Question
Consider the two triangles.
Triangles W U V and X Z Y are shown. Angles V U W and Y X Z are congruent. Angles U W V and X Z Y are congruent. Angles U V W and Z Y X are congruent. The length of side V W is 60 and the length of side Z Y is 48. The length of side Y X is 40 and the length of V U is 50. The length of side U W is 40 and the length of X Z is 32.
How can the triangles be proven similar by the SAS similarity theorem?
Show that the ratios StartFraction X Y Over V U EndFraction and StartFraction Y Z Over V W EndFraction are equivalent, and ∠U ≅ ∠X.
Show that the ratios StartFraction U V Over X Y EndFraction and StartFraction W V Over Z Y EndFraction are equivalent, and ∠V ≅ ∠Y.
Show that the ratios StartFraction U W Over Z X EndFraction and StartFraction X Y Over W V EndFraction are equivalent, and ∠W ≅ ∠X.
Show that the ratios StartFraction X Z Over W U EndFraction and StartFraction Z Y Over W V EndFraction are equivalent, and ∠U ≅ ∠Z.
Triangles W U V and X Z Y are shown. Angles V U W and Y X Z are congruent. Angles U W V and X Z Y are congruent. Angles U V W and Z Y X are congruent. The length of side V W is 60 and the length of side Z Y is 48. The length of side Y X is 40 and the length of V U is 50. The length of side U W is 40 and the length of X Z is 32.
How can the triangles be proven similar by the SAS similarity theorem?
Show that the ratios StartFraction X Y Over V U EndFraction and StartFraction Y Z Over V W EndFraction are equivalent, and ∠U ≅ ∠X.
Show that the ratios StartFraction U V Over X Y EndFraction and StartFraction W V Over Z Y EndFraction are equivalent, and ∠V ≅ ∠Y.
Show that the ratios StartFraction U W Over Z X EndFraction and StartFraction X Y Over W V EndFraction are equivalent, and ∠W ≅ ∠X.
Show that the ratios StartFraction X Z Over W U EndFraction and StartFraction Z Y Over W V EndFraction are equivalent, and ∠U ≅ ∠Z.
Answers
Answered by
GPT-4o mini
To show that the triangles \( \triangle W U V \) and \( \triangle X Z Y \) are similar by the SAS similarity theorem, we need to establish that two pairs of corresponding sides are in proportion and the included angles between those sides are congruent.
### Given:
1. Angles:
- \( \angle V U W \cong \angle Y X Z \)
- \( \angle U W V \cong \angle X Z Y \)
- \( \angle U V W \cong \angle Z Y X \)
2. Side lengths:
- \( VW = 60 \)
- \( ZY = 48 \)
- \( YX = 40 \)
- \( UV = 50 \)
- \( UW = 40 \)
- \( XZ = 32 \)
### Step 1: Verify Proportions and Angles
**First pair:**
- Sides:
- \( UV = 50 \)
- \( XY = 40 \)
- Ratio:
\[
\frac{XY}{UV} = \frac{40}{50} = \frac{4}{5}
\]
- Corresponding Angle:
\[
\angle U \cong \angle X
\]
**Second pair:**
- Sides:
- \( VW = 60 \)
- \( ZY = 48 \)
- Ratio:
\[
\frac{ZY}{VW} = \frac{48}{60} = \frac{4}{5}
\]
Both ratios are equivalent:
\[
\frac{XY}{UV} = \frac{ZY}{VW} = \frac{4}{5}
\]
And since \( \angle U \cong \angle X \), we satisfy the conditions for SAS similarity.
### Conclusion for SAS:
By SAS similarity theorem, since we have two pairs of corresponding sides that are in proportion \( \left( \frac{XY}{UV} = \frac{4}{5} \text{ and } \frac{ZY}{VW} = \frac{4}{5} \right) \) and included angles that are congruent \( ( \angle U \cong \angle X) \), the triangles \( \triangle W U V \) and \( \triangle X Z Y \) are similar.
### Additional Verification Steps:
We can perform additional checks for the other angles and side pairs as outlined in your question, but the above proves that \( \triangle W U V \sim \triangle X Z Y \) by the SAS similarity theorem:
- Since we’ve established the congruency of \( \angle U \cong \angle X \), you can continue showing that other ratios and angles similarly support similarity.
These confirmations reinforce the triangle similarity, but the core requirement under SAS is already satisfied in the above analysis.
Thus, the triangles \( \triangle W U V \) and \( \triangle X Z Y \) are similar by the SAS similarity theorem.
### Given:
1. Angles:
- \( \angle V U W \cong \angle Y X Z \)
- \( \angle U W V \cong \angle X Z Y \)
- \( \angle U V W \cong \angle Z Y X \)
2. Side lengths:
- \( VW = 60 \)
- \( ZY = 48 \)
- \( YX = 40 \)
- \( UV = 50 \)
- \( UW = 40 \)
- \( XZ = 32 \)
### Step 1: Verify Proportions and Angles
**First pair:**
- Sides:
- \( UV = 50 \)
- \( XY = 40 \)
- Ratio:
\[
\frac{XY}{UV} = \frac{40}{50} = \frac{4}{5}
\]
- Corresponding Angle:
\[
\angle U \cong \angle X
\]
**Second pair:**
- Sides:
- \( VW = 60 \)
- \( ZY = 48 \)
- Ratio:
\[
\frac{ZY}{VW} = \frac{48}{60} = \frac{4}{5}
\]
Both ratios are equivalent:
\[
\frac{XY}{UV} = \frac{ZY}{VW} = \frac{4}{5}
\]
And since \( \angle U \cong \angle X \), we satisfy the conditions for SAS similarity.
### Conclusion for SAS:
By SAS similarity theorem, since we have two pairs of corresponding sides that are in proportion \( \left( \frac{XY}{UV} = \frac{4}{5} \text{ and } \frac{ZY}{VW} = \frac{4}{5} \right) \) and included angles that are congruent \( ( \angle U \cong \angle X) \), the triangles \( \triangle W U V \) and \( \triangle X Z Y \) are similar.
### Additional Verification Steps:
We can perform additional checks for the other angles and side pairs as outlined in your question, but the above proves that \( \triangle W U V \sim \triangle X Z Y \) by the SAS similarity theorem:
- Since we’ve established the congruency of \( \angle U \cong \angle X \), you can continue showing that other ratios and angles similarly support similarity.
These confirmations reinforce the triangle similarity, but the core requirement under SAS is already satisfied in the above analysis.
Thus, the triangles \( \triangle W U V \) and \( \triangle X Z Y \) are similar by the SAS similarity theorem.