How can triangles WUV and XYZ be proven similar using …

Mathematics Questions
Short Answer
Step by Step Solution
Mathematics Questions

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?

Short Answer

To prove triangles WUV and XYZ are similar using the SAS similarity theorem, first identify the congruent angles and then assess the side lengths for proportional relationships. By demonstrating that the ratios of the sides are proportional, it is concluded that the triangles are similar based on the SAS similarity theorem.

Step-by-Step Solution

Step 1: Identify Angle Congruence

To prove that triangles WUV and XYZ are similar using the SAS similarity theorem, first identify the congruent angles. For these triangles, we have the following angle pairs:

  • Angle VUW is congruent to Angle YXZ
  • Angle UWV is congruent to Angle XZY
  • Angle UVW is congruent to Angle ZYX
This congruence of angles is essential for establishing similarity.

Step 2: Assess Side Lengths

Next, evaluate the lengths of the sides of each triangle to check for proportional relationships. The side lengths for triangle WUV are:

  • UW = 40
  • VU = 50
For triangle XYZ, the sides are:
  • XZ = 32
  • YX = 40
Calculate the ratios of the corresponding sides to see if they maintain proportionality.

Step 3: Verify Proportionality and Conclude Similarity

Now, calculate the ratios to confirm that they are indeed proportional. The proportion is derived from:

  • (VU/WU) = (YX/XZ) => (50/40) = (40/32)
Both simplify to (5/4), demonstrating that the sides are proportional. Since we have two proportional sides and their included angles are congruent, we can conclude that triangles WUV and XYZ are similar, confirming the application of the SAS similarity theorem.

Related Concepts

Congruent Angles

Angles that are equal in measure and indicate that the triangles have the same shape

Proportional Relationships

Relationships that exist when the ratios of the lengths of corresponding sides of two triangles are equal

Sas Similarity Theorem

A theorem that states that if two triangles have a pair of corresponding sides in proportion and the included angles are congruent, then the triangles are similar.

Table Of Contents
  1. 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?

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