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Congruent and similar triangles

Level: 101 · for anyone reviewing geometry before precalculus

One line: A triangle has six measurements, and three are enough to fix it, if they are the right three: SSS, SAS and ASA fix a triangle, so equal data means congruent; three angles fix only the shape, which is similarity; and two sides with an angle not between them can fit two different triangles, which is why no book lists SSA.

The idea

Two triangles are congruent if they are the same triangle in two places: every angle and every side of one equals the corresponding angle and side of the other. One could be picked up (and perhaps flipped over) and laid exactly on the other. They are similar if they have the same shape and possibly different sizes: the angles match, and the sides are proportional, every side of one being k times the matching side of the other for one number k, the scale factor. Congruent triangles are the similar ones with k = 1.

Checking all six measurements is more than needed, and the useful question is how few will do. The book's three cases for congruence answer it:

Case Given equal Fixes the triangle?
SSS, side-side-side all three sides yes
SAS, side-angle-side two sides and the angle between them yes
ASA, angle-side-angle two angles and the side between them yes
AAA, angle-angle-angle all three angles no: the shape only, any size. This is the first case for similarity.
SSA, side-side-angle two sides and an angle not between them no: sometimes two triangles fit

"Fixes the triangle" means: build it with ruler and protractor from those three data, and only one triangle can come out. Then two triangles with the same data must be copies of that one, so they are congruent. The program does exactly this construction and counts how many triangles fit.

The word between in SAS and ASA does all the work. Two sides with the angle between them hinge together at that angle, and the third side has no choice. Two sides with some other angle leave the second side free to swing, and it can land in two places.

This page follows Section A.2, Geometry Essentials, objective 3, in Michael Sullivan's Precalculus: Figures 17–21, Example 5 and problems 4, 5, 10–12 and 43–46.

Symbol by symbol

Piece Say it What it means
≅ "is congruent to" same size and shape: all corresponding sides and angles equal.
∼ "is similar to" same shape: corresponding angles equal, corresponding sides proportional.
corresponding "corresponding" matched up: the side of one triangle that plays the same role in the other, found by the angles it sits between, not by where it is drawn.
a, b, c and A, B, C the usual naming: side a is opposite angle A, and so on.
k "scale factor" the one ratio every pair of corresponding sides shares: d/a = e/b = f/c = k.

Similar triangles: one ratio for every pair

For similarity the book lists three cases, which mirror the congruence cases with "equal" weakened to "proportional":

  • AA, angle-angle: two angles equal. (The third then is too, since the angles add to 180°.)
  • SSS, side-side-side: all three pairs of sides in the same ratio, such as 10/30 = 5/15 = 6/18 = 1/3.
  • SAS, side-angle-side: two pairs of sides in the same ratio, and the angles between them equal, such as 4/6 = 12/18 = 2/3 with the included angles equal.

Once two triangles are known to be similar, the angles are simply copied across, and a missing side comes from one proportion. Example 5: the matching sides 3 and 5 give k = 5/3, so the side matching 6 is 6 · 5/3 = 10.

How to match sides. The figures in the book are drawn in different orientations on purpose. The safe way to match is by the angles: the side between the 60° and 90° angles in one triangle matches the side between the 60° and 90° angles in the other, wherever it is drawn. Equivalently, the side opposite the 30° angle matches the side opposite the 30° angle.

And the scale factor reaches beyond lengths. Similar triangles with scale factor k have perimeters in the ratio k and areas in the ratio k², by the rule of the previous lesson.

What the program prints

To build a triangle from an angle takes cosines and sines, which the book reaches in its trigonometry chapters; here they are only the machinery inside the construction, and the page never asks you to use them. Sides and angles print to two decimals. The similarity ratios are exact fractions.

Verified output of congruent_and_similar_triangles.py — regenerated by tools/run_examples.py, never hand-typed.

1. THREE MEASUREMENTS THAT FIX A TRIANGLE: FIGURE 18
   Each set of three is fed to the construction; one triangle comes out.
     (a) ASA: angles 80 and 40, side 10 between them
         sides  11.37   7.42  10.00   angles  80.00  40.00  60.00
     (b) SSS: sides 15, 20, 8
         sides  15.00  20.00   8.00   angles  41.68 117.55  20.77
     (c) SAS: sides 7 and 8, angle 40 between them
         sides   8.00   7.00   5.22   angles  80.38  59.62  40.00
   The other three measurements are forced. Two triangles with the same
   three data are therefore the same triangle in two places: congruent.

2. THREE ANGLES DO NOT FIX A TRIANGLE: PROBLEM 5
   Angles 30, 50, 100 (Figure 17), with the side between 30 and 50 set to 1, 2, 5:
         sides   0.51   0.78   1.00   angles  30.00  50.00 100.00
         sides   1.02   1.56   2.00   angles  30.00  50.00 100.00
         sides   2.54   3.89   5.00   angles  30.00  50.00 100.00
   Same angles, sides in the ratio 1 : 2 : 5. The angles fix the shape and
   nothing else, so Angle-Angle-Angle is the answer to problem 5: it is not a
   case for congruence. It is the first case for similarity.
   And the third angle is never news: it is 180 minus the other two, so AAA
   is really only two measurements.

3. TWO SIDES AND AN ANGLE NOT BETWEEN THEM: THE CASE THAT IS MISSING
   Side a = 8 opposite a 40 degree angle A, and side b = 10:
         sides   8.00  10.00   2.90   angles  40.00 126.54  13.46
         sides   8.00  10.00  12.42   angles  40.00  53.46  86.54
   Two different triangles fit the same three data: side a can swing to
   meet the base in two places. So Side-Side-Angle is not a congruence case,
   unless the angle is 90 degrees, where the swing has one place to land:
         sides  13.00   5.00  12.00   angles  90.00  22.62  67.38
   That special case is Pythagoras: two sides of a right triangle fix the third.
   With a = 5 too short to reach, no triangle fits at all:
         0 triangles

4. SIMILAR MEANS ONE RATIO FOR EVERY PAIR OF SIDES: FIGURES 19 AND 20
     Figure 19  d/a, e/b, f/c       ratios 2, 2, 2
     Figure 20b 10/30, 5/15, 6/18   ratios 1/3, 1/3, 1/3
     Figure 20c 4/6, 12/18 (SAS)    ratios 2/3, 2/3
   Every ratio is the same number, the scale factor. Lengths multiply by it.

5. EXAMPLE 5 AND PROBLEMS 43-46: FIND x AND THE ANGLES
   Match the sides by the angles they sit between, then one ratio gives x.
             known pair  scale k  matches x  x = k * side   A, B, C
     Ex 5        3 -> 5      5/3          6            10   90, 60, 30
     43          4 -> 8        2          2             4   90, 60, 30
     44         12 -> 6      1/2         16             8   30, 75, 75
     45        20 -> 30      3/2         45  135/2 = 67.5   60, 95, 25
     46         10 -> 8      4/5         50            40   50, 125, 5
   The angles are copied, not computed: similar triangles have equal angles.
   Only a length needs the scale factor.

6. TRUE OR FALSE: PROBLEMS 10-12
     10  sides 30, 29, 10 and 30, 29, 10: all three equal, SSS: congruent. True.
     11  angles 25 and 100 in both: two equal angles, AA: similar. True.
     12  120 between 3 and 2, and between 4 and 3: ratios 4/3 = 1.3333
         and 3/2 = 1.5000 differ, so SAS fails: not similar. False.
         Try the other pairing, 4/2 and 3/3: 2 and 1, worse.

7. A SCALE FACTOR k MULTIPLIES AREA BY k^2
     legs 3, 4 scaled by   2: area 6 ->  24, ratio 4
     legs 3, 4 scaled by   3: area 6 ->  54, ratio 9
     legs 3, 4 scaled by 1/2: area 6 -> 3/2, ratio 1/4
   Similar triangles with sides in the ratio 1 : 3 have areas in the ratio 1 : 9.

Section 3 is the one no textbook box shows. With a = 8 opposite a 40° angle and b = 10, the side a can swing to meet the base at two points, making an obtuse triangle and an acute one, both with exactly the three given measurements. That is the whole reason SSA is missing from the list. The exception is the right angle: with a 90° angle and the hypotenuse given, the swing has only one place to land, and the third side follows from Pythagoras. Geometry books call that special case HL, hypotenuse-leg.

Questions

1. Fill in (problem 4): two triangles are ___ if corresponding angles are equal and the lengths of the corresponding sides are proportional.

Answer Similar.

2. Problem 5: which is not a case for congruent triangles: (a) ASA, (b) SAS, (c) AAA, (d) SSS?

Answer (c) AAA. Equal angles fix the shape only; a triangle and its double have the same angles and different sides. Section 2 of the program builds three sizes.

3. Problem 12: one triangle has sides 3 and 2 with 120° between them, the other 4 and 3 with 120° between them. Similar?

Answer No. SAS similarity needs 4/3 = 3/2, and 1.333… is not 1.5. Pairing them the other way, 4/2 = 2 and 3/3 = 1, is worse. False.

4. Problem 43: a right triangle with angles 60° and 30° has short leg 2 and hypotenuse 4. A similar triangle has hypotenuse 8. Find the short leg x.

Answer k = 8/4 = 2, so x = 2 · 2 = 4, and the angles are 90°, 60°, 30° as before.

5. Two triangles each have a 50° angle and a 70° angle. Are they congruent? Similar?

Answer Similar, certainly (AA). Congruent, not necessarily: nothing says anything about size. If also one pair of corresponding sides is equal, then yes, by ASA or AAS.

6. Why is it enough, for similarity, to check two angles rather than three?

Answer The angles of a triangle add up to 180°, so the third angle is 180° minus the other two, and is equal automatically.

7. Two sides 8 and 10 and a 40° angle opposite the 8. Why can two different triangles have these three measurements?

Answer The angle is not between the given sides, so the side of length 8 can swing about its end and meet the base line in two places. Both triangles have sides 8 and 10 and a 40° angle opposite the 8; their third sides are 2.90 and 12.42. This is SSA, the missing case.

8. Similar triangles have scale factor 3. The small one has area 5. What is the area of the large one?

Answer 5 · 3² = 45, not 15. Area scales by k².

9. Is every pair of right triangles similar? Every pair of equilateral triangles?

Answer Right triangles: no. They share one angle, 90°, and need a second. A 3-4-5 triangle and a 1-1-√2 triangle are both right and not similar. Equilateral triangles: yes, always, since all their angles are 60°.

10. Problem 45: triangle 1 has side 20 (between the 60° and 95° angles) and side 45 (between the 95° and 25° angles). In triangle 2 the side between the 60° and 95° angles is 30. Find x, the side between the 95° and 25° angles.

Answer Match by angles: 20 ↔ 30, so k = 3/2 and x = 45 · 3/2 = 67.5.

11. Why does the book warn that the figures are drawn in different orientations?

Answer Because matching "the top side with the top side" is the usual mistake. Correspondence is by angle, not by position on the page. Rotating or flipping a triangle changes how it looks and nothing about which sides correspond.

12. A 6-foot person casts a 4-foot shadow while a tree casts a 30-foot shadow. How tall is the tree?

Answer The sun's rays make the same angle with the ground at both, and both person and tree stand at 90°, so the two triangles are similar by AA. k = 30/4 = 7.5, and the tree is 6 · 7.5 = 45 feet. Thales is said to have measured the height of a pyramid this way.

How to practise

  1. Name the case before deciding. For every pair of triangles, say SSS, SAS, ASA, AA, or "not enough", out loud, before writing a number.
  2. Match by angles, then write one proportion. Problems 43–46: list the matching pairs of sides by the angles they sit between, then set up a single ratio. Check against section 5 of the output.
  3. Draw the SSA case once. Draw a 40° angle, a side of 10 along one arm, and swing a compass of radius 8 from the far end: it cuts the other arm twice. Once seen, never forgotten.
  4. The cards, especially the trap tag.

Flashcards

The page as a deck of Anki cards: congruent_and_similar_triangles.txt. Import with File → Import. Tags: definition, congruence, similarity, problems, trap.

Where this goes next

Similar triangles are why the slope of a line is one number: any two points on a line make a right triangle with the same shape, so rise over run comes out the same whichever two you pick. That is lines and slope, the third section of the book's first chapter. And in trigonometry, sine, cosine and tangent are ratios of sides of a right triangle, which only makes sense because all right triangles with the same angle are similar.

Po polsku, w skrócie

Trójkąty przystające to ten sam trójkąt w dwóch miejscach: wszystkie boki i kąty równe. Podobne mają ten sam kształt, ale mogą mieć inny rozmiar: kąty są równe, a boki proporcjonalne, każdy k razy dłuższy, gdzie k to skala podobieństwa. Trójkąt ma sześć wymiarów, ale trzy wystarczą, jeśli to te właściwe: trzy boki (bbb), dwa boki i kąt między nimi (bkb), dwa kąty i bok między nimi (kbk). Wtedy da się zbudować tylko jeden trójkąt, więc dwa trójkąty o tych samych danych są przystające.

Trzy kąty (kkk) ustalają tylko kształt, nie rozmiar, i to jest cecha podobieństwa, nie przystawania. Dwa boki i kąt, który nie leży między nimi, to pułapka: drugi bok może się „wahnąć" i trafić w podstawę w dwóch miejscach, więc pasują dwa różne trójkąty. Program to buduje i liczy. Przy podobieństwie odpowiadające sobie boki dobiera się po kątach, między którymi leżą, a nie po położeniu na rysunku; pola figur podobnych rosną jak k².

Auf Deutsch: Stichwörter

SSS, SWS, WSW legen ein Dreieck fest, WWW und SSW nicht; ähnliche Dreiecke haben einen Streckfaktor und proportionale Seiten.

Stichwörter: kongruent (deckungsgleich), ähnlich, Kongruenzsätze SSS, SWS, WSW, Streckfaktor, proportionale Seiten, Strahlensatz, Flächen im Verhältnis k².

See also