# TPREP101: SAT Prep

Unit 6: Geometry   The SAT Mathematics section contains many questions devoted to testing geometric concepts as well.  A clear understanding of these concepts, as well as shortcuts that can be used to more quickly solve a geometric problem, will vastly increase your math score.  Work through the topics below to ensure that you have mastered each concept, using the practice problems to test your knowledge.

This unit should take approximately 4 hours to complete.

☐    Subunit 6.1: 1.25 hours

☐    Subunit 6.2: 1 hour

☐    Subunit 6.3: 0.25 hours

☐    Subunit 6.4: 0.25 hours

☐    Subunit 6.5: 0.5 hours

☐    Subunit 6.6: 0.75 hours

Unit6 Learning Outcomes
Upon successful completion of this unit, the student will be able to: - Use relationships between lines and angles to identify the value of an unknown angle. - Find the value of unknown sides or angles in a triangle. - Find the value of an unknown line or angle within a circle. - Find the value of unknown sides or angles in other polygons. - Find perimeter, area, and volume of a geometrical shape.

6.1 Angles and Lines   - Reading: ErikTheRed.com: Erik Jacobsen’s “SAT Math Facts and Formulas” Link: ErikTheRed.com: Erik Jacobsen’s “SAT Math Facts and Formulas” (PDF)

`````` Instructions: Please click on the link above and select the link to
the “SAT Math Facts and Formulas” PDF.  Read the sections entitled
“Lines,” “Triangles,” “Circles,” “Rectangles and Friends,” and
“Solids” (pages 4-6).  Make notes or flashcards of any concepts you
need to review in order to remember for the test.  This resource
will cover the topics outlined in subunits 6.1-6.5, as well as any
inclusive sub-subunits.

Reading these sections and taking notes should take approximately 1
hour.

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6.1.1 Parallel and Perpendicular Lines   Note: This topic is covered by the “SAT Math Facts and Formulas” reading assigned below subunit 6.1.  In particular, review the formula for slope, and the formulas for writing an equation in ‘point-slope’ and ‘slope-intercept’ form, as described under the “Lines (Linear Functions)” heading.

6.1.2 Corresponding, Complementary, and Supplementary Angles   Note: This topic is covered by the reading assigned below subunit 6.1.  Corresponding angles are angles with the same degree measure.  Complementary angles are two angles that add up to 90 degrees.  Supplementary angles are two angles that add up to 180 degrees.

Instructions: The basics of finding missing angles are pretty straightforward.  Please click on the link above and watch this short video to brush up.  Watch this video as many times as needed to understand how to find missing angles.

Watching this video and pausing to take notes should take less than 15 minutes.

6.2 Triangles   6.2.1 Right Triangles   Note: This topic is covered by the reading assigned below subunit 6.1.  Make sure you have a good handle on the ‘Special Right Triangles’ (mentioned on the worksheet and reviewed in videos below) and the Pythagorean triples (explained below).  Knowing these values off the top of your head will help a lot with shortcuts and speed when you encounter right triangles on the SAT.

Instructions: Please click on the link above, and view this video for a brush-up on how to use the Pythagorean Theorem to find the length of a missing side on a right triangle.  Simplifying square roots is also covered – understanding how to simplify roots which are not perfect squares is important in order to select the correct answer on the SAT, since the answers are designed to be found without a calculator (and so will be written as simplified roots, rather than decimals).

Watching this video and pausing to take notes should take less than 15 minutes.

• Reading: Math Open Reference’s “Pythagorean Triples” Link: Math Open Reference’s “Pythagorean Triples” (HTML)

Instructions: This page explains and lists the basic Pythagorean triples – the lowest values that fulfill the Pythagorean Theorem (and will be found as sides of right triangles) when all sides are whole numbers.  Become familiar with the Pythagorean triples and their multiples so that you can quickly solve simple right triangle problems, and recognize right triangles by the values of the sides.

Reading this webpage should take less than 15 minutes.

Instructions: Watch these two videos for a review of the “Special Triangles” and examples of how to work these types of problems.

Watching these videos and pausing to take notes should take approximately 30 minutes.

6.2.2 Equilateral Triangles   Note: This topic is covered by the “SAT Math Facts and Formulas” reading assigned below subunit 6.1.  In particular, review the description of an equilateral triangle, as described under the “Triangles” heading.

6.2.3 Isosceles Triangles   Note: This topic is covered by the “SAT Math Facts and Formulas” reading assigned below subunit 6.1.  In particular, review the description of an isosceles triangle, as described under the “Triangles” heading.

6.2.4 Congruent vs. Similar   Note: This topic is covered below subunit 6.1.  Congruent triangles are basically identical.  They have the same angle measures and are the same size (the lengths of their sides are the same).  Similar triangles have the same angle measures, but may be different sizes.  The lengths of their sides may be different, but they are proportionately the same (e.g. if triangle A’s hypotenuse is 3, then a twice as large similar triangle would have a hypotenuse of 6).

6.3 Circles   6.3.1 Circumference and Diameter   Note: This topic is covered by the “SAT Math Facts and Formulas” reading assigned below subunit 6.1.  In particular, review the formulas under the “Circles” heading.  Remember that diameter (D) is the length of a straight line passing from one point on a circle, through the midpoint, to a point on the opposite side of the circle.  It is therefore twice the length of the circle’s radius (r).  The circumference of a circle is equal to πD.

6.3.2 Midpoint and Angles   Note: This topic is covered by the “SAT Math Facts and Formulas” reading assigned below subunit 6.1.  In particular, review the formulas for the length of an arc and area of a sector under the “Circles” heading.  Try drawing a circle, and choosing some values for the arc or angle or sector area, leaving one unknown you can practice finding by using the appropriate equation.

`````` Instructions: The SAT will often ask a question or two about
finding distances or angles of chords or shapes inscribed inside a
problems.  Be sure to take notes about any rules proven here that

Watching these videos should take approximately 15 minutes.

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6.4 Polygons   - Web Media: Khan Academy’s “Sum of Interior Angles of a Polygon” and “Sum of Exterior Angles of a Convex Polygon” Link: Khan Academy’s “Sum of Interior Angles of a Polygon” (YouTube) and “Sum of Exterior Angles of a Convex Polygon” (YouTube)

`````` Instructions: The SAT also likes to ask questions that require you
to understand how to find interior or exterior angles of a polygon.
Please click on the links above and watch these videos to find out
how.  Be sure to take notes so that you can review this information
later.

Watching these videos should take approximately 15 minutes.

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6.5 Perimeter, Area, and Volume   Note: List the formulas from “SAT Math Facts and Formulas” (under subunit 6.1) to find area and volume for each shape.  Review and practice these formulas.  The more quickly you can remember them on test day, the more quickly you can complete the problems.

• Web Media: Khan Academy’s “Finding the Area and Perimeter of a Non-Standard Polygon” Link: Khan Academy’s “Finding the Area and Perimeter of a Non-Standard Polygon” (YouTube)

Instructions: Please click on the link above and watch this video to learn how to find the area of a polygon without its own specific area formula.

Watching this video and pausing to take notes should take less than 15 minutes.

• Web Media: Khan Academy’s “Solid Geometry Volume” and “Cylinder Volume and Surface Area” Link: Khan Academy’s “Solid Geometry Volume” (YouTube) and “Cylinder Volume and Surface Area” (YouTube)

Instructions: Please watch these videos for a quick tutorial review of the use of the volume equations found on the “SAT Math Facts and Formulas” PDF (subunit 6.1).  The first video includes finding the volume of a cube (starting at 2:30) and the second video covers finding the volume of a cylinder.

Watching these videos and pausing to take notes should take less than 15 minutes.

6.6 Geometry Practice Problems   - Assignment: cK-12: Jason Shah’s “SAT Prep - Math Questions” and “SAT Prep - Math Questions with Explanations” Link: cK-12: Jason Shah’s “SAT Prep - Math Questions” (HTML) and “SAT Prep - Math Questions with Explanations” (HTML)

`````` Instructions: Please click on the first link above for “Math
Questions” and complete questions 6-8 to apply your new knowledge
and to practice geometry questions.

Once you have completed the questions, please click on the second
link above for “Math Questions with Explanations” and review the
correct answers for questions 6–8.  For any questions you missed,
makes sense to you.  You may want to look back at the previous
sections in this unit for help.  Once you have studied the correct
answer, think about how you will approach a similar problem in the
future and will avoid making a similar mistake.

Completing this assignment should take approximately 15 minutes.

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• Assignment: cK-12: Jason Shah’s “SAT Prep II - Math Questions” and “SAT Prep II - Math Questions with Explanations” Link: cK-12: Jason Shah’s “SAT Prep II - Math Questions” (HTML) and “SAT Prep II - Math Questions with Explanations” (HTML)

Instructions: Please click on the first link above for “Math Questions” and complete questions 6-8 to apply your new knowledge and to practice geometry questions.

Once you have completed the questions, please click on the second link above for “Math Questions with Explanations” and review the correct answers for questions 6-8.  For any questions you missed, read and review the provided explanations until the correct answer makes sense to you.  You may want to look back at the previous sections in this unit for help.  Once you have studied the correct answer, think about how you will approach a similar problem in the future and will avoid making a similar mistake.

Completing this assignment should take less than 15 minutes.