Triangles · Learner pages

Lesson 1 · Match corresponding parts

Use AA, SAS and SSS similarity with correct correspondence.

Name or initials: ____________   Date: ____________

Triangle angles add to 180°. Similar figures have the same shape, with a common scale factor for corresponding lengths.

1. Match angles first. If ∠A = ∠P and ∠B = ∠Q, then ∠C = ∠R because each triangle totals 180°.
2. Write the order carefully. Write △ABC ∼ △PQR so A matches P, B matches Q and C matches R.
3. Match side ratios. The corresponding ratios are AB/PQ = BC/QR = AC/PR. Do not mix unmatched sides.
4. Choose a valid criterion. AA uses two equal angles; SSS uses three proportional side pairs; SAS uses two proportional pairs and the equal included angle.

Worked example: Triangles with side lengths 3, 4, 5 and 6, 8, 10 are similar by SSS because every larger/smaller ratio is 2. They are not congruent because corresponding lengths differ.

1. Two triangles each have angles 40° and 60°. What can we conclude?

   1. They are similar by AA · 2. They must have equal side lengths · 3. Their third angles differ

2. A triangle has angles 50° and 70°. Find the third angle.

   1. 50° · 2. 70° · 3. 60°

3. Triangles have side lengths 4, 6, 8 and 6, 9, 12. Which criterion establishes their similarity?

   1. No criterion applies · 2. SSS · 3. AA from the listed side data alone

Discuss: Draw a triangle and a copy enlarged by factor 3. Label matching vertices and write all three corresponding side ratios.

Lesson 2 · Use a parallel line inside a triangle

Apply the basic proportionality theorem and its converse.

Name or initials: ____________   Date: ____________

A line parallel to one side creates equal corresponding angles.

1. State the geometry. D lies on AB, E lies on AC and DE is parallel to BC.
2. Use the divided-side ratio. The theorem gives AD/DB = AE/EC. With AD = 3, DB = 2 and AE = 6, write 3/2 = 6/EC.
3. Solve the proportion. Cross-multiply: 3EC = 12, so EC = 4.
4. Check and distinguish the converse. 3/2 = 6/4 checks. Conversely, if points on the two sides divide them equally in this ratio, the converse theorem allows us to conclude DE ∥ BC.

Worked example: If AD = 4, DB = 6, AE = 2 and EC = 3, then AD/DB = 2/3 = AE/EC. With D and E on the two sides, the converse theorem gives DE ∥ BC.

1. In △ABC, D is on AB, E is on AC and DE ∥ BC. If AD = 3, DB = 2, AE = 6, find EC.

   1. 4 · 2. 5 · 3. 6

2. In △ABC, D is on AB, E is on AC and DE ∥ BC. If AD = 4, DB = 3, AE = 8, find EC.

   1. 7 · 2. 8 · 3. 6

3. In △ABC, D is on AB, E is on AC and DE ∥ BC. If AD = 2, DB = 5, AE = 4, find EC.

   1. 12 · 2. 10 · 3. 11

Discuss: Draw the diagram and change DB while keeping AD fixed. Predict whether EC must grow or shrink if AE stays fixed.

Lesson 3 · Use similarity to measure indirectly

Justify the similar triangles before setting up a scale ratio.

Name or initials: ____________   Date: ____________

A proportional calculation needs a geometric reason for the proportion.

1. State assumptions. A vertical 2 m pole and a tree stand on level ground under the same sunlight. The shadows are 3 m and 12 m.
2. Justify the shape match. Both triangles have right angles, and parallel sunlight gives the same acute sun angle. They are similar by AA.
3. Match height to shadow. h/12 = 2/3, so h = 12 × 2/3 = 8 m.
4. Check the result and assumptions. The shadow is four times as long, so the height is four times 2 m. Sloping ground or a leaning pole would require a different model.

Worked example: A 1.5 m reference stick has a 2 m shadow. A wall has a 6 m shadow at the same time. Height = 6 × 1.5/2 = 4.5 m, under the stated level-ground and vertical-object assumptions.

1. At the same time on level ground, a 2 m pole has a 3 m shadow. A tree has a 12 m shadow. Assuming parallel sunlight, find the tree height.

   1. 8 m · 2. 9 m · 3. 10 m

2. At the same time on level ground, a 3 m pole has a 4 m shadow. A tree has a 20 m shadow. Assuming parallel sunlight, find the tree height.

   1. 16 m · 2. 17 m · 3. 15 m

3. At the same time on level ground, a 1 m pole has a 2 m shadow. A tree has a 14 m shadow. Assuming parallel sunlight, find the tree height.

   1. 9 m · 2. 7 m · 3. 8 m

Discuss: Use safe tabletop objects and a single lamp to test which conditions are needed before comparing shadows. Record measurements and assumptions.

Final check · Set 1

Try one set after the lessons. Use the other set after reviewing any errors.

1. For SAS similarity, two corresponding side pairs are proportional. What angle condition is also required?

   Any two angles may add to 180° / No angle condition is needed / The included angles must be equal

2. Similar triangles have AB = 4, PQ = 6 and BC = 8, with A↔P, B↔Q, C↔R. Find QR.

   16 / 12 / 10

3. In △ABC, D is on AB, E is on AC and DE ∥ BC. If AD = 6, DB = 3, AE = 4, find EC.

   3 / 4 / 2

4. In △ABC, D is on AB, E is on AC and DE ∥ BC. If AD = 3, DB = 4, AE = 9, find EC.

   14 / 12 / 13

5. At the same time on level ground, a 4 m pole has a 3 m shadow. A tree has a 9 m shadow. Assuming parallel sunlight, find the tree height.

   13 m / 14 m / 12 m

6. At the same time on level ground, a 3 m pole has a 2 m shadow. A tree has a 10 m shadow. Assuming parallel sunlight, find the tree height.

   17 m / 15 m / 16 m

Final check · Set 2

Try one set after the lessons. Use the other set after reviewing any errors.

1. Two triangles have one equal angle but no further measurements. Is that alone enough for similarity?

   No / Yes, by AA / Yes, by SSS

2. Corresponding side lengths in two similar triangles are 5 and 15. What is the larger/smaller length scale factor?

   9 / 10 / 3

3. In △ABC, D is on AB, E is on AC and DE ∥ BC. If AD = 2, DB = 3, AE = 8, find EC.

   12 / 13 / 14

4. In △ABC, D is on AB, E is on AC and DE ∥ BC. If AD = 4, DB = 5, AE = 12, find EC.

   16 / 17 / 15

5. At the same time on level ground, a 2 m pole has a 4 m shadow. A tree has a 18 m shadow. Assuming parallel sunlight, find the tree height.

   9 m / 10 m / 11 m

6. At the same time on level ground, a 5 m pole has a 6 m shadow. A tree has a 24 m shadow. Assuming parallel sunlight, find the tree height.

   21 m / 22 m / 20 m