Geometry Big Ideas Math Chapter 5

Study Notes

Triangle Types by Sides

Scalene Triangle: No sides are congruent.
Isosceles Triangle: Two sides are congruent.
Equilateral Triangle: All three sides are congruent.

Triangle Types by Angles

Obtuse Triangle: Contains one angle greater than 90°.
Right Triangle: Contains one angle equal to 90°.
Acute Triangle: All three angles are less than 90°.
Equiangular Triangle: All three angles are exactly 60°.

Classifying Triangles

Measure distances of all sides using the distance formula.
Keep distances in square root format for accuracy.
Classify as:
- Scalene: All sides different.
- Isosceles: Two sides the same.
- Equilateral: All three sides the same.
Use the Pythagorean theorem for right triangle classification:
- Right Triangle: ( leg^2 + leg^2 = hypotenuse^2 )
- Acute Triangle: ( leg^2 + leg^2 > hypotenuse^2 )
- Obtuse Triangle: ( leg^2 + leg^2 < hypotenuse^2 )

Key Triangle Theorems

Triangle Sum Theorem: The interior angles of a triangle sum to 180°.
Exterior Angles Theorem: An exterior angle equals the sum of two non-adjacent interior angles.
Third Angles Theorem: If two triangles have two congruent angles, the third angles are also congruent.

Triangle Congruence Criteria

S.A.S. (Side-Angle-Side): Two sides and included angle are congruent.
S.S.S. (Side-Side-Side): All three sides are congruent.
Hypotenuse-Leg: Hypotenuse and one leg are congruent in right triangles.
A.S.A. (Angle-Side-Angle): Two angles and included side are congruent.
A.A.S. (Angle-Angle-Side): Two angles and a non-included side are congruent.

Non-allowed Congruence Functions

A.S.S. and S.S.A. configurations are not valid for proving triangle congruence.

Angle Types Within Triangles

Base Angles: Angles connected to both a leg and the base of the triangle.
Vertex Angle: Angle formed by the two legs of the triangle.

Base Angles Theorem and Converse

Base Angles Theorem: If two sides are congruent, then the angles opposite those sides (base angles) are also congruent.
Base Angles Theorem Converse: If the two base angles are congruent, then the sides opposite those angles are congruent.

Congruence of Polygons

If all corresponding parts (sides and angles) of polygons are congruent, then the polygons themselves are congruent.

Medians and Altitudes of Triangles

Median: Segment from a vertex to the midpoint of the opposite side.
Altitude: Segment from a vertex to the opposite side, forming right angles with that side.

Auxiliary Lines

Lines, rays, or segments not in the original triangle figure, added for geometric proof or analysis.

Description

Test your understanding of triangle properties with this quiz based on Chapter 5 of the Big Ideas Math curriculum. Focus on the names and classifications of triangle sides and angles, including scalene, isosceles, equilateral, obtuse, right, and acute triangles.