Question

modeling with mathematics in exercises 25–28, use the diagram based on the color wheel. the 12 triangles in the diagram are isosceles triangles with congruent vertex angles.

1. complementary colors lie directly opposite each other on the color wheel. explain how you know that the yellow triangle is congruent to the purple triangle.

Explanation:

Step1: Identify Triangle Properties

The 12 triangles are isosceles with congruent vertex angles. So, all triangles have two equal sides (legs) and equal vertex angles.

Step2: Check Congruence Condition

For the yellow and purple triangles:

• They are isosceles (given: all 12 triangles are isosceles).
• Their vertex angles are congruent (given: all vertex angles are congruent).
• The sides forming the vertex angle (legs) are equal because the color wheel is a circle, so the radii (legs of the isosceles triangles) are equal.

By the SAS (Side - Angle - Side) congruence criterion, if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent. Here, the two legs (radii) and the included vertex angle of the yellow triangle are equal to those of the purple triangle.

Answer:

The yellow and purple triangles are congruent by the SAS congruence criterion. All 12 triangles are isosceles (two equal legs, as radii of the circle) with congruent vertex angles. So, two sides (legs) and the included angle of the yellow triangle equal those of the purple triangle, satisfying SAS.