This geometry similar triangles worksheet gives 24 problems on similar triangles and proportions, ordered from easy to hard, with a separate answer key.
These 24 problems practice similar triangles and proportions (G-SRT.B.4 and G-SRT.B.5). You will prove similarity with AA, SAS and SSS, use the triangle proportionality theorem and its converse, and apply geometric means in right triangles.
The last problems use similar triangles to measure trees, poles and rivers. Show every proportion before solving it.
Geometry Similar Triangles Worksheet: problems 1–5
Problem 1: Similar by SSS
The side lengths of triangles ABC and DEF are shown. Show that the triangles are similar, name the criterion, give the scale factor from ABC to DEF, and write a similarity statement.
Problem 2: Missing sides
Triangle ABC is similar to triangle DEF with \(AB=6,\ BC=8,\ AC=10\) and \(DE=9\). Find \(EF\) and \(DF\).
Problem 3: Similar by SAS
In the figure, \(AB=6,\ AC=9,\ DE=10,\ DF=15\) and \(m\angle A=m\angle D=50^{\circ}\). Are the triangles similar? Justify your answer and give the scale factor from ABC to DEF.
Problem 4: SSS or not
Decide whether each pair of triangles, given by side lengths, is similar. Show the ratios.
- 4, 6, 8 and 6, 9, 12
- 4, 6, 8 and 6, 9, 13
- 5, 12, 13 and 15, 36, 39
Problem 5: Parallel line and AA
In triangle ABC, \(DE\) ∥ \(BC\), \(AD=4,\ DB=2\) and \(AE=5\). Explain why triangle ADE is similar to triangle ABC, then find \(AC\) and \(EC\).
Geometry Similar Triangles Worksheet: problems 6–10
Problem 6: Side-splitter: find EC
In the figure \(DE\) ∥ \(BC\). Find \(EC\).
Problem 7: Side-splitter: find AD
In the figure \(DE\) ∥ \(BC\). Find \(AD\).
Problem 8: Converse: parallel
Points D and E lie on sides AB and AC with \(AD=3,\ DB=5,\ AE=4.5,\ EC=7.5\). Is DE parallel to BC? Justify.
Problem 9: Converse: not parallel
Points D and E lie on sides AB and AC with \(AD=4,\ DB=6,\ AE=5,\ EC=8\). Is DE parallel to BC? Justify.
Problem 10: Using the whole side
In the figure \(DE\) ∥ \(BC\), \(AD=3,\ DB=5\) and \(DE=4.5\). Find \(BC\).
Geometry Similar Triangles Worksheet: problems 11–15
Problem 11: Three parallel lines
Three parallel lines are cut by two transversals as shown. Find \(x\).
Problem 12: Geometric means
Find the geometric mean of each pair of numbers. Simplify radicals.
- 4 and 9
- 5 and 20
- 6 and 10
- 3 and 12
Problem 13: Altitude to the hypotenuse
Triangle ABC has a right angle at C, and CD is the altitude to the hypotenuse. If \(AD=4\) and \(DB=9\), find \(CD\).
Problem 14: Leg lengths
In right triangle ABC (right angle at C), the altitude CD splits the hypotenuse into \(AD=3\) and \(DB=9\). Find the legs \(AC\) and \(BC\).
Problem 15: Altitude, hypotenuse and legs
In right triangle ABC (right angle at C), altitude \(CD=12\) and \(AD=9\). Find \(DB\), \(AB\), \(AC\) and \(BC\).
Geometry Similar Triangles Worksheet: problems 16–20
Problem 16: Pythagorean theorem proof
Complete the reason for each step in this proof of the Pythagorean theorem.
| Statements | Reasons |
|---|---|
| Right triangle ABC with right angle at C; CD is the altitude to AB | Given |
| Triangle ACD is similar to triangle ABC, and triangle CBD is similar to triangle ABC | ? |
| \(\frac{AB}{AC}=\frac{AC}{AD}\) and \(\frac{AB}{BC}=\frac{BC}{DB}\) | ? |
| \(AC^{2}=AB\cdot AD\) and \(BC^{2}=AB\cdot DB\) | ? |
| \(AC^{2}+BC^{2}=AB\cdot AD+AB\cdot DB=AB(AD+DB)\) | ? |
| \(AC^{2}+BC^{2}=AB^{2}\) | ? |
Problem 17: Check the leg formula
A right triangle has legs \(AC=5\) and \(BC=12\) and hypotenuse \(AB=13\). The altitude from C meets AB at D. Use the leg formulas to find \(AD\) and \(DB\), then show that they add to \(AB\).
Problem 18: Shadow of a tree
Jamal is 5.5 feet tall and casts a 4 foot shadow. At the same time a tree casts a 28 foot shadow. How tall is the tree?
Problem 19: Mirror method
Maya’s eyes are 5 feet above the ground. She stands 6 feet from a mirror lying flat on the ground and sees the top of a flagpole in it. The mirror is 30 feet from the base of the pole. How tall is the pole?
Problem 20: Width of a river
Diego wants the distance TA across a river, where T is a tree on the far bank and A is the point directly across on his bank. He walks 24 feet along the bank to B, then 12 feet farther to C. He turns away from the river and walks until D is in line with B and T. Then \(CD=18\) feet. Find TA.
Geometry Similar Triangles Worksheet: problems 21–24
Problem 21: Solve with algebra
In triangle ABC, \(DE\) ∥ \(BC\), \(AD=x,\ DB=x+3,\ AE=6\) and \(EC=9\). Find \(x\) and the length of \(AB\).
Problem 22: Flower beds
Two triangular flower beds are similar with a ratio of 3 to 5. The smaller bed has a perimeter of 21 feet and a shortest side of 6 feet. Find the perimeter and the shortest side of the larger bed.
Problem 23: Sails
A small triangular sail has sides 6 ft, 8 ft and 10 ft. A larger sail is similar, and its longest side is 25 ft. Find the other two sides and the perimeter of the larger sail.
Problem 24: Lamppost shadow
Noah is 6 feet tall and stands 18 feet from the base of a lamppost. His shadow is 9 feet long. How tall is the lamppost?
How to use this geometry similar triangles worksheet
Print this geometry similar triangles worksheet as a PDF, solve every problem on paper, then compare with the answer key.
See every chapter on the Geometry math page.
