This chapter establishes properties of angles formed when lines intersect, with special focus on parallel lines cut by a transversal.
- Basic Terms:
- Line segment: A part of a line with two endpoints.
- Ray: A part of a line with one endpoint, extending infinitely in one direction.
- Angle: Formed by two rays sharing a common endpoint (vertex).
- Complementary angles: Sum = 90 degrees.
- Supplementary angles: Sum = 180 degrees.
- Adjacent angles: Share a common vertex and arm, with no overlap.
- Linear pair: Adjacent angles whose non-common arms form a straight line; sum = 180 degrees.
- Vertically opposite angles: Formed when two lines intersect; they are equal.
- Transversal and Parallel Lines:
- When a transversal cuts two parallel lines:
- Corresponding angles are equal (F-shape).
- Alternate interior angles are equal (Z-shape).
- Co-interior (same-side interior) angles are supplementary (sum = 180 degrees).
- Key Theorems:
- The sum of angles in a triangle = 180 degrees.
- An exterior angle of a triangle = sum of two non-adjacent interior angles.
Two angles are supplementary. One is 65 degrees. Find the other.
Supplementary means sum = 180. Other angle = 180 - 65 = 115 degrees.
Two lines intersect and one angle is 40 degrees. Find all four angles.
The vertically opposite angle = 40 degrees.
The adjacent angles form a linear pair: 180 - 40 = 140 degrees each.
Four angles: 40, 140, 40, 140 degrees.
A transversal cuts two parallel lines. One corresponding angle is 75 degrees. Find the alternate interior angle on the same transversal.
Corresponding angles are equal (75 degrees). Alternate interior angles are also equal to corresponding angles with the parallel line, so alternate interior angle = 75 degrees.
In a triangle, two angles are 60 degrees and 70 degrees. Find the third.
Third angle = 180 - 60 - 70 = 50 degrees.
An exterior angle of a triangle is 110 degrees, and one non-adjacent interior angle is 45 degrees. Find the other non-adjacent interior angle.
Other = 110 - 45 = 65 degrees.
Lines AB and CD are parallel. A transversal makes co-interior angles of (3x + 10) and (2x + 20) degrees. Find x.
Co-interior angles are supplementary: (3x + 10) + (2x + 20) = 180.
5x + 30 = 180 → 5x = 150 → x = 30.
Two lines meet at a point. If one angle is (2y + 15) and the vertically opposite angle is (3y - 10), find y.
Vertically opposite angles are equal: 2y + 15 = 3y - 10 → y = 25.
Common mistakes
Students mix up alternate interior and co-interior angles. Alternate interior angles (Z-pattern) are equal; co-interior angles (C-pattern) are supplementary. Also, vertically opposite angles are only equal when two straight lines intersect — not for any two angles at a point.
Summary
Pairs of angles (complementary, supplementary, linear pair, vertically opposite) follow specific rules. Parallel lines and transversals generate equal corresponding and alternate angles, and supplementary co-interior angles. These form the toolkit for all angle-based proofs.