Argue informally about triangle angle sums, exterior angles, parallel lines cut by a transversal and the angle-angle criterion

Learning goal: Use informal arguments to establish facts about the angles created when parallel lines are cut by a transversal, about the angle sum and exterior angle of a triangle, and about the angle-angle criterion for similarity of triangles, and explain each argument in your own words.

What an informal argument is. It is a chain of reasons, each one already accepted, that ends at the fact you want. It is not a measurement: measuring one triangle shows that one triangle's angles sum to 180°; an argument shows why every triangle's angles must. The arguments below rest on two things already established: rigid motions preserve angle measure and parallelism, and the angles along a straight line add to 180°.

Parallel lines cut by a transversal. A transversal is a line that crosses two other lines. When the two lines are parallel, the crossing creates eight angles, and they come in at most two sizes (exactly one size, 90°, when the transversal is perpendicular to the lines). Suppose one of the angles at the first crossing is 112°. The angle beside it, on the same straight line, is 180° − 112° = 68°. The angle across the crossing from the 112° angle, its vertical angle, also forms a straight line with that same 68° neighbor, so it is 180° − 68° = 112° as well. The two remaining angles at that crossing are each 180° − 112° = 68° by the same reasoning.

Now slide the first crossing along the transversal until it lands on the second crossing. Because the two lines are parallel, the translation takes the first line onto the second line, and it takes the transversal onto itself. A translation preserves angle measure, so each angle at the first crossing is carried onto an angle of the same measure at the second. The angles in matching positions, called corresponding angles, are equal: 112° matches 112° and 68° matches 68°. From that, two more facts follow. Alternate interior angles, the pair between the lines on opposite sides of the transversal, are equal, because each is the vertical angle of a corresponding angle. Same-side interior angles, the pair between the lines on the same side of the transversal, add to 180°, because one of them forms a straight line with the corresponding angle of the other: 112° + 68° = 180°. The condition is essential: if the lines are not parallel, the translation along the transversal does not carry one line onto the other, and the corresponding angles differ.

The angle sum of a triangle. Take a triangle with angles 40° at A, 60° at B and 80° at C, with AB as the base. Draw the line through C parallel to AB. That line and the two sides AC and BC create angles at C. Side AC is a transversal of the two parallel lines, so the angle between the new line and AC at C is an alternate interior angle to the 40° angle at A, and equals 40°. Side BC is another transversal, so the angle between the new line and BC at C equals the 60° angle at B. At C, the three angles 40°, 80° and 60° lie side by side along the straight line, and angles along a straight line add to 180°. So the triangle's three angles add to 180°. Nothing in the argument used the specific numbers: for any triangle, the two base angles are copied up to the apex by alternate interior angles, where they fill out a straight line together with the apex angle.

The three-copies experiment and its argument. Cut three copies of triangle ABC and lay them around the vertex C so that one copy of each angle meets there; the three top edges appear to form a single straight line. The argument that they really do: place the second copy by rotating the first 180° about the midpoint M of side AC. A half turn about the midpoint of a segment swaps the segment's two ends, because each end is carried to the point the same distance away on the opposite side of the midpoint, so A lands on C and C on A, and the copy's angle A now sits at C. A half turn takes a line through the center to itself and every other line to a line parallel to it, and AB does not pass through M, so the copy's base is a line through C parallel to AB. Place the third copy by rotating the original 180° about the midpoint N of BC: B lands on C, the copy's angle B sits at C, and its base is again a line through C parallel to AB. There is only one line through C parallel to AB, so both copied bases lie on the same line, and the three angles at C, the copies of A and B on either side of the original angle C, fill that straight line: A + B + C = 180°. The tiling is the experiment; the half-turn argument is why the experiment cannot fail.

The exterior angle of a triangle. Extend the base AB beyond B. The angle between the extension and side BC is the exterior angle at B. It forms a straight line with the interior angle at B, so exterior + 60° = 180° and the exterior angle is 120°. The interior angles also sum to 180°: 40° + 60° + 80° = 180°. Removing the 60° from both facts leaves exterior angle = 40° + 80° = 120°. In general the exterior angle at a vertex equals the sum of the two interior angles at the other vertices, the remote interior angles, because both the exterior angle and that sum are what is left of 180° after removing the interior angle at the vertex.

The angle-angle criterion for similarity. If two angles of one triangle equal two angles of another triangle, the triangles are similar. First, the third angles are equal as well, because each is 180° minus the same two angles: a triangle with angles 40° and 60° has third angle 80°, and so does any other triangle with angles 40° and 60°. Second, dilate the first triangle by the ratio of a pair of corresponding sides, say the sides opposite the 40° angles. Dilation keeps every angle, so the dilated triangle still has angles 40°, 60°, 80°, and now one of its sides equals the corresponding side of the second triangle. A triangle is fixed by one side and the two angles at its ends: lay the equal sides together and the other two sides must run off at the given angles and meet at a single point. So the dilated triangle can be carried onto the second by rigid motions, and the first triangle is similar to the second by the definition of similarity. Two equal angles are enough; one is not. Two triangles that each have a 40° angle can have their other angles 60° and 80° in one and 20° and 120° in the other, and those are not similar.

The misconception. Students often believe angle facts are true because a protractor said so, and then become unsure when a drawing is slightly off. The measurement is an experiment that suggests the fact; the argument is what establishes it. Conversely, an argument that uses "the lines are parallel" cannot be applied to lines that are not, and an argument that uses "the angle sum is 180°" applies to triangles, not to quadrilaterals, whose angle sum is 360° because a diagonal lying inside the quadrilateral (every quadrilateral has at least one) splits it into two triangles.

Worked transversal: Two parallel lines are cut by a transversal and one angle measures 112°. Its vertical angle is 112°; the two other angles at that crossing are 180° − 112° = 68°. At the second crossing, corresponding angles match the first crossing exactly, so the eight angles are four of 112° and four of 68°. The two alternate interior angles are 112° and 112° (or 68° and 68°), and the two same-side interior angles are 112° and 68°, adding to 180°.

Worked exterior angle: A triangle has interior angles 35° and 55° at two vertices. The third angle is 180° − 35° − 55° = 90°. The exterior angle at the 90° vertex is 180° − 90° = 90°, and the sum of the remote interior angles is 35° + 55° = 90°. The two calculations agree, as the argument says they must.

Worked angle-angle similarity: Triangle 1 has angles 50° and 70°, so its third angle is 60°. Triangle 2 has angles 70° and 60°, so its third angle is 50°. The triangles share the angles 50°, 60°, 70°, so they are similar by the angle-angle criterion. If the side opposite the 50° angle is 5 cm in triangle 1 and 7.5 cm in triangle 2, the scale factor from triangle 1 to triangle 2 is 7.5/5 = 1.5, and every other side of triangle 2 is 1.5 times its partner.

Worked unknown angles: A triangle's angles are x, 2x and 3x. Since they add to 180°, 6x = 180 and x = 30, so the angles are 30°, 60° and 90°. The exterior angle at the 30° vertex is 180° − 30° = 150°, which equals 60° + 90°, the remote interior angles.

Recording task: Draw a triangle with angles 40°, 60° and 80° using a protractor, with the 80° angle at the top. Through the top vertex draw a line parallel to the base, using a set square or by measuring equal distances from the base at two points. Measure the two angles the parallel line makes with the sloping sides and record them next to the base angles they should copy (40° and 60°). Write the argument beneath the drawing in three sentences: which angles are alternate interior angles and why they are equal, which three angles lie along the straight line, and what the sum must be.

Independent performance: Produce a written argument for each of the four facts: corresponding angles are equal when parallel lines are cut by a transversal; the angles of a triangle sum to 180°; an exterior angle equals the sum of the two remote interior angles; and two triangles with two pairs of equal angles are similar. Each argument must name the earlier fact it relies on (rigid motions preserve angle measure; angles along a straight line add to 180°; dilation preserves angles) and must state its condition (lines are parallel; the figure is a triangle). Include one numerical example per fact and one non-example: a transversal across non-parallel lines with corresponding angles 40° and 50°, and two triangles sharing only one angle. Success criteria: no argument relies on a measurement; every condition is stated; every numerical example checks; and the non-examples show which condition failed.