Trigonometry · Trigonometry: Similar Triangles, Right Triangles and Acute TrianglesInvestigating Similar Triangles: AA, SSS and SAS Similarity
1.1 1.2About 70 minutes
Learning goals- I can verify by measuring that similar triangles have equal corresponding angles and proportional corresponding sides.
- I can use the AA, SSS and SAS similarity conditions to decide whether two triangles are similar.
- I can explain why a line parallel to one side of a triangle creates a similar triangle.
- I can write a similarity statement with the vertices in the correct order.
Words to know
AA similarity
If two angles of one triangle equal two angles of another, the triangles are similar.
SSS similarity
If all three pairs of corresponding sides have the same ratio, the triangles are similar.
SAS similarity
If two pairs of corresponding sides have the same ratio and the angles between them are equal, the triangles are similar.
contained angle
The angle formed between two given sides of a triangle.
corresponding angles (parallel lines)
Angles in matching positions where a transversal crosses two parallel lines. They are equal.
alternate angles
Angles on opposite sides of a transversal, between two parallel lines (a Z pattern). They are equal.
Do you really need to check all six?
Last lesson, similar triangles needed three equal angles and three proportional sides. That is six facts to check. A carpenter checking two roof trusses, or a surveyor checking a triangle of land, does not want to measure everything. Good news: for triangles, a few facts are enough to guarantee the rest. In this lesson you will investigate why, and learn three shortcuts.
Investigation: same angles, different sizes
Below are three triangles. Each one was drawn with angles of 40° at A, 60° at B and 80° at C, but with a different base length AB. If you draw them and measure the other sides with a ruler, you get the results in the table.
| Triangle | BC (opposite 40°) | CA (opposite 60°) | AB (opposite 80°) | BC ÷ AB | CA ÷ AB |
|---|
| 1 | 3.3 cm | 4.4 cm | 5.0 cm | 0.66 | 0.88 |
| 2 | 4.9 cm | 6.6 cm | 7.5 cm | 0.65 | 0.88 |
| 3 | 6.5 cm | 8.8 cm | 10.0 cm | 0.65 | 0.88 |
What we notice: when the angles are the same, the ratio of any two sides is the same in every triangle. That means the sides are proportional, so the triangles are similar. Keep this table in mind: in Lesson 5 these fixed ratios get names (sine, cosine and tangent).
Three ways to prove triangles are similar
1. AA similarity (angle-angle). If two angles of one triangle equal two angles of another, the triangles are similar.
Why only two? The angles of every triangle add to 180°. If two pairs match, the third angles must be 180° minus the same total, so they match too.
2. SSS similarity (side-side-side). If all three pairs of corresponding sides have the same ratio, the triangles are similar.
3. SAS similarity (side-angle-side). If two pairs of corresponding sides have the same ratio and the contained angles (the angles between those sides) are equal, the triangles are similar.
Compare these with the congruence conditions from last lesson: for congruence the sides must be equal; for similarity they only need to be proportional.
Worked exampleExample 1: AA similarity
In △ABC, ∠A = 50° and ∠B = 70°. In △DEF, ∠D = 50° and ∠F = 60°. Are the triangles similar? If so, write a similarity statement.
- Find the missing angles. △ABC: ∠C = 180° − 50° − 70° = 60°. △DEF: ∠E = 180° − 50° − 60° = 70°.
- Match equal angles: ∠A = ∠D = 50°, ∠B = ∠E = 70°, ∠C = ∠F = 60°.
- Two (in fact three) pairs of angles are equal, so the triangles are similar by AA.
- Write the vertices in matching order: A ↔ D, B ↔ E, C ↔ F.
Worked exampleExample 2: SSS similarity
△PQR has sides 6 cm, 8 cm and 10 cm. △XYZ has sides 15 cm, 9 cm and 12 cm. Are they similar?
- Match the sides by size: shortest with shortest, middle with middle, longest with longest.
- Shortest: 9 ÷ 6 = 1.5. Middle: 12 ÷ 8 = 1.5. Longest: 15 ÷ 10 = 1.5.
- All three ratios are equal, so the triangles are similar by SSS, with scale factor 1.5.
Yes, the triangles are similar (SSS), k = 1.5.
Worked exampleExample 3: SAS similarity
In △PQR, PQ = 4 cm, PR = 6 cm and ∠P = 40°. In △STU, ST = 6 cm, SU = 9 cm and ∠S = 40°. Are the triangles similar?
- Compare the sides around the 40° angle: ST ÷ PQ = 6 ÷ 4 = 1.5 and SU ÷ PR = 9 ÷ 6 = 1.5.
- The ratios are equal, and the angles between these sides (∠P and ∠S) are both 40°.
- So the triangles are similar by SAS.
△PQR ~ △STU (SAS), k = 1.5.
A line parallel to one side
In Grade 8 you learned that when a transversal crosses two parallel lines, the corresponding angles are equal (an F pattern) and the alternate angles are equal (a Z pattern). This gives a very common way to find similar triangles.
In △ABC, draw a segment DE parallel to BC, with D on AB and E on AC. Then:
- ∠ADE = ∠ABC (corresponding angles, transversal AB)
- ∠AED = ∠ACB (corresponding angles, transversal AC)
- ∠A is in both triangles.
So △ADE ~ △ABC by AA. The small triangle at the top is a reduction of the big one.
The "bow tie"
If two segments AE and BD cross at C, and AB ∥ DE, you get two triangles that touch at a point. The angles at C are vertically opposite, so they are equal. The angles at A and E are alternate angles (transversal AE), so they are equal. That is AA again: △ABC ~ △EDC. Notice the order: A matches E because their angles are equal.
Worked exampleExample 4: Writing a short proof
In the bow tie above, prove that △ABC ~ △EDC.
- ∠ACB = ∠ECD because they are vertically opposite angles.
- ∠BAC = ∠DEC because they are alternate angles and AB ∥ DE.
- Two pairs of angles are equal, so the triangles are similar.
△ABC ~ △EDC by AA. A clear proof gives a reason for each equal pair.
Just for funWhy did the triangle bring only two angles to its similarity test?
It knew the third one would follow. (Thanks, 180°!)
Hands-on activityParallel-line triangles on paper
You need: lined paper (or graph paper), a ruler, a protractor and a pencil.
- The lines on lined paper are parallel. Choose one line as BC near the bottom of the page. Mark B and C about 12 cm apart.
- Mark a point A about 10 lines higher, roughly above the middle. Draw AB and AC.
- Pick a lined-paper line about halfway up. Where it crosses AB call it D; where it crosses AC call it E. DE is parallel to BC.
- Measure AD, AB, AE, AC, DE and BC to the nearest millimetre.
- Work out AD ÷ AB, AE ÷ AC and DE ÷ BC. Then measure ∠ADE and ∠ABC.
What to look for: the three ratios should be almost equal (within about 0.02), and ∠ADE should equal ∠ABC. That is the evidence that △ADE ~ △ABC. Try a second line DE closer to A: the ratios get smaller but stay equal to each other.
Your turn
1In △ABC, ∠A = 35° and ∠B = 85°. In △XYZ, ∠X = 85° and ∠Y = 60°. Are the triangles similar? If so, write a similarity statement.
∠C = 180° − 35° − 85° = 60° and ∠Z = 180° − 85° − 60° = 35°. Matching: A ↔ Z, B ↔ X, C ↔ Y. Yes: △ABC ~ △ZXY (AA).
2Triangle 1 has sides 4, 6, 7. Triangle 2 has sides 12, 18, 21. Similar?
12 ÷ 4 = 3, 18 ÷ 6 = 3, 21 ÷ 7 = 3. Yes, by SSS (k = 3).
3Triangle 1 has sides 3, 4, 6. Triangle 2 has sides 6, 8, 10. Similar?
6 ÷ 3 = 2, 8 ÷ 4 = 2, but 10 ÷ 6 ≈ 1.67. No.
4In △ABC, AB = 5, AC = 8 and ∠A = 55°. In △DEF, DE = 7.5, DF = 12 and ∠D = 55°. Similar? Which condition?
7.5 ÷ 5 = 1.5 and 12 ÷ 8 = 1.5, and the contained angles are both 55°. Yes, by SAS.
5In △PQR, S is on PQ and T is on PR with ST ∥ QR. Explain why △PST ~ △PQR.
∠P is shared. ∠PST = ∠PQR because they are corresponding angles (ST ∥ QR, transversal PQ). Two pairs of equal angles, so the triangles are similar by AA.
6Two right triangles each have a 32° angle. Must they be similar? Explain.
Yes. Each has a 90° angle and a 32° angle, so two pairs of angles are equal (AA).
7One triangle has angles 40° and 70°. Another has angles 40° and 80°. Are they similar?
Third angles: 180 − 40 − 70 = 70°, and 180 − 40 − 80 = 60°. Angle sets {40°, 70°, 70°} and {40°, 80°, 60°} differ, so no.
More practice: warm-up
1How many pairs of equal angles do you need to prove two triangles similar?
Two (AA). The third pair is then automatically equal.
2A triangle has angles of 45° and 65°. What is the third angle?
3What does SSS similarity require?
All three pairs of corresponding sides have the same ratio.
4Are triangles with sides 2, 3, 4 and 4, 6, 8 similar?
5In SAS similarity, which angle must be equal?
The contained angle: the angle between the two pairs of proportional sides.
More practice: on your own
1△ABC has AB = 9, BC = 12, CA = 15. △DEF has DE = 6, EF = 8, FD = 10. Are they similar? State the scale factor from △DEF to △ABC.
9 ÷ 6 = 1.5, 12 ÷ 8 = 1.5, 15 ÷ 10 = 1.5. Yes (SSS), △ABC ~ △DEF and k = 1.5.
2Segments MQ and NP cross at O, and MN ∥ PQ. Prove that △MNO ~ △QPO.
∠MON = ∠QOP (vertically opposite). ∠NMO = ∠PQO (alternate angles, MN ∥ PQ, transversal MQ). So △MNO ~ △QPO by AA.
3A right triangle has legs 5 cm and 12 cm. Another has legs 7.5 cm and 18 cm. Are they similar? Explain.
7.5 ÷ 5 = 1.5 and 18 ÷ 12 = 1.5, and the angle between the legs is 90° in both. Yes, by SAS.
4Are all isosceles triangles with a 40° angle between the equal sides similar? Explain.
Yes. The base angles are each (180° − 40°) ÷ 2 = 70°, so every such triangle has angles 40°, 70°, 70°. They are similar by AA.
5Are all isosceles triangles similar? Explain with an example.
No. An isosceles triangle with angles 40°, 70°, 70° and one with angles 100°, 40°, 40° have different angle sets, so they are not similar.
Challenge
1In △ABC, D is on AB and E is on AC with AD = 3, AB = 9, AE = 4 and AC = 12. Prove that △ADE ~ △ABC, and then explain why DE ∥ BC.
AD ÷ AB = 3 ÷ 9 = 1/3 and AE ÷ AC = 4 ÷ 12 = 1/3, and ∠A is the contained angle in both, so △ADE ~ △ABC by SAS. Similar triangles have equal corresponding angles, so ∠ADE = ∠ABC. These are corresponding angles for lines DE and BC with transversal AB; since they are equal, DE ∥ BC.
2In right △ABC, ∠C = 90°. CD is drawn from C perpendicular to the hypotenuse AB. Prove that △ADC ~ △ACB.
∠A is in both triangles. ∠ADC = 90° (CD ⊥ AB) and ∠ACB = 90°. Two pairs of equal angles, so △ADC ~ △ACB by AA (A ↔ A, D ↔ C, C ↔ B).
3A triangle with sides 8, x and 12 is similar to a triangle with sides 6, 9 and y, where the sides are listed in matching order. Find x and y.
k = 8 ÷ 6 = 4/3 (from the second triangle to the first). x = (4/3)(9) = 12. 12 = (4/3)y, so y = 12 × 3/4 = 9.
Quiz
1. Which information is enough to prove two triangles similar?
2. One triangle has angles 30° and 70°. Another has angles 70° and 80°. Are they similar?
3. Are triangles with sides 5, 6, 8 and 10, 12, 15 similar?
4. In △ABC, DE ∥ BC with D on AB and E on AC. Which angles are equal?
5. SAS similarity needs two pairs of proportional sides and:
6. In a "bow tie" made by two crossing segments, the two angles at the crossing point are equal because they are:
Big idea: Expectation 1.1 asks students to verify, through investigation, that similar triangles have equal corresponding angles and proportional corresponding sides. The AA, SSS and SAS conditions are the shortcuts used in every later lesson.
How to help: Do the lined-paper activity together; the measured ratios being nearly equal is convincing. Ask your teen to say the reason for every pair of equal angles ("shared angle", "corresponding angles", "vertically opposite").
Common difficulties:
- Writing the similarity statement in the wrong order. Match vertices by equal angles, not by where they sit on the page.
- Matching sides by position instead of by size (for SSS, sort each triangle's sides from shortest to longest).
- For SAS, using an angle that is not between the two sides.
- Mixing up congruence (equal sides) and similarity (proportional sides).