Trigonometry · Trigonometry: Similar Triangles, Right Triangles and Acute TrianglesCongruent and Similar Figures
1.2 1.1About 65 minutes
Learning goals- I can explain the difference between congruent figures and similar figures.
- I can name corresponding angles and sides from a congruence or similarity statement.
- I can find a scale factor and use it to decide whether two figures are similar.
- I can state the SSS, SAS and ASA conditions for congruent triangles.
Words to know
congruent
Exactly the same shape and size. Corresponding sides are equal and corresponding angles are equal. Symbol: ≅.
similar
Exactly the same shape but not necessarily the same size. Corresponding angles are equal and corresponding sides are proportional. Symbol: ~.
corresponding parts
Angles or sides that are in matching positions in two figures.
proportional
Having equal ratios. Sides are proportional when every pair of corresponding sides gives the same ratio.
scale factor (k)
The number every length is multiplied by to get from one figure to a similar figure: k = image length ÷ original length.
enlargement / reduction
A similar copy that is bigger (k > 1) or smaller (0 < k < 1) than the original.
Same shape, different size
Open a photo on a phone and pinch to zoom. The picture gets bigger, but a face does not get wider without getting taller. Everything grows by the same factor, so the shape stays the same. Now think of two hockey cards printed from the same plate. They are identical: same shape and same size.
Mathematicians have a word for each situation. The zoomed photo is similar to the original. The two hockey cards are congruent. This whole unit (similar triangles, then trigonometry) is built on these two ideas, so we start here.
Remember from Grades 8 and 9
In Grade 8 you made scale drawings and performed dilations on a grid with the rule (x, y) → (kx, ky). You saw that a dilation keeps every angle the same and multiplies every length by k, so the image is similar to the original. You also learned the Pythagorean theorem, and in Grade 9 you used it in real-life and composite-shape problems. In Grade 10 you will use similarity to prove facts about triangles, and then build trigonometry on top of it.
Congruent figures
Two figures are congruent if one can be placed exactly on top of the other. You may need to slide, turn or flip it first, but you may not stretch or shrink it.
For congruent triangles:
- all three pairs of corresponding sides are equal, and
- all three pairs of corresponding angles are equal.
We write △ABC ≅ △PQR. The order of the letters matters: A matches P, B matches Q and C matches R. So ∠A = ∠P, ∠B = ∠Q, ∠C = ∠R, and AB = PQ, BC = QR, CA = RP.
Similar figures
Two figures are similar if one is an exact enlargement or reduction of the other. For similar triangles:
- all three pairs of corresponding angles are equal, and
- all three pairs of corresponding sides are proportional: each side of one triangle is the same multiple, k, of its partner.
We write △ABC ~ △DEF (the symbol ~ is read "is similar to"). Again, the order tells you which parts match.
The scale factor is
k = (length in the new figure) ÷ (matching length in the original)
If k > 1 the new figure is an enlargement. If 0 < k < 1 it is a reduction.
Worked exampleExample 1: Reading a similarity statement
△ABC ~ △DEF. AB = 4 cm, BC = 6 cm, CA = 5 cm and DE = 10 cm. ∠B = 48°.
(a) Find the scale factor from △ABC to △DEF. (b) Find EF and FD. (c) Which angle in △DEF equals 48°?
- Match the letters in order: A ↔ D, B ↔ E, C ↔ F. So AB ↔ DE, BC ↔ EF and CA ↔ FD.
- (a) We know one matching pair: AB = 4 and DE = 10. k = DE ÷ AB = 10 ÷ 4 = 2.5.
- (b) EF = 2.5 × BC = 2.5 × 6 = 15 cm. FD = 2.5 × CA = 2.5 × 5 = 12.5 cm.
- (c) B matches E, so ∠E = 48°. Angles do not get multiplied by the scale factor.
k = 2.5, EF = 15 cm, FD = 12.5 cm and ∠E = 48°.
Worked exampleExample 2: Similar or not?
Rectangle 1 is 4 cm by 6 cm. Rectangle 2 is 6 cm by 9 cm. Rectangle 3 is 6 cm by 8 cm. Which rectangles are similar to Rectangle 1?
- All rectangles have four 90° angles, so the angles always match. The question is whether the sides are proportional.
- Rectangle 2: short sides 6 ÷ 4 = 1.5; long sides 9 ÷ 6 = 1.5. Same ratio, so it is similar (k = 1.5).
- Rectangle 3: short sides 6 ÷ 4 = 1.5; long sides 8 ÷ 6 ≈ 1.33. Different ratios, so it is not similar. It is relatively wider and shorter.
- Notice: equal angles alone were not enough to make rectangles similar. (For triangles it is different, as you will see next lesson.)
Only Rectangle 2 is similar to Rectangle 1.
When are two triangles congruent?
You do not have to check all six pairs of parts. Any one of these is enough:
- SSS (side-side-side): all three pairs of sides are equal.
- SAS (side-angle-side): two pairs of sides are equal and the angles between them (the contained angles) are equal.
- ASA (angle-side-angle): two pairs of angles are equal and the sides between them are equal.
Be careful: AAA is not a congruence condition. Two triangles with the same three angles have the same shape, but one can be much bigger. That is exactly the idea of similarity!
| Congruent (≅) | Similar (~) |
|---|
| Shape | same | same |
| Size | same | can differ |
| Corresponding angles | equal | equal |
| Corresponding sides | equal | proportional (all multiplied by k) |
| Scale factor | k = 1 | any k > 0 |
| Everyday example | two copies of the same key | a photo and its zoomed-in version |
Just for funWhat did the small triangle say to its enlargement?
"We're so alike, people say we have the same angle on everything."
Hands-on activityMake congruent and similar triangles
You need: two sheets of paper, a ruler, a protractor, a pencil and scissors (take care with scissors; a grown-up can help).
- Stack the two sheets. On the top sheet draw a triangle with sides 6 cm, 8 cm and 10 cm. Cut through both sheets at once. Lay one triangle on the other, turning or flipping it if needed. They are congruent.
- On the leftover paper, draw a triangle with sides 3 cm, 4 cm and 5 cm. This is a reduction of your first triangle. What is the scale factor?
- Measure the three angles of each triangle with your protractor. Record them in a table.
- Write one sentence about the angles and one about the sides.
What to look for: the scale factor is 3 ÷ 6 = 0.5. Both triangles have angles of about 37°, 53° and 90°, even though one is half the size of the other. (Small differences of 1° or 2° come from measuring.)
Your turn
1△KLM ≅ △XYZ. List the three pairs of equal angles and the three pairs of equal sides.
∠K = ∠X, ∠L = ∠Y, ∠M = ∠Z; KL = XY, LM = YZ, MK = ZX.
2△ABC ~ △PQR with AB = 6 cm and PQ = 9 cm. What is the scale factor from △ABC to △PQR? Is △PQR an enlargement or a reduction?
k = 9 ÷ 6 = 1.5. Since k > 1, △PQR is an enlargement.
3In the triangles of the last question, BC = 8 cm. Find QR.
4△ABC ~ △PQR and ∠C = 70°. Which angle in △PQR is 70°?
C matches R, so ∠R = 70°.
5A rectangle is 5 cm by 8 cm. Is a 10 cm by 18 cm rectangle similar to it? Explain.
No. 10 ÷ 5 = 2 but 18 ÷ 8 = 2.25. The ratios are not equal, so the sides are not proportional.
6True or false: all congruent triangles are similar. Explain.
True. Congruent triangles have equal angles and sides in the ratio 1 : 1, which is proportional with k = 1.
7Two triangles each have angles of 30°, 60° and 90°. Must they be congruent? Explain.
No. Equal angles guarantee the same shape, not the same size. One could be an enlargement of the other. (AAA is not a congruence condition.)
More practice: warm-up
1What does the symbol ≅ mean? What does ~ mean?
≅ means is congruent to. ~ means is similar to.
2A figure is enlarged with scale factor 3. A side was 4 cm. How long is it now?
3A figure is reduced with scale factor 0.5. An angle was 64°. What is it now?
Still 64°. Angles do not change in a similar figure.
4Name the three congruence conditions for triangles from this lesson.
5Are any two squares similar? Explain.
Yes. All angles are 90°, and all four sides of each square are equal, so every side is multiplied by the same factor (side of one ÷ side of the other).
More practice: on your own
1△DEF ~ △STU with DE = 12, EF = 15, FD = 9 and ST = 8. Find TU and US.
k = ST ÷ DE = 8 ÷ 12 = 2/3. TU = (2/3)(15) = 10; US = (2/3)(9) = 6.
2A photo 10 cm wide and 15 cm tall is enlarged to be 25 cm wide. How tall is it now if it stays similar?
k = 25 ÷ 10 = 2.5, so height = 2.5 × 15 = 37.5 cm.
3Triangle 1 has sides 5, 7, 9. Triangle 2 has sides 10, 14, 17. Are they similar? Explain.
No. 10 ÷ 5 = 2 and 14 ÷ 7 = 2, but 17 ÷ 9 ≈ 1.89. Not all ratios are equal.
4△ABC and △DEF have AB = DE = 7 cm, ∠B = ∠E = 50° and BC = EF = 9 cm. Which condition shows they are congruent?
SAS. Two pairs of equal sides and the contained angles (∠B and ∠E, which are between those sides) are equal.
5Explain the difference between congruent and similar to someone who missed this lesson, using a real example.
Sample: Congruent figures are exact copies: same shape and same size, like two identical keys. Similar figures have the same shape but can be different sizes, like a poster and a small print of the same picture. In similar figures the angles are equal and every side is multiplied by the same scale factor.
Challenge
1△ABC ~ △DEF with scale factor 3 (from △ABC to △DEF). The perimeter of △ABC is 14 cm. Find the perimeter of △DEF and explain why.
Each side of △DEF is 3 times its partner, so the sum of the sides is 3 times as big: perimeter = 3 × 14 = 42 cm. (a + b + c becomes 3a + 3b + 3c = 3(a + b + c).)
2A triangle has sides 6, 8 and 10. A similar triangle has a longest side of 25. Find its other two sides and its scale factor.
The longest sides match: k = 25 ÷ 10 = 2.5. Other sides: 2.5 × 6 = 15 and 2.5 × 8 = 20.
3Kai says, "If two triangles have two pairs of equal sides, they must be congruent." Give a counterexample in words or a sketch.
Sample: Triangle 1 has sides 5 cm and 5 cm with a 30° angle between them; Triangle 2 has sides 5 cm and 5 cm with a 100° angle between them. Two pairs of sides are equal, but the third sides are different lengths, so they are not congruent. You need SSS, SAS (with the contained angle) or ASA.
Quiz
1. Which statement is true for similar triangles?
2. △JKL ~ △MNO. Which side corresponds to KL?
3. A triangle with a side of 8 cm is enlarged by a scale factor of 1.25. How long is the new side?
4. Which is not a condition that proves two triangles congruent?
5. Rectangle A is 3 by 5. Which rectangle is similar to it?
6. Two congruent figures have a scale factor of:
Big idea: Expectation 1.2 asks students to describe and compare similarity and congruence; 1.1 begins the investigation of the properties of similar triangles (equal angles, proportional sides).
How to help: Look for similar and congruent shapes at home: two identical tiles (congruent), a photo and its enlargement (similar), nested measuring cups (roughly similar). Ask "Same shape? Same size?" and "What is the scale factor?"
Common difficulties:
- Matching the wrong parts. The letter order in △ABC ~ △DEF tells you the matches. Have your teen write the three pairs (A↔D, B↔E, C↔F) before doing anything else.
- Multiplying angles by the scale factor. Only lengths change.
- Thinking equal angles always mean similar shapes. This is true for triangles, but not for other polygons (Example 2 shows two rectangles with equal angles that are not similar).
- Dividing the wrong way for k. The scale factor is new ÷ original.