Question

Sketch and label two similar right triangles \(A B C\) and \(D E F\) with right angles at \(C\) and \(F\). Let the measures of angles \(A\) and \(D\) be 30 . Name the corresponding sides that are proportional.

Short answer

\(AB \sim DE\), \(BC \sim EF\), and \(AC \sim DF\) are proportional sides.

Step-by-step solution

Understand the Problem

We are asked to sketch two similar right triangles, both having one angle of 30 degrees and the corresponding sides proportional. Similar triangles have equal corresponding angles and proportional sides. Right triangle properties will also be utilized here.

Sketch Triangle ABC

Draw a right triangle \(\triangle ABC\) with a right angle at \(C\). Let angle \(A\) be 30 degrees. Thus, angle \(B\) becomes 60 degrees because the sum of angles in a triangle is 180 degrees. Label the triangle's vertices as \(A, B,\) and \(C\).

Sketch Similar Triangle DEF

Draw another right triangle \(\triangle DEF\) with a right angle at \(F\) and angle \(D\) also 30 degrees. By the same reasoning, angle \(E\) will be 60 degrees. Label the triangle's vertices as \(D, E,\) and \(F\).

Identify Proportional Sides

In similar triangles, corresponding sides are proportional. Therefore, the hypotenuse \(AB\) in \(\triangle ABC\) is proportional to \(DE\) in \(\triangle DEF\), the side opposite the 30-degree angle \(BC\) is proportional to \(EF\), and the side adjacent to the 30-degree angle \(AC\) is proportional to \(DF\).

Label the Proportional Sides

Clearly label the proportional sides in your sketches: \(AB \sim DE\), \(BC \sim EF\), and \(AC \sim DF\).

Key concepts

Right Triangle Properties

Right triangles are a special type of triangle where one of the angles is always 90 degrees. This 90-degree angle makes for some interesting properties which are unique to right triangles:

• The side opposite the right angle is the hypotenuse. It's always the longest side.
• Other two sides are called the legs of the triangle. They make up the right angle.
• Using the Pythagorean theorem is a hallmark feature because it relates the lengths of the sides: \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse.

By knowing one angle, aside from the 90-degree one, we can find the other angle since all angles in a triangle sum to 180 degrees. For example, if one angle is 30 degrees, the third one must be 60 degrees because 30 + 60 + 90 = 180.
This predictable relationship allows for the consistent use of trigonometric ratios like sine, cosine, and tangent that help find missing side lengths.

Angle Relationships

In similar right triangles, the angles remain consistent across any size. This is because similar triangles retain their shape, differing only in size. The angles are crucial because they ensure the sides are proportional.
When you know one angle beside the right angle in a triangle like \( riangle ABC \), if angle \(A = 30°\), then automatically angle \(B = 60°\) since the angles in the triangle should sum up to 180°.

• Angle consistency is what makes two triangles similar.
• Similar triangles have all congruent corresponding angles.
• This means that if one triangle has angles 30°, 60°, and 90°, the other one will have exactly the same angles.

This relationship also means that corresponding sides in similar triangles will be proportional, creating a harmonious balance between angles and sides.

Proportional Sides

An intriguing property of similar triangles is that their corresponding sides are proportional. This means that every side of one triangle is in the same ratio as the corresponding side of another triangle.
For example, if triangle \( riangle ABC \) is similar to triangle \( riangle DEF \), then:

• The hypotenuse \(AB\) of triangle \( riangle ABC \) relates proportionally to \(DE\) of triangle \( riangle DEF\).
• The side opposite the 30-degree angle, \(BC\) is proportional to \(EF\).
• The side adjacent to the 30-degree angle, \(AC\), is proportional to \(DF\).

This proportionality comes in handy not only in mathematics but also in real-world applications that involve scaling shapes. If you know the lengths of some sides in one triangle, this property allows you to easily find the lengths of the corresponding sides in another similar triangle by solving simple proportions.