### 6. Parallel Lines and Similar and Congruent Triangles

Theorems 6.1, 2,
3, 4, 5,
6, 7, 8,
9, 10, 11, 12, 13.

**Theorem 6.1**: If two
parallel lines are transected by a third, the alternate interior
angles are the same size.

**Theorem 6.2**: If a
line intersects two other lines then the following conditions are
equivalent.

- a) The alternate interior angles are the same size.
- b) The corresponding angles are the same size
- c) The opposite interior angles are supplementary.

**Theorem 6.3**: The
measures of the angles in a triangle always add up to* *180^{o}.

**Theorem 6.4**: If two
lines are crossed by a third, then the following conditions are
equivalent.

- a) The alternate interior angles are the same size
- b) The corresponding angles are the same size
- c) The opposite interior angles are supplementary.
- d) The two lines are parallel.
^{1}

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**Theorem 6.5**: Two line
segments are congruent if and only if they have the same length.

**Theorem 6.6**: Two
angles are congruent if and only if they have the same size.

**Theorem 6.7 Side-Side-Side:** (SSS)
Two triangles are congruent if and only if their corresponding sides all have the same lengths.^{2}

**Theorem 6.8 Side-Angle-Side:** (SAS)
Two triangles are congruent if and only if two sides and the angle between them in one triangle are congruent to the two sides and the angle between them in the other triangle.

**Theorem 6.9**: Two
triangles are similar if and only if the ratios of their
corresponding sides are all the same.^{3}

**Theorem 6.10: Angle-Side-Angle:** (ASA)
Two triangles are congruent if and only if two angles and the side between them in one triangle are congruent to two angles and the side between them in a second triangle, then the triangles are
congruent.^{4}

**Theorem 6.11: Angle-Angle-Side:** (AAS)
Two triangles are congruent if and only if two
angles and a side adjacent to one of them in one triangle are congruent to the corresponding two angles and side in a second triangle, then the triangles are
congruent.^{5}

**Theorem 6.12: Hupoteneuse-Side:** (HS) Two right triangles are congruent if and only if their hypoteneuses and one other side are congruent.

**Theorem 6.13:** Corresponding parts of congruent triangles are congruent.^{6}

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Analytic Foundations of Geometry

R. S. Wilson