This geometry similar triangles test answer key gives the full solution and the scoring guide for each of the 9 test questions on similar triangles and proportions.
Use this key to score the similar triangles test out of 20 points. Each solution shows the proportion or reason expected, and the scoring note explains how partial credit is given.
Geometry Similar Triangles Test Answer Key: questions 1–5
Question 1: Similarity by SAS (2 points)
- \(\frac{DE}{AB}=\frac{12}{8}=\frac{3}{2}\) and \(\frac{DF}{AC}=\frac{18}{12}=\frac{3}{2}\), and the included angles are congruent (\(65^{\circ}\)). The triangles are similar by SAS.
- ABC ~ DEF with scale factor \(\frac{3}{2}\).
Similar by SAS; ABC ~ DEF; k = 3/2.
Scoring: 1 point for both ratios and the criterion SAS; 1 point for a correctly ordered statement and k = 3/2.
Question 2: Similar by SSS (2 points)
- \(\frac{12}{9}=\frac{16}{12}=\frac{20}{15}=\frac{4}{3}\): similar by SSS.
- \(\frac{10}{5}=2,\ \frac{12}{6}=2,\ \frac{15}{7}\) ≈ \(2.14\): the ratios are not all equal, so not similar.
(a) similar, ratio 4/3; (b) not similar.
Scoring: 1 point for each correct decision supported by all three ratios.
Question 3: Triangle proportionality (3 points)
- \(\frac{9}{6}=\frac{12}{y}\), so \(9y=72\) and \(y=8\).
- \(AB=15\), so \(\frac{AD}{AB}=\frac{9}{15}=\frac{3}{5}=\frac{DE}{20}\) and \(DE=12\).
- Angle A is shared and \(\angle ADE\) ≅ \(\angle ABC\) (corresponding angles), so AA.
EC = 8; DE = 12; AA.
Scoring: 1 point for EC = 8; 1 point for DE = 12 using the ratio of whole sides; 1 point for AA with a reason.
Question 4: Converse of the side-splitter theorem (2 points)
- \(\frac{6}{10}=\frac{3}{5}\) and \(\frac{9}{15}=\frac{3}{5}\): parallel by the converse.
- \(\frac{5}{7}\) ≈ \(0.714\) and \(\frac{8}{11}\) ≈ \(0.727\): not equal, so not parallel.
(a) parallel; (b) not parallel.
Scoring: 1 point for each correct conclusion with both ratios shown.
Question 5: Right triangle altitude (3 points)
- \(CD^{2}=4\cdot 16=64\), so \(CD=8\).
- \(AB=20\) and \(AC^{2}=AD\cdot AB=4\cdot 20=80\), so \(AC=\sqrt{80}=4\sqrt{5}\).
- \(BC^{2}=DB\cdot AB=16\cdot 20=320\), so \(BC=\sqrt{320}=8\sqrt{5}\).
Check: \(80+320=400=20^{2}\). CD = 8, AC = \(4\sqrt{5}\), BC = \(8\sqrt{5}\).
Scoring: 1 point for each length, with the correct geometric mean setup.
Geometry Similar Triangles Test Answer Key: questions 6–9
Question 6: Complete a similarity proof (3 points)
| Statements | Reasons |
|---|---|
| \(AB\) ∥ \(DE\) | Given |
| \(\angle BAC\) ≅ \(\angle DEC\) | Alternate interior angles are congruent (transversal AE) |
| \(\angle ACB\) ≅ \(\angle ECD\) | Vertical angles are congruent |
| Triangle ABC is similar to triangle EDC | AA similarity |
Scoring: 1 point for each correct reason in steps 2, 3 and 4.
Question 7: Indirect measurement (2 points)
- \(\frac{h}{36}=\frac{6}{8}\)
- \(8h=216\), so \(h=27\).
The flagpole is 27 feet tall.
Scoring: 1 point for a correct proportion with matching quantities; 1 point for h = 27 ft.
Question 8: Similar triangles and algebra (2 points)
\(\frac{PQ}{ST}=\frac{QR}{TU}\): \(\frac{x+2}{10}=\frac{9}{15}\). Cross-multiply: \(15(x+2)=90\), so \(x+2=6\) and \(x=4\).
\(PQ=4+2=6\). x = 4 and PQ = 6.
Scoring: 1 point for the correct proportion; 1 point for x = 4 and PQ = 6.
Question 9: Geometric mean (1 point)
\(\sqrt{5\cdot 20}=\sqrt{100}=10\). 10
Scoring: 1 point for 10.
Scoring summary
20 points: add the points of every question of this geometry similar triangles test answer key.
How to use this geometry similar triangles test answer key
Use this geometry similar triangles test answer key to score the test fairly and to see which skills need more practice.
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Geometry Similar Triangles and Proportions test answer key
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