Chapter Summary

This chapter discusses triangle congruence theorems and their applications. Two triangles are congruent if the following parts of one are congruent to corresponding parts of the other: SSS (three sides), SAS (two sides and the included angle), ASA (two angles and the included side), AAS (two angles and a non-included side), SsA (two sides and the angle opposite the larger side), and HL (in a right triangle, hypotenuse and leg).

These theorems also indicate the conditions sufficient to determine a unique triangle. For instance, the SSS congruence theorem indicates that all triangles with three given sides lengths are congruent.

From the triangle congruence theorems we deduced a variety of theorems about geometric figures.

    1. if a triangle has two congruent angles, then it has tow congruent sides.
    2. Parallelograms have 2-fold rotation symmetry, both pairs of opposite sides and opposite angles are congruent and their diagonals have the same midpoint.
    3. If a quadrilateral has two pairs of congruent opposite sides, two pairs of congruent opposite angles, one pair of congruent and parallel opposite sides, or diagonals with the same midpoint, then the quadrilateral is a parallelogram

When congruent copies of the same figure cover the plane without gaps or overlapping, a tessellation is formed. The figure is a fundamental region for the tessellation. Tessellation’s are important in manufacturing and building, because if a figure tessellates, then copies may be cut out from a larger piece without waste. Any triangle or quadrilateral can be a fundamental region, but not all pentagons can. No one knows if we have found all the pentagons, which tessellate, it is an unsolved problem in mathematics.

It is also useful to know when figures are not congruent. The measure of an exterior and of a triangle equals the sum of the measures of the nonadjacent interiors. So an exterior angle has a measure greater than that of either nonadjacent interior angle. From this theorem, we proved that in a triangle, larger angles are opposite longer sides, and longer sides are opposite larger angles.

 

 

 

Ryan Brumback

Chapter Vocabulary

 

Section 7-1

Sufficient condition

Section 7-2

SSS, SAS, ASA, and AAS congruence theorems

Included angle

Included side

Section 7-3

Isosceles Triangle Base Angles Converse theorem

Section 7-4

Overlapping figures

Nonoverlapping figures

Section 7-5

SSA condition

Legs, hypotenuse

HL condition

HL congruence theorem

SsA congruence theorem

Section 7-6

Tessellation, tessellates

Fundamental region

Section 7-7

Properties of a Parallelogram theorem

Parallelogram Symmetry theorem

Section 7-8

Sufficient conditions for a Parallelogram theorem

Section 7-9

Exterior angle

Interior angle

Exterior angle theorem

Exterior angle inequality

Unequal sides theorem

Unequal angles theorem

Trisect

Ryan Brumback