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Transversal t intersects two parallel lines, a and b.
In geometry, a transversal line (abbreviated 'trans.')is a line that passes through two or more other coplanar lines at different points.
In Euclidean geometry if lines a and b are parallel, and line t intersects lines a and b, then corresponding angles formed by line t and the parallel lines are congruent.
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Alternate angles
A transversal line creates alternate angles. The alternate interior angles are on opposite sides of the transversal line and inside the lines being transversed. Alternate exterior angles are on opposite sides of the transversal line and outside the lines being transversed. For it to be a true transversal line, it must also intersect two parallel lines. This creates several angle patterns.
Theorems
There are at least eight geometrical theorems concerning transversals. They are as follows: Theorem 9 If a transversal intersects two parallel lines, then the alternate interior angles are congruent. Theorem 10 If a transversal intersects two parallel lines, then the corresponding angles are congruent. Theorem 11 If a transversal intersects two parallel lines, then the alternate exterior angles are congruent. Theorem 12 If a transversal intersects two parallel lines, then the interior angles on the same side of the transversal are supplementary.
The next four theorems are converses of the previous four theorems. Theorem 13 If a transversal intersects two lines so that alternate interior angles are congruent, then the lines are parallel. Theorem 14 If a transversal intersects two lines so that corresponding angles are congruent, then the lines are parallel. Theorem 15 If a transversal intersects two lines so that alternate exterior angles are congruent, then the lines are parallel. Theorem 16 If a transversal intersects two lies so that interior angles on the same side of the transversal are supplementary, then the lines are parallel.
Proofs
Theorem 9
Underlined words are the Statements; double underlined words are the Reasons.
Given: y ll z; transversal t intersects line y and z at A and B; O is the midpoint of line AB.
Prove: angle 1 is congruent to angle 3; angle 2 is congruent to angle 4.
1. y ll z; transversal intersects lines y and z at A and B; O is the midpoint of line AB. Given.
2. Through O, draw line CD perpendicular to z. In a plane, only one line can be drawn through a point perpendicular to a line. (P.8) (P. means Postulate)
3. Line CD is perpendicular to y. If a line is perpendicular to one of two ll lines, then it is perpendicular to the other also. (T.8) (T. means Theorem)
4. Angle ACO and BDO are right triangles. Definition of perpendicular lines.
5. ΔACO and BDO are right triangles. Definition of right triangles.
6. Angle 5 is congruent to angle 6. Vertical angles are congruent. (P.12)
7. Line AO is congruent to line BO. Definition of midpoint.
8. ∴ ΔACO is congruent to ΔBDO. HA Theorem. (T.4)
9. ∴ Angle 1 is congruent to angle 3. CPCTC. (P.18)
10. Angle 2 is supplementary to angle 1; angle 3 is supplementary to angle 4. If one line meets another line, the adjacent angles formed are always supplementary. (P.9)
11. ∴ Angle 2 is congruent to angle 4. Angles that are supplements of congruent angles are congruent. (P.11)
Theorem 10
Given: Transversal t intersects lines m and n; m ll n.
Prove: Angle 1 is congruent to angle 2.
1. Transversal t intersects lines m and n; m ll n. Given.
2. Angle 1 is congruent to angle 3. Vertical angles are congruent. (P.12)
3. Angle 3 is congruent to angle 2. If a transversal intersects two parallel, then the alternate interior angles are congruent. (T.9)
4. ∴ Angle 1 is congruent to angle 2. Trasitive Property.
Theorem 11
Given: Transversal t intersects lines m and n; m ll n.
Prove: Angle 1 is congruent to angle 7.
1. Transversal t intersects lines m and n; m ll n. Given.
2. Angle 3 is congruent to angle 5. If a transversal intersects two parallel lines, then the alternate itnerior angles are congruent. T.9
3. Angle 1 is congruent to angle 3; angle 5 is congruent to angle 7. Vertical angles are congruent. (P. 12)
4. ∴ Angle 1 is congruent to angle 7. Substitution Property
Theorem 12
Given: Transversal t intersects lines m' and n; m ll n.
Prove: Angle 1 is supplementary to angle 2.
1. Transversal t intersects lines m' and n; m ll n. Given.
2. Angle 2 is congruent to angle 3. If a transversal intersects two parallel lines, then the alternate interior angles are congruent.
3. Angl;e 1 is supplementary to angle '3. If one line meets another line, the adjacent angles form are always supplementary. (P.9)
4. ∴ Angle 1 is supplementary to angle 2. Substitution Property
Theorems 13-16 (Under Construction)
External links
- Transversal and Angles Interactive transversal with color coded angles (corresponding, alternate interior etc..)
- [http://www.mathopenref.com/transversal.html Definitiointeractive animation
- Corresponding angles associated with a transversal line With interactive animations
- Exterior angles associated with a transversal line With interactive animation
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