Geometry Similar Triangle Theorems Worksheet (PDF)

Geometry math worksheet: Similar Triangle Theorems – Maths PDF

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This similar triangles worksheet for Geometry covers similar triangle theorems (Common Core G-SRT.B.4, G-SRT.B.5), with a free PDF to print.

These 20 problems practice the Triangle Proportionality Theorem, similarity tests and geometric means (Common Core G-SRT.B.4, G-SRT.B.5). The first problems are quick one-step checks of the definitions and formulas from the lesson. The middle problems ask for diagrams, equations and careful reasoning in the style of a high school Geometry class. The last six problems are applications with mirrors, ladders, sails and land parcels.

Geometry · Common Core State Standards: G-SRT.B.4, G-SRT.B.5

Similar triangles worksheet: exercises 1 to 10

Exercise 1 – Side splitter

A line parallel to BC splits both sides in the same ratio.

DE is parallel to BC. AD = 4, DB = 6 and AE = 6. Find EC.

Exercise 2 – Find AD

In triangle ABC, DE is parallel to BC with D on AB and E on AC. \(AD = x\), \(DB = 10\), \(AE = 6\) and \(EC = 15\). Find x.

Exercise 3 – Find BC

In triangle ABC, DE is parallel to BC. AD = 8, AB = 20 and DE = 6. Find BC.

Exercise 4 – Parallel? yes

Is DE parallel to BC?

Is DE parallel to BC if AD = 5, DB = 7.5, AE = 4 and EC = 6? Explain.

Exercise 5 – Parallel? no

Is DE parallel to BC if AD = 6, DB = 9, AE = 5 and EC = 8? Explain.

Exercise 6 – Proportion with x

DE is parallel to BC. \(AD = x + 1\), \(DB = 6\), \(AE = x + 3\) and \(EC = 9\). Find x, AD and AE.

Exercise 7 – SAS similarity

SAS similarity: the included angles are equal and the sides are proportional (6/9 = 8/12).

Triangle ABC has legs 6 and 8. Triangle DEF has legs 9 and 12. Both have a right angle between the legs. Are they similar? Name the test and the scale factor.

Exercise 8 – SSS similarity

Triangle 1 has sides 4, 5, 6. Triangle 2 has sides 10, 12.5, 15. Triangle 3 has sides 8, 10, 13. Which triangles are similar to Triangle 1?

Exercise 9 – Altitude to the hypotenuse

Right triangle with legs 6 and 8.

Right triangle ABC has legs 6 and 8. Find the hypotenuse, the altitude h to the hypotenuse and the two segments AD and DB.

Exercise 10 – Geometric mean

Find the geometric mean of 4 and 9.

Exercises 11 and up

Exercise 11 – Altitude from segments

AD = 4 and DB = 9.

The altitude to the hypotenuse of a right triangle splits the hypotenuse into segments of length 4 and 9. Find the altitude.

Exercise 12 – Legs from segments

The hypotenuse of a right triangle is 25 and the altitude to it splits the hypotenuse into 9 and 16. Find both legs.

Exercise 13 – Pythagorean check

Use the legs 15 and 20 and hypotenuse 25 from the previous problem to verify \(a^{2} + b^{2} = c^{2}\) and connect it to \(pc + qc\).

Exercise 14 – Overlapping triangles

DE parallel to BC.

DE is parallel to BC, AD = 6, AB = 15 and BC = 20. Find DE.

Exercise 15 – Ratio of perimeters

Two similar triangles have corresponding sides 6 and 15. The smaller has perimeter 28. Find the perimeter of the larger and the ratio of their areas.

Exercise 16 – Three parallel lines

Three parallel lines cut two transversals. On one transversal the segments are 3 and 5. On the other the segments are x and 10. Find x.

Exercise 17 – Mirror problem

Reflection method: two similar right triangles.

Liam places a mirror on the ground 20 feet from a building and stands 4 feet behind the mirror. His eye is 6 feet above the ground and he sees the top of the building in the mirror. Find the height of the building.

Exercise 18 – A-frame ladder

An A-frame ladder has legs 10 feet long. A crossbar parallel to the floor is attached 4 feet from the top along each leg. The legs are 5 feet apart at the floor. Find the length of the crossbar.

Exercise 19 – Sail altitude

Sail with legs 15 and 20.

A right triangular sail has legs 15 feet and 20 feet. Find the length of the altitude from the right angle to the long edge.

Exercise 20 – Land parcels

Two triangular parcels are similar with scale factor 3 : 5. The smaller parcel has an area of 54 square yards. Find the area of the larger parcel.

Key facts behind this similar triangles worksheet

Key facts

Triangle Proportionality: DE parallel to BC gives AD/DB = AE/EC.
Triangle similarity tests: AA, SAS~ and SSS~.
Altitude to the hypotenuse: h squared = p times q.
Leg relationship: a squared = p times c.
Similar triangles let you measure heights and widths indirectly.

How to use this similar triangles worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

Download the similar triangles worksheet as a PDF with the button below and keep it with your notes.

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