#### The following fill-in-the-blanks question, explained here in hopes of helping geometry students in Maryland and Illinois prepare for the PARCC test near the end of this school year, appears on the released version of PARCC’s Spring 2015 test in geometry, here:

In the three right triangles shown,

Complete the statements about the triangles by dragging the correct choices into the proper locations. Not all choices will be used.

Choices: AA similarity, adjacent leg, ASA, congruent, cosine, CPCTC, GH/GI, GI/GH, hypotenuse, similar, sine, tangent.

Because each triangle contains a right angle and a 36° angle, the triangles are ____ by ____, and AB/AC = DE/DF = ____. The proportion shows that the ratio of the length of the leg opposite the 36° angle to the length of the ____ will be the same for any right triangle with a 36° angle. The value of the ratio is defined to be the ____ of 36°.

## Resources for further study

**Integrated Publishing** begins a small proof that two right triangles that share the measure of one of the acute angles are similar to each other:

- The right angle in the first triangle is equal to the right angle in the second, since all right angles are equal.
- The sum of the angles of any triangle is 180°. Therefore, the sum of the two acute angles in a right triangle is 90°.

At the point, we note that, having shown that two of the angles have the same measure, AA (Angle-Angle) similarity applies. If two angles are equal in two different triangles, then the triangles are similar. The triangles would be congruent if the side between the two congruent angles were also congruent, by ASA (Angle-Side-Angle) congruency, but that’s not the case here.

For the second part, there are numerous sources on the Web and in textbooks to explain that the tangent of an angle in a right triangle is equal to the ratio of the side opposite the angle to the side adjacent to the angle, neither side being the hypotenuse of the right triangle.

Chapter 8, Section 8.3, of the book * Geometry for Enjoyment and Challenge* by Richard Rhoad

*et al*, all teachers from Illinois, says the AA similarity theorem follows directly from the No-Choice theorem: If two angles of one triangle are congruent to two angles of a second triangle, then the third angles are congruent.

(Richard Rhoad, George Milauskas, and Robert Whipple. *Geometry for Enjoyment and Challenge*, new edition. Evanston, Ill.: McDougal Littell, a division of Houghton Mifflin Company, 1991. The book is used in several geometry classes taught in Illinois high schools.)

## Analysis of this question and online accessibility

The question measures knowledge of the Common Core standards it purports to measure and tests students’ understanding of the definition of a circle and its radius. It is considered to have a low cognitive demand.

The question can be tested online and should yield results that are as valid and reliable as those obtained on paper.

No special accommodation challenges can be identified with this question, so the question is considered fair.

## Challenge

Given that line segment LP is perpendicular to line segment EA, N is the midpoint of line segment LP, and P and R trisect line segment EA, prove that

## Purpose of this series of posts

Voxitatis is developing blog posts that address every algebra 1 question released to the public by the Partnership for Assessment of Readiness for College and Careers, or PARCC, in order to help students prepare to take the test this spring.

Our total release will run from February 27 through March 15, with one or two questions discussed per day. Then we’ll move to geometry at the end of March, algebra 2 during the first half of April, and eighth grade during the last half of April.