02.04 Triangle Proofs

Name it by sides and by angles

Every triangle gets two names. Angles: acute (all under 90°), right (one 90°), obtuse (one over 90°), equiangular (all 60°). Sides: scalene (no matches), isosceles (at least two matches), equilateral (three matches).

Equal sides sit across from equal angles. The longest side sits across from the largest angle.

Equilateral forces 60-60-60, so it cannot be right or obtuse. Equilateral is also isosceles, because "at least two" includes three.

Triangle Sum: 180°

The three interior angles add to 180°. If a line through the triangle creates a smaller triangle, peel off a straight-line neighbor first (180° − the given adjacent angle), then subtract inside the small triangle.

The proof Canvas wants: draw a line through a vertex parallel to the opposite side. The three angles along that straight line add to 180° (straight angle). The two outer ones are alternate interior to the two base angles of the triangle, so substitution puts the three triangle angles on the straight line.

Reasons you will see: By Construction, Angle Addition, Alternate Interior Angles, Definition of a Straight Angle, Substitution.

Triangle Inequality

Any two sides added must beat the third. Check all three pairings. If 6, 7, and 14 show up, 6 + 7 = 13, which is not greater than 14, so no triangle.

The short why: the shortest path from a point to a line is perpendicular. That makes each of two pieces of the base shorter than a slanted side. Add those pieces and you get the whole base, still shorter than the sum of the other two sides.

Isosceles, both directions

Isosceles Triangle Theorem: if two sides match, the angles opposite those sides match. Those angles are the base angles. They do not have to sit on the bottom of the drawing. The leftover angle is the vertex angle.

Converse: if two angles match, the sides opposite those angles match.

Worked number: sides AC ≅ BC, m∠A = 58°. Then m∠B = 58° too, so m∠C = 180 − 58 − 58 = 64°.

Converse proof sketch: given ∠OLN ≅ ∠LNO. Draw the perpendicular bisector of LN from O. Right angles match, halves of the base match, so ASA, then the legs match by CPCTC.

Midsegment

Join the midpoints of two sides. That segment is parallel to the third side and half as long.

On a grid, put the third side on the x-axis, take midpoints with the midpoint formula, then distance. The y-coordinates of the midpoints match, so the midsegment is horizontal, same as the base, hence parallel. Its length is half the base.

Four centers, one concurrency they prove

A median joins a vertex to the midpoint of the opposite side. The three medians meet at the centroid. That meeting is the Concurrency of Medians Theorem. The centroid is always inside.

The other three centers show up so you do not mix the pictures:

• Incenter: angle bisectors. Always inside. Center of the inscribed circle.
• Circumcenter: perpendicular bisectors of the sides. Inside (acute), on the hypotenuse (right), outside (obtuse).
• Orthocenter: altitudes. Inside (acute), at the right-angle vertex (right), outside (obtuse).

On an equilateral triangle the four centers stack as one point. On isosceles they sit on the line of symmetry.