaded Angle with x=30°
(3(30)+5)=95°
Step 1
Use Angle Sum Theorem that interior angles of triangle ALWAYS sum to 180°
Step 2
Write Equation
55+3x+5+x=180 =
180°

4. THM 4.2-THIRD ANGLE THEOREM If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent.
HEXAHEDRON: This is your standard cube, but "Hex" comes from the Greek meaning "six."  About the only fascinating thing about this solid is that when you rotate it to where a point, apex, or vertex is perpendicular to your nose, you should notice that the outer edges form a "hexagon."  And, this solid was associated with the "Earth" for some reason. Now, when this solid is constructed as a frame, without welding or solder, it is very unstable.  The "Earth" is a very unstable thing because of its volcanoes and earthquakes.
TETRAHEDRON:
The word "Tetra" is a Greek word and it has something to do with the number "four." Now, with a tetrahedron you have an equilateral triangle as its base. Then you place an equilateral triangle at each edge of the base then they all come up to a point, apex, or vertex if you will. So, with "tetra" meaning "four", you have four surfaces. The tetrahedron was also associated somehow with "Fire" in Plato's day.

5. EXTERIOR ANGLES
Exterior Angle
These are the Five Platonic Solids   Now, each of the solids have planes or flat surfaces.  These flat surfaces are called faces but I do not like to word face as others apply it.  I like the word surface better because when I think of the word face I picture that an object has a front and a back with the front being the face.  When you rotate or turn a flat surface on a polyhedra perpendicular to your nose, that becomes its new "face" therfore its new front.   So, I make a feeble effort to try to eliminate this confusion by using the word surface.  Even though I use the words point, apex, and vertex redundantly here, actually a point on a geometric solid is called a vertex for the singular and vertices for the plural.
Exterior Angle: the angles formed by one side of a triangle and the extension of another side.

6. REMOTE INTERIOR ANGLES
These are the Five Platonic Solids   Now, each of the solids have planes or flat surfaces.  These flat surfaces are called faces but I do not like to word face as others apply it.  I like the word surface better because when I think of the word face I picture that an object has a front and a back with the front being the face.  When you rotate or turn a flat surface on a polyhedra perpendicular to your nose, that becomes its new "face" therfore its new front.   So, I make a feeble effort to try to eliminate this confusion by using the word surface.  Even though I use the words point, apex, and vertex redundantly here, actually a point on a geometric solid is called a vertex for the singular and vertices for the plural.
Remote InteriorAngles: the interior angles of the triangle NOT adjacent to a given exterior angle.

7. THM 4.3-EXTERIOR ANGLE THEOREM
The measure of an exterior angle is equal to the sum of the measures of the two remote interior angles.
HEXAHEDRON: This is your standard cube, but "Hex" comes from the Greek meaning "six."  About the only fascinating thing about this solid is that when you rotate it to where a point, apex, or vertex is perpendicular to your nose, you should notice that the outer edges form a "hexagon."  And, this solid was associated with the "Earth" for some reason. Now, when this solid is constructed as a frame, without welding or solder, it is very unstable.  The "Earth" is a very unstable thing because of its volcanoes and earthquakes.
TETRAHEDRON:
The word "Tetra" is a Greek word and it has something to do with the number "four." Now, with a tetrahedron you have an equilateral triangle as its base. Then you place an equilateral triangle at each edge of the base then they all come up to a point, apex, or vertex if you will. So, with "tetra" meaning "four", you have four surfaces. The tetrahedron was also associated somehow with "Fire" in Plato's day.

8. EXAMPLE-EXTERIOR ANGLE THEOREM
Find the measure of the exterior angle x
Step 1
Use Exterior Angle Theorem to write Equation.
Step 2
Use Algebra to Solve Equation x
20°
80°
HEXAHEDRON: This is your standard cube, but "Hex" comes from the Greek meaning "six."  About the only fascinating thing about this solid is that when you rotate it to where a point, apex, or vertex is perpendicular to your nose, you should notice that the outer edges form a "hexagon."  And, this solid was associated with the "Earth" for some reason. Now, when this solid is constructed as a frame, without welding or solder, it is very unstable.  The "Earth" is a very unstable thing because of its volcanoes and earthquakes.
TETRAHEDRON:
The word "Tetra" is a Greek word and it has something to do with the number "four." Now, with a tetrahedron you have an equilateral triangle as its base. Then you place an equilateral triangle at each edge of the base then they all come up to a point, apex, or vertex if you will. So, with "tetra" meaning "four", you have four surfaces. The tetrahedron was also associated somehow with "Fire" in Plato's day.

9. The End Homework Worksheet Page 192: 1-5 ALL
Worksheet Page 193: 1(a-d)
A STUDY IN SOLID GEOMETRY Polyhedra