McGraw Hill Reveal Geometry Chapter 5

interior angle of a triangle

the angle at a vertex inside the triangle

Theorem 5.1 Triangle Angle-Sum Theorem

The sum of the measures of the interior angles of a triangle is 180°.

Exterior angle of a triangle

is formed by one side of the triangle and the extension of an adjacent side.

remote interior angles

the angles of a triangle that are not adjacent to a given exterior angle

Theorem 5.2: Exterior Angle Theorem

The measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote interior (nonadjacent interior angles).

corollary

a theorem with a proof that follows as a direct result of another theorem.

Corollary 5.1: Corollary to the Triangle Sum Theorem

The acute angles of a right triangle are complementary.

Corollary 5.2 Corollary to the Base Angles Theorem

There can be at most one right or obtuse angle in a triangle.

principle of superposition

states that two figures are congruent if and only if there is a rigid motion or series of rigid motions that maps one figure exactly onto the other

congruent polygons

all the parts of one polygon are congruent to the corresponding parts

corresponding parts

matching parts of congruent polygons

Congruent Triangles

2 triangles are congruent if and only if all pairs of corresponding sides and angles are congruent

Theorem 5.3: Third Angles Theorem

If two angles of one triangle are congruent to two angles of a second triangle, then the third angles of the triangles are congruent.

Theorem 5.4: Properties of Triangle Congruence Theorem

if two angles in one triangle are congruent to two angles in another triangle, then their third angles are also congruent.

Reflexive Property of Triangle Congruence

△ABC ≅ △ABC

Symmetric Property of Triangle Congruence

If △ABC ≅ △EFG, then △EFG ≅ △ABC

Transitive Property of Triangle Congruence

If △ABC ≅ △EFG and △EFG ≅ △JKL, then △ABC ≅ △JKL

Postulate 5.1: Side-Side-Side (SSS) Congruence

If three sides of one triangle are congruent to three sides of a second triangle, then the triangles are congruent.

included angle.

The interior angle formed by two adjacent sides of a triangle

Postulate 5.2: Side-Angle-Side (SAS) Congruence

If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent.

included side

the side of a triangle between two angles.

Postulate 5.3: Angle-Side-Angle (ASA) Congruence

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.

Theorem 5.5: Angle-Angle-Side (AAS) Congruence

If two angles and the nonincluded side of one triangle are congruent to the corresponding two angles and nonincluded side of a second triangle, then the two triangles are congruent.

Theorem 5.6: Leg-Leg (LL) Congruence

If the legs of one right triangle are congruent to the corresponding legs of another right triangle, then the triangles are congruent.

Theorem 5.7: Hypotenuse-Angle (HA) Congruence

If the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and the corresponding acute angle of another right triangle, then the triangles are congruent.

Theorem 5.8: Leg-Angle (LA) Congruence

If one 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.

Theorem 5.9: Hypotenuse-Leg (HL) Congruence

If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and the corresponding leg of another right triangle, then the triangles are congruent.

isosceles triangle

a triangle with at least two sides congruent.

vertex angle of an isosceles triangle

The angle formed by the legs of an isosceles triangle.

base angles of an isosceles triangle

the angles opposite the legs of an isosceles triangle.

Theorem 5.10: Isosceles Triangle Theorem

If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

Theorem 5.11: Converse of the Isosceles Triangle Theorem

If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

auxiliary line

An extra line or segment drawn in a figure to help complete a proof

Corollary 5.3 Equilateral Triangles

A triangle is equilateral if and only if it is equiangular.

Corollary 5.4 Equilateral Triangles

Each angle of an equilateral triangle measures 60 degrees

coordinate proof

uses figures in the coordinate plane and algebra to prove geometric concepts

Coordinate Proof Step 1

Place the figure on the coordinate plane

Coordinate Proof Step 2

Label the coordinates of the vertices of the figure

Coordinate Proof Step 3

Use algebra to prove properties or theorems