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This section examines some essential angle relationships in the triangle. | |
Triangle Angle Sum:
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Theorem 4-1 Angle Sum Theorem |
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The sum of the measures of the angles of a triangle is 180.
 A + B + C = 180 |
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| The above diagram attempts to show how the three angles of a triangle add up to 180 degrees.
The animation demonstrates how if you place all three angles together they create a straight line or angle. A straight
angle is exactly 180 degrees. A more formal proof is provided in the book. |
This one idea will be used literally hundreds of times over the year. Know this fact well!!!!
Third Angle Congruence:
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Theorem 4-2 Third Angle Theorem |
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If two angles of one triangle are congruent to two angles of a second triangle,
then the third angles of the triangles are congruent. |
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| The basis of this proof is that if both triangles total 180 degress, and both
triangles have two identical angles... then the third angle would be the same remainder. Third angle congruence shows
up throughout many units.. remember it. |
Triangle Exterior Angles & its Corollaries:
Theorem 4-3  Exterior Angle Theorem - The measure of an exterior angle of a trianlge is equal to
the sum of the measures of the
two remote interior angles. In the figure to the left, BCD is an exterior angle of triangle ABC. An EXTERIOR ANGLES is formed by one side of a
triangle and the extension of another side. The interior angles of the triangle not adjacent to a given exterior angle are
called REMOTE INTERIOR ANGLE. Thus in the above case, A and B are the remote angles to the exterior BCD.
Corollary 4-1 -  The acute angles of a right triangle are complementary.
Corollary 4-2 - There can be at most one right or obtuse angle in a triangle.
By the way.... what is a corollary? A COROLLARY is a statement that can be easily
proved using a theorem..... A better way of saying this... is that a corollary is a fact or statement that directly
falls from a given theorem.
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This section looks at the important properties of the Isosceles Triangle. | |
Analyzing Isosceles Triangles:
Earlier we defined an Isosceles Triangles as a triangle that had at least 2 congruent sides. These
two congruent sides help us to prove/show another very important property of the Isosceles Triangle.
| Theorem 4-6 Isosceles Triangle Theorem (ITT) |
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If two sides of a triangle are congruent, then the angles opposite those
sides are congruent.
Summary - In other words if you have two congruent sides, you have two congruent base angles. |
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| Theorem 4-7 Converse of the ITT |
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If two angles of a triangle are congruent, then the sides opposite those angles
are congruent.
Summary - If you have two congruent angles, then you have two congruent legs.
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| The ITT & its converse are used in just about everything because
its information is hidden unless you notice equal sides or equal angles.... keep your eyes open!! |
Corollary 4-3 - A triangle is equilateral if and only if it is equiangular.
Corollary 4-4 - Each angle of an equilateral triangle measures 60 degrees.
Both of the Corollaries follow directly from theorem 4-6 & 4-7.. they are quite easy proofs. Try them!!
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This section looks at what it means for two triangles to be congruent. | |
Congruent Trinagles:
When two triangles are congruent two each  other then......
When two triangle are congruent there are SIX pieces of information that come from that statement. 3 congruent
corresponding sides and 3 congruent corresponding angles. The next couple of sections show us how the prove two triangles
to be congruent..... thus giving allowing us to know that these 6 facts are then true.
Definition of Congruent Triangles (CPCTC) -
Two triangles are congruent if and only if their corresponding parts are congruent.
If triangle ABC  triangle DEF then
If there is no diagram..... there is a way to determine what segments and angle correspond. Notice that in
triangle ABC,  corresponds in triangle DEF, to  . The order that the letters come in always display the corresponding relationship. Thus if our two congruent
triangles were DRG and HIK, then segment RG would correspond to segment IK or angle G would correspond to angle K.
Theorem 4-4 - Congruence of triangles is reflexive, symmetric, and transitive.
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These sections explain the minimum requirements for triangle congruence SSS, SAS, ASA
& AAS. | |
Summary Notes
Triangle Congruence Postulates:
Proving two triangles to be congruent does not require you to determine that all six parts of the triangles
are congruent (3 angles & 3 sides). There are a few short cuts.... each short cut only requires you to find a particular
three things.... and if you do... you prove congruence.
Postulate 4-1 SSS (Side - Side - Side) - If the sides of
one triangle are congruent to the sides of a second triangle, then the triangles are congruent.
Postulate 4-2 SAS (Side - Included Angle - Side) - If two sides and the
INCLUDED angle of one triangle are congruent to two sides and the INCLUDED angle of another triangle, then the triangles are
congruent.
Postulate 4-3 ASA (Angle - Included Side - Angle) - If two angles
and the INCLUDED side of one triangle are congruent to two angles and the INCLUDED side of another triangle, then the triangles
are congruent.
Postulate 4-4 AAS (Angle - Angle - Side) - If two angles and a NON-INCLUDED side of one triangle are congruent to the corresponding two angles and side of a second
triangle, the two triangles are congruent.
Hint - Summary - This is a modification of ASA.... because if you have two corresponding congruent angles,
then the third angles are also congruent. Therefore if you have the information for ASA... you can find the third angle
and solve by AAS. In the same way if you have the information for AAS... you can find the third angle and solve by ASA.
(AAS & ASA are to be used when the given information lends itself to one method or the other.)
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This section introduces new congruence postulates for triangles for the specific case
- the right triangle. | |
Congruence of Right Triangles:
Congruence of Right Triangles is a quick process because only two pieces of informatin are needed to
prove congruence. This occurs because we already know that one pair of corresponding angles are congruent.... the right
angles. Some of these theorems are redundant. We will find that LL is the same as SAS, and LA is the same as ASA
or AAS, and that HA is the same as AAS. The only NEW congruence postualte is HL.
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Theorem 5-5 LL (Leg - Leg) |
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If the legs of one right triangle are congruent to the corresponding legs of another right triangle,
then the triangles are congruent. |
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Hint |
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LL (Leg - Leg) is the same as SAS (Side - Right Angle - Side). This theorem is redundant!! |
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Theorem 5-6 HA (Hypotenuse - Angle) |
| If the hypotenuse and an acute angle of one right triangle
are congruent to the hypotenuse and corresponding acute angle of another right triangle, then the two triangles are congruent. |
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| Hint |
| HA (Hypotenuse - Angle) is the same as AAS (Angle - Angle - Side). This
theorem is redundant!! |
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Theorem 5-7 LA (Leg - Angle) |
| If the leg and an acute angle of one right triangle are congruent
to the corresponding leg and acute angle of another right triangle, then the triangles are congruent. |
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| Hint |
| LA (Leg - Angle) is the same as AAS and ASA depending on the information given.
This theorem is also redundant!! |
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Postulate 5-1 HL (Hypotenuse -Leg) |
| If the hypotenuse and a leg of one right triangle are congruent
to the hypotenuse and corresponding leg of another right triangle, then the triangles are congruent. |
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| Hint |
| HL (Hypotenuse - Leg) is not like any of the previous congruence postulates...
actually if it was given a name it would be ASS or SSA and earlier we found that this was NOT a congruence postulate.
HL works ONLY BECAUSE IT IS A RIGHT TRIANGLE!!!!! |
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These section explains inequality relationships in triangles and the method of indirect
proof. | |
Summary Notes for Textbook Section 5-3
Indirect Proof:
You have used direct reasoning in the proofs you have encountered up to this point. When using
direct reasoning, you start with a true hypothesis and prove that the conclusion is true. With indirect reasoning, you
assume that the conclusion is false and then show that this assumption leads to a contradiction of the hypothesis or some
other accepted fact, like a postulate, theorem, or corollary. Then, since you assumption has been proved false, the
conclusion must be true.
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Indirect Proof - Steps |
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Assume that the conclusion is false.
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Show that the assumption leads to a contradiction of the hypothesis or some other fact, such as a postulate,
theorem, or corollary.
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Point out that the assumption must be false and, therefore, the conclusion must be true. |
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Theorem 5-8 Exterior Angle Inequality Theorem |
| If an angle is an exterior angle of a triangle, then its measure is greater
than the measure of either of its corresponding remote interior angles. |
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| Hint |
| Obviously by teaching you indirect proofs... this is the method to
use to prove this theorem. Also the major idea comes from the exterior angle theorem stated in an earlier chapter...
find it. |
| Definition of Inequality |
| For any real numbers a and b, a > b if and only if there is a
positive number c such that a = b + c. |
Summary Notes for Textbook Section 5-4
Side & Angle Inequalities:
| Theorem 5-9 |
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If one side of a triangle is longer than another side, then the angle opposite the longer side has a
greater measure than the angle opposite the shorter side. |
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| Theorem 5-10 |
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If one angle of a triangle has a greater measure than another angle, then the side opposite the greater
angle is longer than the side opposite the lesser angle. |
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| Theorem 5-11 |
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The perpendicular segment from a point to a line is the shortest segment from the point to the line. |
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| Theorem 5-12 |
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The perpendicular segment from a point to a plane is the shortest segment from the point to the plane. |
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This section introduces the triangle inequality theorem. What are the side requirements
to form a triangle? | |
Triangle Inequality Theorem:
The Triangle Inequality is a very common sense theorem. It basicaly says that you
need long enough sides so that they reach each other. The diagram below might be a bit confusing so what it carefully.
It is showing the growth of two segments until they meet. Before they meet no triangle is formed
Theorem 5-12 Triangle Inequality Theorem- The sum of the lengths of any two sides of a triangle is greater than
the length of the third side.
Hint - For example would sides of length 4, 5 and 6 form a triangle....?
- 4 + 5 > 6 , 4 + 6 > 5, 5 + 6 > 11
How about sides of length 4, 11, and 7
- 4 + 11 > 7, 11 + 7 > 4, 4 + 7 = 11 Thus it is not a triangle!!
A observant student mentioned last year that all you have to do is add the two smallest sides to see if it is larger
than the other.... Nicely Done!
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