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Geometry Unit 2: Parallel lines & polygon angles

When a transversal crosses parallel lines it creates predictable angle pairs. These lead to the angle-sum facts for triangles and polygons.

Covers: Transversals, corresponding, alternate interior and same-side angles, triangle angle sum, exterior angles, polygon angle sums

Study guide

When a transversal crosses two lines, it forms eight angles with names based on their positions. Corresponding angles sit in matching positions at each intersection. Alternate interior angles are between the lines on opposite sides of the transversal; alternate exterior angles are outside the lines on opposite sides. Same-side (consecutive) interior angles are between the lines on the same side.

If the two lines are parallel, corresponding, alternate interior and alternate exterior angles are equal, and same-side interior angles are supplementary. The converses are also true, so you can prove lines are parallel by showing one of these relationships.

The angles of a triangle always add to 180°. An exterior angle of a triangle (formed by extending one side) equals the sum of the two remote interior angles, which is often faster than finding the third interior angle.

For any polygon with n sides, the interior angles add to (n − 2)·180°, because the polygon can be split into n − 2 triangles. In a regular polygon all angles are equal, so each one is (n − 2)·180°/n. The exterior angles of any convex polygon, one at each vertex, always add to 360°, so each exterior angle of a regular n-gon is 360°/n.

In a parallelogram, opposite sides are parallel and congruent, opposite angles are congruent, consecutive angles are supplementary and the diagonals bisect each other. Any one of these, or one pair of sides that is both parallel and congruent, is enough to prove a quadrilateral is a parallelogram. A rectangle is a parallelogram with congruent diagonals or a right angle. A rhombus is a parallelogram with perpendicular diagonals, or with diagonals that bisect its angles, or with four congruent sides. A square is both a rectangle and a rhombus.

In a trapezoid, the two angles along each leg (between the parallel bases) are supplementary. An isosceles trapezoid has congruent legs, congruent base angles and congruent diagonals, and its median (midsegment) is the average of the bases.

Key ideas

• Corresponding and alternate interior angles are equal.
• Same-side interior angles are supplementary.
• Triangle interior angles sum to 180°.
• An exterior angle equals the sum of the two remote interior angles.
• Polygon interior sum = (n − 2)·180°; exterior angles always sum to 360°.

Formulas and rules

Common mistakes

• Assuming lines are parallel without being told.
• Using n instead of n − 2 in the polygon formula.

Words to know

Corresponding angles

Angles in matching positions at two parallel lines; equal.

Same-side interior angles

Between parallel lines on the same side of the transversal; supplementary.

Exterior angle theorem

An exterior angle equals the sum of the two remote interior angles.

Interior angle sum of an n-gon

(n − 2)·180°

Sum of exterior angles of any convex polygon

360°

Transversal

A line that crosses two or more lines.

Alternate interior angles

Between the lines, on opposite sides of the transversal; equal if lines are parallel.

Alternate exterior angles

Outside the lines, on opposite sides of the transversal; equal if lines are parallel.

Triangle angle sum

The angles of a triangle add to 180°.

Regular polygon interior angle

(n − 2)·180°/n

Converse (parallel lines)

If corresponding or alternate interior angles are equal, the lines are parallel.

Sample practice questions

39 more questions like these, from level 1 to 5, are in the practice set.