C-17 Triangle Sum Conjecture
The sum of the angles in a triangle is180º.
Class investigations involved drawing triangles and measuring each angle, or cutting off the corners and lining them up to make a straight line. This conjecture can be proven by constructing a line parallel to the base of the triangle through the 3rd vertex, and using alternate interior angles to show that the three angles must add up to 180º.
C-18 Third Angle Conjecture
C-18 Third Angle Conjecture
If two angles of one triangle have the same measure as two angles of another triangle, then the third angles in each triangle also have the same measure.
The third angle conjecture is easily proven using algebra.
HW: WS 4.1 All
HW: WS 4.1 All
Definition of terms related to isosceles triangles
- The congruent sides are the
legs
- The remaining side is called the base
- The vertex angle is the angle between the congruent sides
- The base angles are between the base and the congruent sides
- The remaining side is called the base
- The vertex angle is the angle between the congruent sides
- The base angles are between the base and the congruent sides
C-19 Isosceles Triangle Conjecture
If a triangle is isosceles, then its base angles are congruent.
C-20 Converse of the Isosceles Triangle Conjecture
If a triangle has two congruent angles, then it is an isosceles triangle.
HW TE
4.2 p206-208 1-7, 10, 11, 4.1 p201-202 4-9
HW p218 19, 20
HW p218 19, 20
C-21 Triangle
Inequality
Conjecture
In any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.
C-22
Side-Angle Inequality Conjecture
If one side of a triangle is longer than another side, then the measure of the angle opposite the longer side is greater than the measure of the angle opposite the other side.
Definitions of terms related to triangles
- An exterior angle is the angle between the side of a triangle and a line extending through the base.
- The adjacent interior angle is the angle in the triangle that forms a linear pair with the exterior angle.
- The remote interior angles are the angles in the triangle that are not adjacent to the exterior angle.
- The adjacent interior angle is the angle in the triangle that forms a linear pair with the exterior angle.
- The remote interior angles are the angles in the triangle that are not adjacent to the exterior angle.
C-23 Triangle Exterior
Angle Conjecture
The measure of the exterior angle of a triangle is equal to the sum of the measures of the remote interior angles.
HW WS 4.3 1-18
Two triangles are congruent if their corresponding sides and angles are congruent.
Do we need to compare all six parts to establish congruence, or is there some reduced set of measures that can be used?
It is obvious that a single matching side or angle are not sufficient.
It can easily be shown that a pair of matching sides, angles, or one of each are also insufficient.
The investigations in this section will look at combinations of three measurements to see which, if any can be used as shortcuts to show congruence. The possibilities are SSS, SAS, SSA, ASA, AAS, and AAA.
An included angle is the angle at the vertex between adjacent sides.
An included side is the side shared by adjacent angles.
C-24 SSS Congruence Conjecture
Do we need to compare all six parts to establish congruence, or is there some reduced set of measures that can be used?
It is obvious that a single matching side or angle are not sufficient.
It can easily be shown that a pair of matching sides, angles, or one of each are also insufficient.
The investigations in this section will look at combinations of three measurements to see which, if any can be used as shortcuts to show congruence. The possibilities are SSS, SAS, SSA, ASA, AAS, and AAA.
An included angle is the angle at the vertex between adjacent sides.
An included side is the side shared by adjacent angles.
C-24 SSS Congruence Conjecture
If the measures of three sides of one triangle are equal to the measures of three sides of a second triangle, then
the two triangles are congruent.
C-25 SAS Congruence Conjecture
If the measures of two sides and the included angle of one triangle are equal to the measures of two sides and the included angle of a second triangle, then
the two triangles are congruent.
SSA is not a congruence shortcut because it is possible to draw a counter-example for two sides and a non-included angle.
HW TE 4.4 p222-223 1-6, 8-17, 21
C-26 ASA Congruence Conjecture
If the measures of two angles and the included side of one triangle are equal to the measures of two angles and the included side of a second triangle, then
the two triangles are congruent.
C-27 SAA Congruence Conjecture
If the measures of two angles and a non-included side of one triangle are equal to the measures of two angles and the corresponding non-included side of a second triangle, then
the two triangles are congruent.
Note that AAS follows directly from ASA and the Third Angle Conjecture because if two angles in a triangle are congruent, then the third angle must also be congruent, leading to congruent triangles by the ASA Conjecture.
AAA is not a congruence shortcut because it is possible to draw a counter-example for two triangles with the same three angles but different side lengths.
HW p227-228 1-15
AAA is not a congruence shortcut because it is possible to draw a counter-example for two triangles with the same three angles but different side lengths.
HW p227-228 1-15
If two triangles are congruent, their corresponding parts are congruent.
The acronym CPCTC, stands for Corresponding Parts of Congruent Triangles are Congruent.
CPCTC will commonly be used to prove that two angles or segments of two different triangles are congruent.
An auxiliary line can sometimes be drawn on a figure to create two congruent triangles if there are no apparent congruent triangles in the initial figure.
HW p231-233 1-9
The acronym CPCTC, stands for Corresponding Parts of Congruent Triangles are Congruent.
CPCTC will commonly be used to prove that two angles or segments of two different triangles are congruent.
An auxiliary line can sometimes be drawn on a figure to create two congruent triangles if there are no apparent congruent triangles in the initial figure.
HW p231-233 1-9
4.7 More Proofs (Book section Flowchart Thinking)
This class will not be doing flowchart proofs, concentrating
instead on two column proofs.
The homework from this section includes six proofs that are to be done as two column proofs. The flowchart proofs in the book may be helpful as a guide if you get stuck, but try to do them on your own first.
HW Worksheet on 4.7 1-9 (TE p245-246)
The homework from this section includes six proofs that are to be done as two column proofs. The flowchart proofs in the book may be helpful as a guide if you get stuck, but try to do them on your own first.
HW Worksheet on 4.7 1-9 (TE p245-246)
In a scalene triangle, the altitude, median, and angle bisector are three distinct segments.
The investigation and proofs in this section show that these three segments become one in an isosceles triangle.
The vertex angle bisector is a line of symmetry for an isosceles triangle.
C-28 Vertex Angle Bisector Conjecture
The investigation and proofs in this section show that these three segments become one in an isosceles triangle.
The vertex angle bisector is a line of symmetry for an isosceles triangle.
C-28 Vertex Angle Bisector Conjecture
In an isosceles triangle, the bisector of the vertex angle is also
the altitude and the median to the base.
C-29 Equilateral/Equiangular Triangle Conjecture
Every equilateral triangle is equiangular, and conversely, every equiangular triangle is equilateral.
HW p249-251
10-32 Even (Worksheet)
Chapter Test
Wednesday January 11 (B) and Thursday
January 12 (G)