Question

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.
2. the measure of the vertex angle of the yellow triangle is 30°. find the measures of the base angles.
3. trace the color wheel. then form a triangle whose vertices are the midpoints of the bases of the red, yellow, and blue triangles. (these colors are the primary colors.) what type of triangle is this?

Explanation:

Problem 25

Step1: Identify Triangle Type

The triangles in the color wheel are isosceles with congruent vertex angles (given). Yellow and purple triangles are opposite (complementary colors), so their vertex angles are equal (from color wheel symmetry, complementary triangles have same vertex angle as they are opposite in a symmetric wheel).

Step2: Check Congruence Conditions

For isosceles triangles, if two triangles have equal vertex angles and equal side lengths (since the color wheel is a regular polygon - like structure, the sides from the center to the vertices (legs of isosceles triangles) are equal, and the base sides (along the wheel's edge) are equal as the triangles are congruent in the wheel's design), by SAS (Side - Angle - Side) congruence criterion, the yellow and purple triangles are congruent. Also, since all the isosceles triangles in the wheel have congruent vertex angles and equal leg lengths (radii of the circle - like wheel), triangles opposite each other (complementary color triangles) will have equal legs and equal vertex angles, hence congruent.

Step1: Recall Isosceles Triangle Angle Sum

In an isosceles triangle, the sum of interior angles is $180^{\circ}$, and the two base angles are equal. Let the measure of each base angle be $x$.

Step2: Set Up Equation

The vertex angle is $30^{\circ}$, so the equation is $30^{\circ}+x + x=180^{\circ}$.

Step3: Solve for $x$

Simplify the equation: $30^{\circ}+2x = 180^{\circ}$. Subtract $30^{\circ}$ from both sides: $2x=180^{\circ}- 30^{\circ}=150^{\circ}$. Then divide both sides by 2: $x=\frac{150^{\circ}}{2} = 75^{\circ}$.

Step1: Analyze Primary Color Triangles

The red, yellow, and blue triangles are isosceles with congruent vertex angles (from the wheel's design, as all triangles have congruent vertex angles). The midpoints of their bases: since the red, yellow, and blue triangles are equally spaced (as they are primary colors, likely separated by equal angles in the wheel), the distances between the midpoints of their bases will be equal.

Step2: Determine Triangle Type

In a circle (the color wheel is circular), the midpoints of the bases of three equally - spaced isosceles triangles (red, yellow, blue) will form a triangle where all sides are equal. Because the arcs between the primary colors are equal (due to the wheel's symmetry), the chords (distances between midpoints) will be equal. A triangle with all sides equal is an equilateral triangle. Also, since the angles between the lines connecting the midpoints will be equal (due to the symmetry of the primary colors' positions), the triangle formed is equilateral.

Answer:

The yellow and purple triangles are isosceles with congruent vertex angles (given the wheel's triangles have congruent vertex angles) and equal leg lengths (radii of the circular wheel). By SAS congruence (equal legs, equal vertex angle, equal included angle), or by the wheel's symmetry (opposite triangles have same side lengths and angles), the yellow triangle is congruent to the purple triangle.

Problem 26