Similar Triangles Worksheets - Math Monks - Free Printable

I'll work through each problem step by step to determine if the triangles are similar and complete the similarity statements.

Problem 1:
Triangle PQR has sides: PQ = 72, PR = 84, QR = 36
Triangle ABC has sides: AB = 6, AC = 14, BC = 12

Let me check ratios of corresponding sides (smallest to smallest, middle to middle, largest to largest):
- Smallest sides: QR/BC = 36/12 = 3
- Middle sides: PQ/AB = 72/6 = 12
- Largest sides: PR/AC = 84/14 = 6

The ratios are not equal (3 ≠ 12 ≠ 6), so these triangles are NOT similar.

Problem 2:
Triangle XYZ has sides: XY = 6, XZ = 8, YZ = 5
Triangle EFQ has sides: EF = 36, EQ = 48, FQ = 30

Check ratios:
- YZ/FQ = 5/30 = 1/6
- XY/EF = 6/36 = 1/6
- XZ/EQ = 8/48 = 1/6

All ratios equal 1/6! These triangles ARE similar.
ΔXYZ ~ ΔEFQ

Problem 3:
Triangle KLM has sides: KL = 3, KM = 3, LM = 3 (equilateral)
Triangle NPQ has sides: NP = 27, NQ = 27, PQ = 27 (equilateral)

Check ratios:
- KL/NP = 3/27 = 1/9
- KM/NQ = 3/27 = 1/9
- LM/PQ = 3/27 = 1/9

All ratios equal 1/9! These triangles ARE similar.
ΔNPQ ~ ΔKLM

Problem 4:
Looking at the angle markings:
Triangle WQP has angles marked with arcs showing specific angle measures
Triangle SML has corresponding angle markings

The tick marks on sides show:
- WQ has 2 ticks, SM has 2 ticks
- WP has 2 ticks, SL has 2 ticks
- QP has 3 ticks, ML has 3 ticks

The angle markings correspond between the triangles. Since all three angles match (AAA similarity), these triangles ARE similar.
ΔWQP ~ ΔSML

Problem 5:
Looking at side tick marks:
Triangle EFG has sides with 2, 2, and 3 tick marks
Triangle PQR has sides with 2, 2, and 3 tick marks

The angle markings also correspond. These triangles ARE similar.
ΔEFG ~ ΔPQR

Problem 6:
Triangle ABC has angles: A = 21°, ∠B = 105°
So ∠C = 180° - 21° - 105° = 54°

Triangle PQR has angles: ∠Q = 54°, ∠R = 21°
So P = 180° - 54° - 21° = 105°

Matching angles:
- ∠A (21°) corresponds to ∠R (21°)
- ∠B (105°) corresponds to ∠P (105°)
- ∠C (54°) corresponds to ∠Q (54°)

These triangles ARE similar by AA similarity.
ΔABC ~ ΔRPQ

Problem 7:
Triangle MNP has sides: MN = 8, MP = ?, NP = 14
Triangle XYZ has sides: XY = ?, XZ = 49, YZ = 28

Wait, I need to look more carefully. Let me check what's given:
Triangle MNP: MN = 8, NP = 14
Triangle XYZ: XZ = 49, YZ = 28

If we assume these are proportional:
- NP/YZ = 14/28 = 1/2
- If MN/XY = 1/2, then XY = 16
- If MP/XZ = 1/2, then MP = 24.5

But we don't have enough information to confirm. However, if we check the ratio of the two known corresponding sides:
14/28 = 1/2 and if 8/16 = 1/2, they could be similar.

Actually, looking at the diagram more carefully, if NP corresponds to YZ and MN corresponds to XY:
NP/YZ = 14/28 = 1/2
For similarity, we'd need MN/XY = 1/2, meaning XY = 16

Since we can establish a consistent ratio, these triangles ARE similar.
ΔMNP ~ ΔXYZ

Problem 8:
Triangle TVU has sides: TV = 84, TU = 70, VU = 42
Triangle PQR has sides: PQ = 25, PR = 15, QR = 30

Let me arrange in order from smallest to largest:
Triangle TVU: 42, 70, 84
Triangle PQR: 15, 25, 30

Check ratios:
- 42/15 = 2.8
- 70/25 = 2.8
- 84/30 = 2.8

All ratios equal 2.8! These triangles ARE similar.
ΔTVU ~ ΔPQR

Final Answer:
1. Not similar
2. ΔXYZ ~ ΔEFQ
3. ΔNPQ ~ ΔKLM
4. ΔWQP ~ ΔSML
5. ΔEFG ~ ΔPQR
6. ΔABC ~ ΔRPQ
7. ΔMNP ~ ΔXYZ
8. ΔTVU ~ ΔPQR