Definition 2.1.1. Vertical Angles.
The opposing angles formed by the intersection of two lines are called vertical angles.
Section 2.1 Triangles
Subsection 2.1.1 Basic Triangle Theorems
Note all theorems in this section can and should be proved without using the parallel postulate.
Definition 2.1.2. Congruent Angles.
Two angles are congruent (\(\angle ABC \cong \angle DEF\)) if and only if their measures are equal (\(m\angle ABC = m\angle DEF\)).
Theorem 2.1.3. Vertical Angle Congruence.
Vertical angles are congruent.
\(A-B-C\) means that the points \(A,B,\) and \(C\) are colinear and \(B\) is between \(A\) and \(C.\)
Theorem 2.1.4. Pasch’s Axiom.
If a line \(\ell\) intersects a triangle \(\triangle ABC\) at a point \(D\) such that \(A-D-B,\) then \(\ell\) must intersect \(\overline{AC}\) or \(\overline{BC}.\)
Theorem 2.1.5. Crossbar.
If \(X\) is a point in the interior of \(\triangle ABC\) then ray \(AX\) intersects \(\overline{BC}\) at a point \(D\) such that \(B-D-C.\)
Definition 2.1.6. Congruent Line Segments.
Two line segments are congruent (\(\overline{AB} \cong \overline{CD}\)) if and only if their measures (length) are equal (\(|AB|=|CD|\)).
Definition 2.1.7. Isosceles.
A triangle is isosceles if and only if two sides are congruent.
Theorem 2.1.8. Isosceles Triangle.
In an isosceles triangle the angles opposite the equal sides are congruent.
Theorem 2.1.9. Perpendicular Bisector.
A point is on the perpendicular bisector of a segment if and only if it is equidistant from the endpoints.
Definition 2.1.10. Exterior Angle.
The supplementary angle formed by extending one side of a triangle is called an exterior angle.
Theorem 2.1.11. Exterior Angle.
The measure of an exterior angle of a triangle is greater than the measure of either of the opposing angles of the triangle.
Definition 2.1.12. Congruent Triangles.
Two triangles are congruent if and only if all their sides and angles are congruent (\(\Delta ABC \cong \Delta DEF\)).
Subsection 2.1.2 Triangle Congruence Theorems
Checkpoint 2.1.13.
Determine if two triangles with two congruent sides and a congruent angle not between the two sides are congruent.
Theorem 2.1.14. Angle-Side-Angle.
Two triangles are congruent if and only if two corresponding angles and the side between them are congruent.
Theorem 2.1.15. Angle-Angle-Side.
Two triangles are congruent if and only if two corresponding angles and a side not between them are congruent.
Theorem 2.1.16. Side-Side-Side.
Two triangles are congruent if and only if all three corresponding sides are congruent.
Theorem 2.1.17. Right Angle-Side-Side.
Two right triangles are congruent if and only if two corresponding sides and a right angle not between those sides are congruent.
Theorem 2.1.18. Converse of Isosceles Triangle.
If two angles of a triangle are congruent then the sides opposite those angles are congruent.
Theorem 2.1.19. Extended Inverse of Isosceles Triangle.
If two sides of a triangle are not congruent then the angles opposite those sides are not congruent. Further the larger angle is opposite the longer side.
Theorem 2.1.20. Triangle Inequality.
The sum of the lengths of any two sides of a triangle is larger than the length of the other side.
