Inside one triangle, the sides and the angles keep the same order. The longest side sits opposite the largest angle, and the shortest side sits opposite the smallest angle. These activities let you test that order, rank parts, and solve for missing measures.
Definitions and Main Ideas
Inside any one triangle, the side lengths and the angle measures keep the same order: the longest side lies opposite the largest angle, and the shortest side lies opposite the smallest angle. That single idea works in both directions — given the angles, you can rank the sides, and given the sides, you can rank the angles. The reliable routine is to label each part Small, Medium, or Large and then answer with the matches.
Vocabulary
Opposite Side
The side of a triangle that does not touch a given angle; the side across from the angle.
Opposite Angle
The angle of a triangle that does not touch a given side; the angle across from the side.
Vertex
The point where two sides of a triangle meet; each angle of the triangle sits at a vertex, and the side opposite that angle is the one side that does not touch the vertex.
Congruent
Having the same measure (for angles) or the same length (for segments).
Main Ideas
The bigger angle is opposite the bigger side, the smaller angle is opposite the smaller side.
If one side of a triangle is longer than another side, then the angle opposite the longer side has a greater measure than the angle opposite the shorter side.
If one angle of a triangle has a greater measure than another angle, then the side opposite the greater angle is longer than the side opposite the lesser angle.
Simply put: the biggest side is opposite the biggest angle, and the smallest side is opposite the smallest angle.
Sides and angles that are congruent come in matching counts — a triangle with two congruent angles has two congruent sides, and vice versa.
Label the parts Small (S), Medium (M), and Large (L) first, then match each side with the angle opposite it.
The two directions above are the two directions of the Inequalities in a Triangle theorem: the sides-to-angles direction and the angles-to-sides direction. When two angles are congruent (equal in measure), their opposite sides are congruent (equal in length), and the other way around.
Ranking Explorer
What to do: Move the two sliders to change two angles of the triangle. The third angle follows from the 180° angle sum. Watch the side lengths and the rank labels change: S = Small, M = Medium, L = Large. Click any side or angle to highlight it together with its opposite part.
Rank Challenge
What to do: Each figure gives the angles or the sides of a triangle. Name the part asked for: the longest or shortest side, the largest or smallest angle, or the sides or angles listed in order. A first miss gives a hint and another try; a second miss names the correct part and explains it.
The Next item button cycles through all ten items in order.
Build the Ranking
What to do: Work each problem one step at a time. First choose the statement the problem calls for, then rank or solve, then answer. The diagram labels a part only after the step that finds it has been answered.
Continue moves to the next step. New Problem cycles through all eight problems in order.
Shared Side Explorer
What to do: Follow one figure through three steps. First label each triangle's sides S, M, or L: S = Small, M = Medium, L = Large. Then use the shared side as the bridge and cancel it as a contender. Then decide the longest or shortest segment the figure actually settles.
Step buttons move through the walkthrough: label the sides, bridge through the shared side, then decide.
Example buttons change the figure; the step stays the same so the same step can be compared across figures.
Shared Side
What to do: Two triangles share side XY. Rank each triangle's sides against XY first, then chain through XY to compare sides from different triangles. Name the longest or shortest segment of the whole figure, or list every segment shorter or longer than XY. A first miss gives a hint and another try; a second miss names the answer and explains it.
The Next item button cycles through all eight items in order.
Your Turn
What to do: Answer each question. A first miss gives a hint and another try; a second miss shows the full explanation. Your score counts first-try answers only.