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AA, SAS, and SSS similarity theorems
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Two triangles are similar if:
Symbol:
If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.
Note: If two angles match, the third must also match (angle sum = 180°)
If two sides of one triangle are proportional to two sides of another triangle AND the included angles are congruent, the triangles are similar.
Two triangles are similar. The sides of the first triangle are 3, 4, and 5 cm. The shortest side of the second triangle is 6 cm. Find the other two sides of the second triangle.
Step 1: Identify corresponding sides: Shortest side of first triangle: 3 cm Shortest side of second triangle: 6 cm
Step 2: Find the scale factor: Scale factor = 6/3 = 2 The second triangle is 2 times larger
Step 3: Apply scale factor to other sides: Second side: 4 × 2 = 8 cm Third side: 5 × 2 = 10 cm
Step 4: Verify the similarity ratio: All ratios should be equal: 6/3 = 2 ✓ 8/4 = 2 ✓ 10/5 = 2 ✓
Answer: The other two sides are 8 cm and 10 cm
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If all three pairs of corresponding sides are proportional, the triangles are similar.
The ratio of corresponding sides:
If with scale factor :
In , and . In , and . Are the triangles similar?
Two angles of are congruent to two angles of :
By AA (Angle-Angle) similarity, the triangles are similar.
Answer: Yes, by AA
Triangle ABC is similar to triangle DEF. If AB = 12, BC = 15, and DE = 8, find EF.
Step 1: Identify corresponding sides: Since △ABC ~ △DEF: AB corresponds to DE BC corresponds to EF
Step 2: Find the scale factor: Scale factor = DE/AB = 8/12 = 2/3
Step 3: Set up proportion for unknown side: BC/EF = AB/DE 15/EF = 12/8
Step 4: Solve for EF: 15/EF = 3/2 3 × EF = 15 × 2 3 × EF = 30 EF = 10
Step 5: Verify with scale factor: EF = BC × (2/3) = 15 × (2/3) = 10 ✓
Answer: EF = 10
If with sides , , and , find and .
Find the scale factor using corresponding sides:
In similar triangles, the ratio of corresponding sides is 3:5. If the perimeter of the smaller triangle is 24 cm, what is the perimeter of the larger triangle?
Step 1: Understand the property: In similar triangles, the ratio of perimeters equals the ratio of corresponding sides
Step 2: Set up the proportion: Perimeter ratio = Side ratio P_small/P_large = 3/5
Step 3: Substitute known value: 24/P_large = 3/5
Step 4: Cross multiply: 3 × P_large = 24 × 5 3 × P_large = 120 P_large = 40
Step 5: Verify: Ratio: 24/40 = 3/5 ✓
Answer: The perimeter of the larger triangle is 40 cm
Triangle ABC has sides 6, 8, and 10. Triangle DEF has sides 9, 12, and 15. Are these triangles similar? If so, what is the scale factor?
Step 1: Order the sides from smallest to largest: Triangle ABC: 6, 8, 10 Triangle DEF: 9, 12, 15
Step 2: Check ratios of corresponding sides: Shortest sides: 9/6 = 3/2 = 1.5 Middle sides: 12/8 = 3/2 = 1.5 Longest sides: 15/10 = 3/2 = 1.5
Step 3: Analyze the ratios: All three ratios are equal to 3/2
Step 4: Conclusion: Since all corresponding side ratios are equal, the triangles ARE similar by SSS Similarity
Step 5: Identify scale factor: Scale factor = 3/2 or 1.5 Triangle DEF is 1.5 times larger than triangle ABC
Answer: Yes, the triangles are similar with scale factor 3/2 (or 1.5)
A tree casts a shadow 24 feet long at the same time a 6-foot person casts a 4-foot shadow. How tall is the tree?
The sun creates similar triangles (same angle).
Set up proportion:
Solve:
Answer: The tree is 36 feet tall
A tree casts a shadow 24 feet long at the same time a 6-foot person casts a shadow 4 feet long. How tall is the tree?
Step 1: Identify the similar triangles: The tree and its shadow form a right triangle The person and their shadow form a right triangle Since the sun angle is the same, these triangles are similar
Step 2: Set up the proportion: Tree height/Tree shadow = Person height/Person shadow h/24 = 6/4
Step 3: Simplify the right side: h/24 = 3/2
Step 4: Cross multiply: 2h = 24 × 3 2h = 72 h = 36
Step 5: Alternative method - find scale factor: Scale factor = 24/4 = 6 Tree height = 6 × 6 = 36 feet
Step 6: Verify: 36/24 = 3/2 ✓ 6/4 = 3/2 ✓ The ratios are equal
Answer: The tree is 36 feet tall
For YZ (corresponds to BC):
For XZ (corresponds to AC):
Answer: ,