Key takeaway: This geometry similar triangles test packs nine questions into 20 points, and most classes finish within 45 minutes. Therefore, students prove similarity, use the side-splitter theorem, find geometric means and measure a flagpole indirectly. Also, one question asks for a short similarity proof. Because every answer needs a proportion, write the ratios you use.
This 45-minute test covers similar triangles and proportions (G-SRT.B.4 and B.5): SAS and SSS similarity, the side-splitter theorem and its converse, geometric means in right triangles, a short proof and indirect measurement. It is worth 20 points.
- Grade: Geometry
- Chapter: Similar Triangles and Proportions
- Common Core: G-SRT.B.4, G-SRT.B.5
- Time: 45 minutes
- Total: 20 points (Question 1: 2 pts; Question 2: 2 pts; Question 3: 3 pts; Question 4: 2 pts; Question 5: 3 pts; Question 6: 3 pts; Question 7: 2 pts; Question 8: 2 pts; Question 9: 1 pt)
Geometry Similar Triangles Test: questions 1–5
1. Similarity by SAS (2 points)
In the figure, \(AB=8,\ AC=12,\ DE=12,\ DF=18\) and \(m\angle A=m\angle D=65^{\circ}\).
- Are the triangles similar? Name the criterion. (1 point)
- Write a similarity statement and the scale factor from ABC to DEF. (1 point)
2. Similar by SSS (2 points)
Decide whether each pair of triangles, given by side lengths, is similar. Justify each answer.
- 9, 12, 15 and 12, 16, 20 (1 point)
- 5, 6, 7 and 10, 12, 15 (1 point)
3. Triangle proportionality (3 points)
In triangle ABC, \(DE\) ∥ \(BC\). Use the figure.
- Find \(EC\). (1 point)
- Find \(DE\). (1 point)
- Name the similarity criterion that shows triangle ADE is similar to triangle ABC. (1 point)
4. Converse of the side-splitter theorem (2 points)
Points D and E lie on sides AB and AC of triangle ABC. Decide whether DE is parallel to BC.
- \(AD=6,\ DB=10,\ AE=9,\ EC=15\) (1 point)
- \(AD=5,\ DB=7,\ AE=8,\ EC=11\) (1 point)
5. Right triangle altitude (3 points)
Triangle ABC has a right angle at C, and CD is the altitude to the hypotenuse. \(AD=4\) and \(DB=16\). Find each length in simplest radical form.
- \(CD\) (1 point)
- \(AC\) (1 point)
- \(BC\) (1 point)
Geometry Similar Triangles Test: questions 6–9
6. Complete a similarity proof (3 points)
In the figure, segments AE and BD intersect at C, and \(AB\) ∥ \(DE\). Write a reason for each step to prove that triangle ABC is similar to triangle EDC.
| Statements | Reasons |
|---|---|
| \(AB\) ∥ \(DE\) | Given |
| \(\angle BAC\) ≅ \(\angle DEC\) | (1 point) ____________ |
| \(\angle ACB\) ≅ \(\angle ECD\) | (1 point) ____________ |
| Triangle ABC is similar to triangle EDC | (1 point) ____________ |
7. Indirect measurement (2 points)
A 6 foot tall person casts an 8 foot shadow at the same time that a flagpole casts a 36 foot shadow. Use similar triangles to find the height of the flagpole.
- Write a proportion. (1 point)
- Solve it. (1 point)
8. Similar triangles and algebra (2 points)
Triangle PQR is similar to triangle STU with \(PQ=x+2,\ QR=9,\ ST=10\) and \(TU=15\). Find \(x\) and \(PQ\).
9. Geometric mean (1 point)
Find the geometric mean of 5 and 20.
Which topics fill the geometry similar triangles test?
Question 1 checks SAS with sides 8 and 12 against sides 12 and 18. Because both ratios equal \(\frac{2}{3}\), the triangles are similar when the included angles match. Then question 2 asks for SSS decisions on pairs such as 9, 12, 15 and a larger triple. Meanwhile, question 3 uses a parallel segment for 3 points. Likewise, question 5 drops an altitude with pieces 4 and 16, so the altitude is 8.
What is a sensible pace for this test?
Questions 1 to 4 are short, so a third of the period is enough. Then spend 12 minutes on question 5 and the proof in question 6. After that, use 10 minutes for question 7, where a 6 foot person with an 8 foot shadow matches a 36 foot shadow. As a result, the flagpole is 27 feet tall. Meanwhile, question 8 sets \(\frac{x+2}{10}=\frac{9}{15}\), so \(x=4\). Finally, leave a few minutes for question 9, whose geometric mean is 10.
What errors cost points?
Often, students match the wrong sides. Also, some use SSA and call it similarity. Instead, check for an included angle. In addition, a few add lengths when they should multiply by the scale factor. Therefore, circle the corresponding sides first. Otherwise, a correct idea earns no credit.
How can you prepare for the similarity test?
List the three criteria and one example of each. Then solve one side-splitter problem and its converse. Also, practice a shadow problem with a drawing. As a result, the proportions will feel automatic.
See every chapter on the Geometry math page.
