High School Geometry Common Core Similarity Criteria , 43% OFF - Free Printable

To solve the problem of determining whether the pairs of triangles are similar and identifying the similarity criteria (SSS, SAS, AA), we need to analyze each pair step by step. Here's a detailed explanation for each part:

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Part 1: Are the following pairs of triangles similar? If they are, then name their similarity criteria.

#### a)
- Triangles: Two right triangles with one angle marked as \(90^\circ\).
- Analysis: Both triangles have a right angle (\(90^\circ\)) and share another angle (since the sum of angles in a triangle is \(180^\circ\)). This means the third angle is also equal.
- Conclusion: The triangles are similar by the AA (Angle-Angle) criterion.
- Answer: Yes, AA.

#### b)
- Triangles: Two triangles with sides labeled as \(5\), \(6\), and \(7\) for one triangle, and \(10\), \(12\), and \(14\) for the other.
- Analysis: The sides of the second triangle are exactly twice the sides of the first triangle (\(10 = 2 \times 5\), \(12 = 2 \times 6\), \(14 = 2 \times 7\)).
- Conclusion: The triangles are similar by the SSS (Side-Side-Side) criterion.
- Answer: Yes, SSS.

#### c)
- Triangles: Two triangles with one angle marked as \(30^\circ\) and another angle marked as \(60^\circ\).
- Analysis: Both triangles have two corresponding angles equal (\(30^\circ\) and \(60^\circ\)). The third angle must also be equal (\(90^\circ\)).
- Conclusion: The triangles are similar by the AA (Angle-Angle) criterion.
- Answer: Yes, AA.

#### d)
- Triangles: Two triangles with one side marked as \(12\) and another side marked as \(18\), but no angles or proportional sides are given.
- Analysis: There is not enough information to determine similarity. We need either more angles or proportional sides.
- Conclusion: The triangles are not necessarily similar.
- Answer: No.

#### e)
- Triangles: Two triangles with one angle marked as \(45^\circ\) and another angle marked as \(45^\circ\).
- Analysis: Both triangles have two corresponding angles equal (\(45^\circ\) and \(45^\circ\)). The third angle must also be equal (\(90^\circ\)).
- Conclusion: The triangles are similar by the AA (Angle-Angle) criterion.
- Answer: Yes, AA.

#### f)
- Triangles: Two triangles with one side marked as \(12\) and another side marked as \(18\), but no angles or proportional sides are given.
- Analysis: There is not enough information to determine similarity. We need either more angles or proportional sides.
- Conclusion: The triangles are not necessarily similar.
- Answer: No.

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Part 2: Are the following pairs of triangles similar? If YES, name the similarity criteria (SSS, SAS, AA) and create a similarity statement. If NO, just circle No.

#### a)
- Triangles: \(\triangle ABC\) and \(\triangle DEF\).
- Given: \(AB = 6\), \(BC = 8\), \(AC = 10\); \(DE = 3\), \(EF = 4\), \(DF = 5\).
- Analysis: The sides of \(\triangle DEF\) are half the sides of \(\triangle ABC\) (\(3 = \frac{6}{2}\), \(4 = \frac{8}{2}\), \(5 = \frac{10}{2}\)).
- Conclusion: The triangles are similar by the SSS (Side-Side-Side) criterion.
- Similarity Statement: \(\triangle ABC \sim \triangle DEF\) by SSS.
- Answer: Yes, SSS; \(\triangle ABC \sim \triangle DEF\).

#### b)
- Triangles: \(\triangle GHI\) and \(\triangle JKL\).
- Given: \(\angle G = \angle J = 45^\circ\), \(\angle H = \angle K = 60^\circ\).
- Analysis: Both triangles have two corresponding angles equal (\(45^\circ\) and \(60^\circ\)). The third angle must also be equal (\(75^\circ\)).
- Conclusion: The triangles are similar by the AA (Angle-Angle) criterion.
- Similarity Statement: \(\triangle GHI \sim \triangle JKL\) by AA.
- Answer: Yes, AA; \(\triangle GHI \sim \triangle JKL\).

#### c)
- Triangles: \(\triangle MNO\) and \(\triangle PQR\).
- Given: \(MN = 5\), \(NO = 10\), \(OP = 15\); \(PQ = 10\), \(QR = 20\), \(RP = 30\).
- Analysis: The sides of \(\triangle PQR\) are twice the sides of \(\triangle MNO\) (\(10 = 2 \times 5\), \(20 = 2 \times 10\), \(30 = 2 \times 15\)).
- Conclusion: The triangles are similar by the SSS (Side-Side-Side) criterion.
- Similarity Statement: \(\triangle MNO \sim \triangle PQR\) by SSS.
- Answer: Yes, SSS; \(\triangle MNO \sim \triangle PQR\).

#### d)
- Triangles: \(\triangle STU\) and \(\triangle VWX\).
- Given: \(\angle S = \angle V = 30^\circ\), \(\angle T = \angle W = 60^\circ\).
- Analysis: Both triangles have two corresponding angles equal (\(30^\circ\) and \(60^\circ\)). The third angle must also be equal (\(90^\circ\)).
- Conclusion: The triangles are similar by the AA (Angle-Angle) criterion.
- Similarity Statement: \(\triangle STU \sim \triangle VWX\) by AA.
- Answer: Yes, AA; \(\triangle STU \sim \triangle VWX\).

#### e)
- Triangles: \(\triangle YZT\) and \(\triangle XYZ\).
- Given: \(\angle Y = \angle X = 45^\circ\), \(\angle Z = \angle Y = 45^\circ\).
- Analysis: Both triangles have two corresponding angles equal (\(45^\circ\) and \(45^\circ\)). The third angle must also be equal (\(90^\circ\)).
- Conclusion: The triangles are similar by the AA (Angle-Angle) criterion.
- Similarity Statement: \(\triangle YZT \sim \triangle XYZ\) by AA.
- Answer: Yes, AA; \(\triangle YZT \sim \triangle XYZ\).

#### f)
- Triangles: \(\triangle LMN\) and \(\triangle OPQ\).
- Given: \(LM = 6\), \(MN = 8\), \(\angle M = 90^\circ\); \(OP = 3\), \(PQ = 4\), \(\angle P = 90^\circ\).
- Analysis: The sides of \(\triangle OPQ\) are half the sides of \(\triangle LMN\) (\(3 = \frac{6}{2}\), \(4 = \frac{8}{2}\)), and both triangles have a right angle.
- Conclusion: The triangles are similar by the SAS (Side-Angle-Side) criterion.
- Similarity Statement: \(\triangle LMN \sim \triangle OPQ\) by SAS.
- Answer: Yes, SAS; \(\triangle LMN \sim \triangle OPQ\).

#### g)
- Triangles: \(\triangle RST\) and \(\triangle UVW\).
- Given: \(RS = 5\), \(ST = 12\), \(RT = 13\); \(UV = 10\), \(VW = 24\), \(UW = 26\).
- Analysis: The sides of \(\triangle UVW\) are twice the sides of \(\triangle RST\) (\(10 = 2 \times 5\), \(24 = 2 \times 12\), \(26 = 2 \times 13\)).
- Conclusion: The triangles are similar by the SSS (Side-Side-Side) criterion.
- Similarity Statement: \(\triangle RST \sim \triangle UVW\) by SSS.
- Answer: Yes, SSS; \(\triangle RST \sim \triangle UVW\).

#### h)
- Triangles: \(\triangle XYZ\) and \(\triangle LMN\).
- Given: \(\angle X = \angle L = 30^\circ\), \(\angle Y = \angle M = 60^\circ\).
- Analysis: Both triangles have two corresponding angles equal (\(30^\circ\) and \(60^\circ\)). The third angle must also be equal (\(90^\circ\)).
- Conclusion: The triangles are similar by the AA (Angle-Angle) criterion.
- Similarity Statement: \(\triangle XYZ \sim \triangle LMN\) by AA.
- Answer: Yes, AA; \(\triangle XYZ \sim \triangle LMN\).

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Final Answer:

\[
\boxed{
\begin{array}{ll}
\text{Part 1:} & \text{a) Yes, AA; b) Yes, SSS; c) Yes, AA; d) No; e) Yes, AA; f) No} \\
\text{Part 2:} & \text{a) Yes, SSS; } \triangle ABC \sim \triangle DEF \\
& \text{b) Yes, AA; } \triangle GHI \sim \triangle JKL \\
& \text{c) Yes, SSS; } \triangle MNO \sim \triangle PQR \\
& \text{d) Yes, AA; } \triangle STU \sim \triangle VWX \\
& \text{e) Yes, AA; } \triangle YZT \sim \triangle XYZ \\
& \text{f) Yes, SAS; } \triangle LMN \sim \triangle OPQ \\
& \text{g) Yes, SSS; } \triangle RST \sim \triangle UVW \\
& \text{h) Yes, AA; } \triangle XYZ \sim \triangle LMN \\
\end{array}
}
\]