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Chapter 5
Triangles and Congruence
Section 5.1 Triangles and Congruence
Triangle – a three sided polygon
Classification of Triangles
By Sides
Scalene – not two sides are congruent
Isosceles – two sides are congruent
Equilateral – all sides are congruent
By Angles
Acute – all angles are acute
Obtuse – one obtuse angle
Right – one right angle
Can you have a triangle with 2 obtuse angles, or two right
Theorem 5.1 Angle-Sum Theorem for Triangles
The sum of the interior angles of a triangle is 180 degrees
Auxiliary Figure – a point, line, ray or segment that is added to a figure to aid in a
proof
Theorem 5.2 Exterior Angle Theorem for Triangles
The measure of an exterior angle of a triangle is equal to the sum of the
measures of the two remote interior angles
Section 5.2 Angles of Polygons
Divide the polygon into triangles; each triangle has 180 degrees, multiply number
of triangles by 180
Theorem 5.3 Angle Sum Theorem for Polygons
The sum of the measures of the angles of a convex polygon with n sides is
(n-2)*180
Corollary to a theorem is a theorem that follows easily from a previously proved
theorem
Corollary 5.4
The measure of each angle of a regular n-gon is
(n  2)180
n
Theorem 5.3 Exterior angle Theorem for a Polygon
The sum of the measures of the exterior angles of a convex polygon, one
angle at each vertex of the polygon is 360
Corollary 5.6
The measure of each exterior angle of a regular n-ogon is
360
n
Section 5.3 Tessellations
Go over ideas from section 4 about transformations in the plane
Talk about Tesselations
This depends on time otherwise wait until Christmas
Chapter 5 Start of Proofs
What does congruent and similar mean
Congruent Triangles
- idea of same shape and size
- More specifically corresponding sides are congruent and corresponding angles
are congruent
We are going to compare correspond sides and angles to try and prove the
triangles congruent
Corresponding Parts
- Sides and angles located in the same part of the polygon, triangle in this
chapter
- Could be congruent, this is what we are trying to prove
Naming congruent triangles
Make sure the congruent vertices are in the same order for the two
triangles
Look for a similar way to go around the triangle to be able to name them
in the same order
Postulate 17 Side Side Side Congruence Postulate SSS
If 3 sides of one triangle are congruent to 3 sides of another triangle then
the 2 triangles are congruent
Look for the markings to show that they are congruent
When angles are congruent
Vertical angles
The two triangles will form a bowtie
Angel bisector – cuts an angle into 2 congruent angles
Perpendicular – forms 4 congruent segments
Segments congruent
Midpoint – this is a point that divides a segment into 2 congruent
segments
Segment Bisector – cuts a segment into 2 congruent segments
Reflexive segments – side of one triangle shared with the side of another
triangle
Postulate 18 Side Angle Side Postulate SAS
If two sides and the included angle of one triangle are congruent to two
sides and the included angle of another triangle then the two triangles are
congruent
- Included angle is the angle formed by the two consecutive sides
Postulate 19 Angle Side Angle Postulate ASA
If two angles and the included side of one triangle are congruent to two
angles and the included side of another triangle, then the two triangles are
congruent
Proofs
These are used to prove objects congruent, triangles
Has a statement and a reason
Draw a large T statements are on the left and reasons are on the right
Statements
1. The given is always written first
2. These are stating sides or angles congruent or even supplementary
3. Final statement is stating the triangles congruent
Reasons
1. First reason is always the give
2. Reason should be no longer than 3 words, if you are starting to explain
something stop and see if you can sum it up int 2 to 3 words
a. Theorems
b. Postulates
c. Definitions
d. Properties
3. Ask yourself 3 questions to help come up with the reason
a. Why are these sides congruent
b. Why are the angles congruent
c. Why can I write this
4. Last reason is always one of the three postulates so far discussed, may be
more added later
When writing a proof
Try and mark either on the picture or the proof itself when you have sides
and angles congruent
It doesn’t matter the order in the proof that you prove the sides and angles
congruent
In the picture when you mark the sides and angles congruent the result
needs to be one of the 3 postulates so far discussed
In order to use sides or angles as congruent they must be stated congruent
someplace in the postulate
Corresponding Parts of Congruent Triangles are Congruent CPCTC
This is used to prove other parts of the triangles congruent once the
triangles are proven congruent themselves
This would then be the last part of the proof, always after the triangles are
stated congruent
Word problems and proofs
Setting up congruent triangles to help calculate measurements or distances
Use the idea of CPCTC to find the missing length you need
Give answers and state why that is the answer
Angle Angle Side Theorem
If 2 angles and a non-included side of one triangle are congruent to 2
angles and a non-included side of another triangle, then the two triangles
are congruent.
Prove this theorem
Why does this work
If you know two angles of a triangle can you figure out the third, so if 2 angles are
congruent then the third angle must be congruent as well
Right Triangles
Right Triangles have many special properties all because of the right angle
We will discuss some of the properties in this chapter and discuss more as the
year goes on
hypotenuse
leg
leg
Hypotenuse is always the longest side and is always the side across from the right
angle
Two other legs of triangle, not necessarily congruent just shorter than hypotenuse
The other two angles are smaller than 90 degrees
Other proofs we can use if they tell use it is a right triangle
We still need to state the angle is a right angle in the proof in order to use
these ideas
Not all theorems but can be proven
Hypotenuse Angle (HA) – if the hypotenuse and one angle or one triangle are
congruent to the hypotenuse and one angle of another triangle then the triangles
are congruent
Hypotenuse Leg (HL) – if the hypotenuse and one leg of one triangle are
congruent to the hypotenuse and one leg of another triangle then the two triangles
are congruent
Pythagoreans Theorem - a 2  b 2  c 2 , add the square of the two legs and
it should equal the square of the hypotenuse
Leg Leg (LL) – If the two legs of one triangle are congruent to the two legs of
another triangle then the two triangles are congruent
Leg Angle LA) – if one leg and one angle of one triangle are congruent to one leg
and one angle of another triangle then the two triangles are congruent
Overlapping Triangles
Overlapping triangles
Now we will have reflexive angles as well as reflexive sides
Make sure you know what two triangles you are trying to probe congruent
Maybe even draw them separately and label them again